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class="vector-toc-link" href="#Uncertainty_principle"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Uncertainty principle</span> </div> </a> <ul id="toc-Uncertainty_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Composite_systems_and_entanglement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Composite_systems_and_entanglement"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Composite systems and entanglement</span> </div> </a> <ul id="toc-Composite_systems_and_entanglement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equivalence_between_formulations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalence_between_formulations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Equivalence between formulations</span> </div> </a> <ul id="toc-Equivalence_between_formulations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symmetries_and_conservation_laws" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symmetries_and_conservation_laws"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Symmetries and conservation laws</span> </div> </a> <ul id="toc-Symmetries_and_conservation_laws-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Free_particle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Free_particle"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Free particle</span> </div> </a> <ul id="toc-Free_particle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Particle_in_a_box" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Particle_in_a_box"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Particle in a box</span> </div> </a> <ul id="toc-Particle_in_a_box-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Harmonic_oscillator" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Harmonic_oscillator"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Harmonic oscillator</span> </div> </a> <ul id="toc-Harmonic_oscillator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mach–Zehnder_interferometer" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mach–Zehnder_interferometer"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Mach–Zehnder interferometer</span> </div> </a> <ul id="toc-Mach–Zehnder_interferometer-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_other_scientific_theories" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relation_to_other_scientific_theories"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Relation to other scientific theories</span> </div> </a> <button aria-controls="toc-Relation_to_other_scientific_theories-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Relation to other scientific theories subsection</span> </button> <ul id="toc-Relation_to_other_scientific_theories-sublist" class="vector-toc-list"> <li id="toc-Classical_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classical_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Classical mechanics</span> </div> </a> <ul id="toc-Classical_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Special_relativity_and_electrodynamics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Special_relativity_and_electrodynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Special relativity and electrodynamics</span> </div> </a> <ul id="toc-Special_relativity_and_electrodynamics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_general_relativity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_to_general_relativity"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Relation to general relativity</span> </div> </a> <ul id="toc-Relation_to_general_relativity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Philosophical_implications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Philosophical_implications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Philosophical implications</span> </div> </a> <ul id="toc-Philosophical_implications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Explanatory_notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Explanatory_notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Explanatory notes</span> </div> </a> <ul id="toc-Explanatory_notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Quantum mechanics</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 137 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-137" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">137 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Kwantummeganika" title="Kwantummeganika – Afrikaans" lang="af" hreflang="af" data-title="Kwantummeganika" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Quantenmechanik" title="Quantenmechanik – Alemannic" lang="gsw" hreflang="gsw" data-title="Quantenmechanik" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%82%E0%A4%9F%E0%A4%AE_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%82%E0%A4%A4" title="क्वांटम सिद्धांत – Angika" lang="anp" hreflang="anp" data-title="क्वांटम सिद्धांत" data-language-autonym="अंगिका" data-language-local-name="Angika" class="interlanguage-link-target"><span>अंगिका</span></a></li><li class="interlanguage-link interwiki-ar badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%8A%D9%83%D8%A7%D9%86%D9%8A%D9%83%D8%A7_%D8%A7%D9%84%D9%83%D9%85" title="ميكانيكا الكم – Arabic" lang="ar" hreflang="ar" data-title="ميكانيكا الكم" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Mecanica_quantica" title="Mecanica quantica – Aragonese" lang="an" hreflang="an" data-title="Mecanica quantica" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D5%94%D5%B8%D6%82%D5%A1%D5%B6%D5%BF%D5%A1%D5%B5%D5%AB%D5%B6_%D5%B4%D5%A5%D5%A3%D5%A1%D5%B6%D5%AB%D5%AF" title="Քուանտային մեգանիկ – Western Armenian" lang="hyw" hreflang="hyw" data-title="Քուանտային մեգանիկ" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target"><span>Արեւմտահայերէն</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%95%E0%A7%8B%E0%A7%B1%E0%A6%BE%E0%A6%A3%E0%A7%8D%E0%A6%9F%E0%A6%BE%E0%A6%AE_%E0%A6%AC%E0%A6%B2%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8" title="কোৱাণ্টাম বলবিজ্ঞান – Assamese" lang="as" hreflang="as" data-title="কোৱাণ্টাম বলবিজ্ঞান" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Mec%C3%A1nica_cu%C3%A1ntica" title="Mecánica cuántica – Asturian" lang="ast" hreflang="ast" data-title="Mecánica cuántica" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Mek%C3%A1nika_ku%C3%A1ntika" title="Mekánika kuántika – Guarani" lang="gn" hreflang="gn" data-title="Mekánika kuántika" data-language-autonym="Avañe'ẽ" data-language-local-name="Guarani" class="interlanguage-link-target"><span>Avañe'ẽ</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Kvant_mexanikas%C4%B1" title="Kvant mexanikası – Azerbaijani" lang="az" hreflang="az" data-title="Kvant mexanikası" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85_%D9%85%DA%A9%D8%A7%D9%86%DB%8C%DA%A9%DB%8C" title="کوانتوم مکانیکی – South Azerbaijani" lang="azb" hreflang="azb" data-title="کوانتوم مکانیکی" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ban mw-list-item"><a href="https://ban.wikipedia.org/wiki/M%C3%A9kanika_kuantum" title="Mékanika kuantum – Balinese" lang="ban" hreflang="ban" data-title="Mékanika kuantum" data-language-autonym="Basa Bali" data-language-local-name="Balinese" class="interlanguage-link-target"><span>Basa Bali</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8B%E0%A6%AF%E0%A6%BC%E0%A6%BE%E0%A6%A8%E0%A7%8D%E0%A6%9F%E0%A6%BE%E0%A6%AE_%E0%A6%AC%E0%A6%B2%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8" title="কোয়ান্টাম বলবিজ্ঞান – Bangla" lang="bn" hreflang="bn" data-title="কোয়ান্টাম বলবিজ্ঞান" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Li%C5%8Dng-ch%C3%BA_le%CC%8Dk-ha%CC%8Dk" title="Liōng-chú le̍k-ha̍k – Minnan" lang="nan" hreflang="nan" data-title="Liōng-chú le̍k-ha̍k" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0%D2%BB%D1%8B" title="Квант механикаһы – Bashkir" lang="ba" hreflang="ba" data-title="Квант механикаһы" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%B0%D0%B2%D0%B0%D1%8F_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0" title="Квантавая механіка – Belarusian" lang="be" hreflang="be" data-title="Квантавая механіка" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%B0%D0%B2%D0%B0%D1%8F_%D0%BC%D1%8D%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0" title="Квантавая мэханіка – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Квантавая мэханіка" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%82%E0%A4%9F%E0%A4%AE_%E0%A4%AE%E0%A5%88%E0%A4%95%E0%A5%87%E0%A4%A8%E0%A4%BF%E0%A4%95%E0%A5%8D%E0%A4%B8" title="क्वांटम मैकेनिक्स – Bhojpuri" lang="bh" hreflang="bh" data-title="क्वांटम मैकेनिक्स" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Mekanikang_kwantum" title="Mekanikang kwantum – Central Bikol" lang="bcl" hreflang="bcl" data-title="Mekanikang kwantum" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Квантова механика – Bulgarian" lang="bg" hreflang="bg" data-title="Квантова механика" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Fuzalmechanik" title="Fuzalmechanik – Bavarian" lang="bar" hreflang="bar" data-title="Fuzalmechanik" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kvantna_mehanika" title="Kvantna mehanika – Bosnian" lang="bs" hreflang="bs" data-title="Kvantna mehanika" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Mekanikerezh_kwantek" title="Mekanikerezh kwantek – Breton" lang="br" hreflang="br" data-title="Mekanikerezh kwantek" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D1%8B%D0%BD_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Квантын механика – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Квантын механика" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Mec%C3%A0nica_qu%C3%A0ntica" title="Mecànica quàntica – Catalan" lang="ca" hreflang="ca" data-title="Mecànica quàntica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BB%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Квантла механика – Chuvash" lang="cv" hreflang="cv" data-title="Квантла механика" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kvantov%C3%A1_mechanika" title="Kvantová mechanika – Czech" lang="cs" hreflang="cs" data-title="Kvantová mechanika" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Mecaneg_cwantwm" title="Mecaneg cwantwm – Welsh" lang="cy" hreflang="cy" data-title="Mecaneg cwantwm" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kvantemekanik" title="Kvantemekanik – Danish" lang="da" hreflang="da" data-title="Kvantemekanik" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Quantenmechanik" title="Quantenmechanik – German" lang="de" hreflang="de" data-title="Quantenmechanik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kvantmehaanika" title="Kvantmehaanika – Estonian" lang="et" hreflang="et" data-title="Kvantmehaanika" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%B2%CE%B1%CE%BD%CF%84%CE%B9%CE%BA%CE%AE_%CE%BC%CE%B7%CF%87%CE%B1%CE%BD%CE%B9%CE%BA%CE%AE" title="Κβαντική μηχανική – Greek" lang="el" hreflang="el" data-title="Κβαντική μηχανική" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Mec%C3%A1nica_cu%C3%A1ntica" title="Mecánica cuántica – Spanish" lang="es" hreflang="es" data-title="Mecánica cuántica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kvantuma_mekaniko" title="Kvantuma mekaniko – Esperanto" lang="eo" hreflang="eo" data-title="Kvantuma mekaniko" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/Mec%C3%A1nica_cu%C3%A1ntica" title="Mecánica cuántica – Extremaduran" lang="ext" hreflang="ext" data-title="Mecánica cuántica" data-language-autonym="Estremeñu" data-language-local-name="Extremaduran" class="interlanguage-link-target"><span>Estremeñu</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Mekanika_kuantiko" title="Mekanika kuantiko – Basque" lang="eu" hreflang="eu" data-title="Mekanika kuantiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%DA%A9%D8%A7%D9%86%DB%8C%DA%A9_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85%DB%8C" title="مکانیک کوانتومی – Persian" lang="fa" hreflang="fa" data-title="مکانیک کوانتومی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Quantum_mechanics" title="Quantum mechanics – Fiji Hindi" lang="hif" hreflang="hif" data-title="Quantum mechanics" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/M%C3%A9canique_quantique" title="Mécanique quantique – French" lang="fr" hreflang="fr" data-title="Mécanique quantique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Meicnic_chandamach" title="Meicnic chandamach – Irish" lang="ga" hreflang="ga" data-title="Meicnic chandamach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Meacanaigs_quantumach" title="Meacanaigs quantumach – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Meacanaigs quantumach" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Mec%C3%A1nica_cu%C3%A1ntica" title="Mecánica cuántica – Galician" lang="gl" hreflang="gl" data-title="Mecánica cuántica" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%96%91%EC%9E%90%EC%97%AD%ED%95%99" title="양자역학 – Korean" lang="ko" hreflang="ko" data-title="양자역학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ha mw-list-item"><a href="https://ha.wikipedia.org/wiki/Kimiyyar_kwantom" title="Kimiyyar kwantom – Hausa" lang="ha" hreflang="ha" data-title="Kimiyyar kwantom" data-language-autonym="Hausa" data-language-local-name="Hausa" class="interlanguage-link-target"><span>Hausa</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%94%D5%BE%D5%A1%D5%B6%D5%BF%D5%A1%D5%B5%D5%AB%D5%B6_%D5%B4%D5%A5%D5%AD%D5%A1%D5%B6%D5%AB%D5%AF%D5%A1" title="Քվանտային մեխանիկա – Armenian" lang="hy" hreflang="hy" data-title="Քվանտային մեխանիկա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A4%BE%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BE_%E0%A4%AF%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%80" title="प्रमात्रा यान्त्रिकी – Hindi" lang="hi" hreflang="hi" data-title="प्रमात्रा यान्त्रिकी" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kvantna_mehanika" title="Kvantna mehanika – Croatian" lang="hr" hreflang="hr" data-title="Kvantna mehanika" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Quantumala_mekaniko" title="Quantumala mekaniko – Ido" lang="io" hreflang="io" data-title="Quantumala mekaniko" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-ig mw-list-item"><a href="https://ig.wikipedia.org/wiki/Quantum_mechanics" title="Quantum mechanics – Igbo" lang="ig" hreflang="ig" data-title="Quantum mechanics" data-language-autonym="Igbo" data-language-local-name="Igbo" class="interlanguage-link-target"><span>Igbo</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Mekanika_kuantum" title="Mekanika kuantum – Indonesian" lang="id" hreflang="id" data-title="Mekanika kuantum" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Mechanica_quantic" title="Mechanica quantic – Interlingua" lang="ia" hreflang="ia" data-title="Mechanica quantic" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Ukuguxazela_kohoyana" title="Ukuguxazela kohoyana – Zulu" lang="zu" hreflang="zu" data-title="Ukuguxazela kohoyana" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target"><span>IsiZulu</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Skammtafr%C3%A6%C3%B0i" title="Skammtafræði – Icelandic" lang="is" hreflang="is" data-title="Skammtafræði" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Meccanica_quantistica" title="Meccanica quantistica – Italian" lang="it" hreflang="it" data-title="Meccanica quantistica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%9B%D7%A0%D7%99%D7%A7%D7%AA_%D7%94%D7%A7%D7%95%D7%95%D7%A0%D7%98%D7%99%D7%9D" title="מכניקת הקוונטים – Hebrew" lang="he" hreflang="he" data-title="מכניקת הקוונטים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/%C3%91%CA%8B%C5%8B_ho%C9%96e" title="Ñʋŋ hoɖe – Kabiye" lang="kbp" hreflang="kbp" data-title="Ñʋŋ hoɖe" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target"><span>Kabɩyɛ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%95%E0%B3%8D%E0%B2%B5%E0%B2%BE%E0%B2%82%E0%B2%9F%E0%B2%AE%E0%B3%8D_%E0%B2%AD%E0%B3%8C%E0%B2%A4%E0%B2%B6%E0%B2%BE%E0%B2%B8%E0%B3%8D%E0%B2%A4%E0%B3%8D%E0%B2%B0" title="ಕ್ವಾಂಟಮ್ ಭೌತಶಾಸ್ತ್ರ – Kannada" lang="kn" hreflang="kn" data-title="ಕ್ವಾಂಟಮ್ ಭೌತಶಾಸ್ತ್ರ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%99%E1%83%95%E1%83%90%E1%83%9C%E1%83%A2%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%9B%E1%83%94%E1%83%A5%E1%83%90%E1%83%9C%E1%83%98%E1%83%99%E1%83%90" title="კვანტური მექანიკა – Georgian" lang="ka" hreflang="ka" data-title="კვანტური მექანიკა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D1%82%D1%8B%D2%9B_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Кванттық механика – Kazakh" lang="kk" hreflang="kk" data-title="Кванттық механика" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Umakanika_kwanta" title="Umakanika kwanta – Swahili" lang="sw" hreflang="sw" data-title="Umakanika kwanta" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Mekanik_kantik" title="Mekanik kantik – Haitian Creole" lang="ht" hreflang="ht" data-title="Mekanik kantik" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/M%C3%A9kanik_kantik" title="Mékanik kantik – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Mékanik kantik" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D1%82%D1%8B%D0%BA_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Кванттык механика – Kyrgyz" lang="ky" hreflang="ky" data-title="Кванттык механика" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Mechanica_quantica" title="Mechanica quantica – Latin" lang="la" hreflang="la" data-title="Mechanica quantica" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Kvantu_meh%C4%81nika" title="Kvantu mehānika – Latvian" lang="lv" hreflang="lv" data-title="Kvantu mehānika" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Kvantin%C4%97_mechanika" title="Kvantinė mechanika – Lithuanian" lang="lt" hreflang="lt" data-title="Kvantinė mechanika" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Kwantummechanica" title="Kwantummechanica – Limburgish" lang="li" hreflang="li" data-title="Kwantummechanica" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Mec%C3%A0nega_quant%C3%ACstega" title="Mecànega quantìstega – Lombard" lang="lmo" hreflang="lmo" data-title="Mecànega quantìstega" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Kvantummechanika" title="Kvantummechanika – Hungarian" lang="hu" hreflang="hu" data-title="Kvantummechanika" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BD%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Квантна механика – Macedonian" lang="mk" hreflang="mk" data-title="Квантна механика" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%95%E0%B5%8D%E0%B4%B5%E0%B4%BE%E0%B4%A3%E0%B5%8D%E0%B4%9F%E0%B4%82_%E0%B4%AC%E0%B4%B2%E0%B4%A4%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%82" title="ക്വാണ്ടം ബലതന്ത്രം – Malayalam" lang="ml" hreflang="ml" data-title="ക്വാണ്ടം ബലതന്ത്രം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Mekkanika_kwantistika" title="Mekkanika kwantistika – Maltese" lang="mt" hreflang="mt" data-title="Mekkanika kwantistika" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AA%E0%A5%81%E0%A4%82%E0%A4%9C_%E0%A4%AF%E0%A4%BE%E0%A4%AE%E0%A4%BF%E0%A4%95%E0%A5%80" title="पुंज यामिकी – Marathi" lang="mr" hreflang="mr" data-title="पुंज यामिकी" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%A7%D9%84%D9%85%D9%8A%D9%83%D8%A7%D9%86%D9%8A%D9%83%D8%A7_%D8%A7%D9%84%D9%83%D9%85%D9%8A%D9%87" title="الميكانيكا الكميه – Egyptian Arabic" lang="arz" hreflang="arz" data-title="الميكانيكا الكميه" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-mzn mw-list-item"><a href="https://mzn.wikipedia.org/wiki/%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85%DB%8C_%D9%81%DB%8C%D8%B2%DB%8C%DA%A9" title="کوانتومی فیزیک – Mazanderani" lang="mzn" hreflang="mzn" data-title="کوانتومی فیزیک" data-language-autonym="مازِرونی" data-language-local-name="Mazanderani" class="interlanguage-link-target"><span>مازِرونی</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Mekanik_kuantum" title="Mekanik kuantum – Malay" lang="ms" hreflang="ms" data-title="Mekanik kuantum" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/Li%C3%B4ng-c%E1%B9%B3%CC%84_l%C4%ADk-h%C5%8Fk" title="Liông-cṳ̄ lĭk-hŏk – Mindong" lang="cdo" hreflang="cdo" data-title="Liông-cṳ̄ lĭk-hŏk" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target"><span>閩東語 / Mìng-dĕ̤ng-ngṳ̄</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA" title="Квант механик – Mongolian" lang="mn" hreflang="mn" data-title="Квант механик" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%80%E1%80%BD%E1%80%99%E1%80%BA%E1%80%90%E1%80%99%E1%80%BA%E1%80%99%E1%80%80%E1%80%B9%E1%80%80%E1%80%84%E1%80%BA%E1%80%B8%E1%80%94%E1%80%85%E1%80%BA" title="ကွမ်တမ်မက္ကင်းနစ် – Burmese" lang="my" hreflang="my" data-title="ကွမ်တမ်မက္ကင်းနစ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kwantummechanica" title="Kwantummechanica – Dutch" lang="nl" hreflang="nl" data-title="Kwantummechanica" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A4%BE%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BE_%E0%A4%AF%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%80" title="प्रमात्रा यान्त्रिकी – Nepali" lang="ne" hreflang="ne" data-title="प्रमात्रा यान्त्रिकी" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%9F%E0%A4%AE_%E0%A4%AE%E0%A5%87%E0%A4%95%E0%A4%BE%E0%A4%A8%E0%A4%BF%E0%A4%95%E0%A5%8D%E0%A4%B8%E0%A5%8D" title="क्वान्टम मेकानिक्स् – Newari" lang="new" hreflang="new" data-title="क्वान्टम मेकानिक्स्" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%A6" title="量子力学 – Japanese" lang="ja" hreflang="ja" data-title="量子力学" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Kwantenmechaanik" title="Kwantenmechaanik – Northern Frisian" lang="frr" hreflang="frr" data-title="Kwantenmechaanik" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kvantemekanikk" title="Kvantemekanikk – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kvantemekanikk" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kvantemekanikk" title="Kvantemekanikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Kvantemekanikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Mecanica_quantica" title="Mecanica quantica – Occitan" lang="oc" hreflang="oc" data-title="Mecanica quantica" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Kvant_mexanika" title="Kvant mexanika – Uzbek" lang="uz" hreflang="uz" data-title="Kvant mexanika" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A9%81%E0%A8%86%E0%A8%82%E0%A8%9F%E0%A8%AE_%E0%A8%AE%E0%A8%95%E0%A9%88%E0%A8%A8%E0%A8%BF%E0%A8%95%E0%A8%B8" title="ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ – Punjabi" lang="pa" hreflang="pa" data-title="ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%DA%A9%D9%88%D8%A7%D9%86%D9%B9%D9%85_%D9%85%DA%A9%DB%8C%D9%86%DA%A9%D8%B3" title="کوانٹم مکینکس – Western Punjabi" lang="pnb" hreflang="pnb" data-title="کوانٹم مکینکس" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%DA%A9%D9%88%D8%A7%D9%86%D9%BC%D9%88%D9%85_%D9%85%DB%8C%D8%AE%D8%A7%D9%86%DB%8C%DA%A9" title="کوانټوم میخانیک – Pashto" lang="ps" hreflang="ps" data-title="کوانټوم میخانیک" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Kuantom_mikianix" title="Kuantom mikianix – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Kuantom mikianix" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Mec%C3%A0nica_qu%C3%A0ntica" title="Mecànica quàntica – Piedmontese" lang="pms" hreflang="pms" data-title="Mecànica quàntica" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Quantenmechanik" title="Quantenmechanik – Low German" lang="nds" hreflang="nds" data-title="Quantenmechanik" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Mechanika_kwantowa" title="Mechanika kwantowa – Polish" lang="pl" hreflang="pl" data-title="Mechanika kwantowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Mec%C3%A2nica_qu%C3%A2ntica" title="Mecânica quântica – Portuguese" lang="pt" hreflang="pt" data-title="Mecânica quântica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Kvant_mexanika" title="Kvant mexanika – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="Kvant mexanika" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Mecanic%C4%83_cuantic%C4%83" title="Mecanică cuantică – Romanian" lang="ro" hreflang="ro" data-title="Mecanică cuantică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0" title="Квантова механіка – Rusyn" lang="rue" hreflang="rue" data-title="Квантова механіка" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0%D1%8F_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Квантовая механика – Russian" lang="ru" hreflang="ru" data-title="Квантовая механика" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0%D0%B9_%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0" title="Квантовай физика – Yakut" lang="sah" hreflang="sah" data-title="Квантовай физика" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Quantum_mechanics" title="Quantum mechanics – Scots" lang="sco" hreflang="sco" data-title="Quantum mechanics" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Mekanika_kuantike" title="Mekanika kuantike – Albanian" lang="sq" hreflang="sq" data-title="Mekanika kuantike" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Micc%C3%A0nica_quant%C3%ACstica" title="Miccànica quantìstica – Sicilian" lang="scn" hreflang="scn" data-title="Miccànica quantìstica" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9A%E0%B7%8A%E0%B7%80%E0%B7%9C%E0%B6%B1%E0%B7%8A%E0%B6%A7%E0%B6%B8%E0%B7%8A_%E0%B6%BA%E0%B7%8F%E0%B6%B1%E0%B7%8A%E0%B6%AD%E0%B7%8A%E2%80%8D%E0%B6%BB_%E0%B7%80%E0%B7%92%E0%B6%AF%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E2%80%8D%E0%B7%80" title="ක්වොන්ටම් යාන්ත්‍ර විද්‍යා‍ව – Sinhala" lang="si" hreflang="si" data-title="ක්වොන්ටම් යාන්ත්‍ර විද්‍යා‍ව" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Quantum_mechanics" title="Quantum mechanics – Simple English" lang="en-simple" hreflang="en-simple" data-title="Quantum mechanics" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%DA%AA%D9%88%D8%A7%D9%86%D9%BD%D9%85_%D9%85%DA%AA%D9%8A%D9%86%DA%AA%D8%B3" title="ڪوانٽم مڪينڪس – Sindhi" lang="sd" hreflang="sd" data-title="ڪوانٽم مڪينڪس" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kvantov%C3%A1_mechanika" title="Kvantová mechanika – Slovak" lang="sk" hreflang="sk" data-title="Kvantová mechanika" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kvantna_mehanika" title="Kvantna mehanika – Slovenian" lang="sl" hreflang="sl" data-title="Kvantna mehanika" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Kwantow%C5%8F_mechanika" title="Kwantowŏ mechanika – Silesian" lang="szl" hreflang="szl" data-title="Kwantowŏ mechanika" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%85%DB%8C%DA%A9%D8%A7%D9%86%DB%8C%DA%A9%DB%8C_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%DB%86%D9%85" title="میکانیکی کوانتۆم – Central Kurdish" lang="ckb" hreflang="ckb" data-title="میکانیکی کوانتۆم" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BD%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Квантна механика – Serbian" lang="sr" hreflang="sr" data-title="Квантна механика" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Kvantna_mehanika" title="Kvantna mehanika – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Kvantna mehanika" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/M%C3%A9kanika_kuantum" title="Mékanika kuantum – Sundanese" lang="su" hreflang="su" data-title="Mékanika kuantum" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kvanttimekaniikka" title="Kvanttimekaniikka – Finnish" lang="fi" hreflang="fi" data-title="Kvanttimekaniikka" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kvantmekanik" title="Kvantmekanik – Swedish" lang="sv" hreflang="sv" data-title="Kvantmekanik" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Mekanikang_quantum" title="Mekanikang quantum – Tagalog" lang="tl" hreflang="tl" data-title="Mekanikang quantum" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AF%81%E0%AE%B5%E0%AE%BE%E0%AE%A3%E0%AF%8D%E0%AE%9F%E0%AE%AE%E0%AF%8D_%E0%AE%87%E0%AE%AF%E0%AE%99%E0%AF%8D%E0%AE%95%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D" title="குவாண்டம் இயங்கியல் – Tamil" lang="ta" hreflang="ta" data-title="குவாண்டம் இயங்கியல்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/Tamikanikt_tasmktant" title="Tamikanikt tasmktant – Tachelhit" lang="shi" hreflang="shi" data-title="Tamikanikt tasmktant" data-language-autonym="Taclḥit" data-language-local-name="Tachelhit" class="interlanguage-link-target"><span>Taclḥit</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0%D1%81%D1%8B" title="Квант механикасы – Tatar" lang="tt" hreflang="tt" data-title="Квант механикасы" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%95%E0%B1%8D%E0%B0%B5%E0%B0%BE%E0%B0%82%E0%B0%9F%E0%B0%82_%E0%B0%AF%E0%B0%BE%E0%B0%82%E0%B0%A4%E0%B1%8D%E0%B0%B0%E0%B0%BF%E0%B0%95_%E0%B0%B6%E0%B0%BE%E0%B0%B8%E0%B1%8D%E0%B0%A4%E0%B1%8D%E0%B0%B0%E0%B0%82" title="క్వాంటం యాంత్రిక శాస్త్రం – Telugu" lang="te" hreflang="te" data-title="క్వాంటం యాంత్రిక శాస్త్రం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C%E0%B8%84%E0%B8%A7%E0%B8%AD%E0%B8%99%E0%B8%95%E0%B8%B1%E0%B8%A1" title="กลศาสตร์ควอนตัม – Thai" lang="th" hreflang="th" data-title="กลศาสตร์ควอนตัม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0%D0%B8_%D0%BA%D0%B2%D0%B0%D0%BD%D1%82%D3%A3" title="Механикаи квантӣ – Tajik" lang="tg" hreflang="tg" data-title="Механикаи квантӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kuantum_mekani%C4%9Fi" title="Kuantum mekaniği – Turkish" lang="tr" hreflang="tr" data-title="Kuantum mekaniği" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0" title="Квантова механіка – Ukrainian" lang="uk" hreflang="uk" data-title="Квантова механіка" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%82%D8%AF%D8%B1%DB%8C_%D9%85%DB%8C%DA%A9%D8%A7%D9%86%DB%8C%D8%A7%D8%AA" title="قدری میکانیات – Urdu" lang="ur" hreflang="ur" data-title="قدری میکانیات" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Mec%C3%A0nega_cuant%C3%ACstega" title="Mecànega cuantìstega – Venetian" lang="vec" hreflang="vec" data-title="Mecànega cuantìstega" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Kvantmehanik" title="Kvantmehanik – Veps" lang="vep" hreflang="vep" data-title="Kvantmehanik" data-language-autonym="Vepsän kel’" data-language-local-name="Veps" class="interlanguage-link-target"><span>Vepsän kel’</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C6%A1_h%E1%BB%8Dc_l%C6%B0%E1%BB%A3ng_t%E1%BB%AD" title="Cơ học lượng tử – Vietnamese" lang="vi" hreflang="vi" data-title="Cơ học lượng tử" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Kvantmekaaniga" title="Kvantmekaaniga – Võro" lang="vro" hreflang="vro" data-title="Kvantmekaaniga" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8" title="量子力學 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="量子力學" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Mekanika_kwantum" title="Mekanika kwantum – Waray" lang="war" hreflang="war" data-title="Mekanika kwantum" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%A6" title="量子力学 – Wu" lang="wuu" hreflang="wuu" data-title="量子力学" data-language-autonym="吴语" 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Quantum mechanics cannot predict the exact location of a particle in space, only the probability of finding it at different locations.<sup id="cite_ref-Born1926_1-0" class="reference"><a href="#cite_note-Born1926-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The brighter areas represent a higher probability of finding the electron.</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol 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class="sidebar-title-with-pretitle"><a class="mw-selflink selflink">Quantum mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1799e4a910c7d26396922a20ef5ceec25ca1871c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.882ex; height:5.509ex;" alt="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }"></span><div class="sidebar-caption" style="font-size:90%;padding-top:0.4em;font-style:italic;"><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a></div></td></tr><tr><td class="sidebar-above hlist nowrap" style="display:block;margin-bottom:0.4em;"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a></li></ul></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Background</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)">Complementarity</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_number" title="Quantum number">Quantum number</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">State</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li></ul></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Experiments</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell's inequality</a></li> <li><a href="/wiki/CHSH_inequality" title="CHSH inequality">CHSH inequality</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Leggett_inequality" title="Leggett inequality">Leggett inequality</a></li> <li><a href="/wiki/Leggett%E2%80%93Garg_inequality" title="Leggett–Garg inequality">Leggett–Garg inequality</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li></ul> </div> <ul><li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a> <ul><li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice</a></li></ul></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed-choice</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Formulations</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Overview</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase-space</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Sum-over-histories (path integral)</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective-collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Advanced topics</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Yakir_Aharonov" title="Yakir Aharonov">Aharonov</a></li> <li><a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">Bell</a></li> <li><a href="/wiki/Hans_Bethe" title="Hans Bethe">Bethe</a></li> <li><a href="/wiki/Patrick_Blackett" title="Patrick Blackett">Blackett</a></li> <li><a href="/wiki/Felix_Bloch" title="Felix Bloch">Bloch</a></li> <li><a href="/wiki/David_Bohm" title="David Bohm">Bohm</a></li> <li><a href="/wiki/Niels_Bohr" title="Niels Bohr">Bohr</a></li> <li><a href="/wiki/Max_Born" title="Max Born">Born</a></li> <li><a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Bose</a></li> <li><a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">de Broglie</a></li> <li><a href="/wiki/Arthur_Compton" title="Arthur Compton">Compton</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Clinton_Davisson" title="Clinton Davisson">Davisson</a></li> <li><a href="/wiki/Peter_Debye" title="Peter Debye">Debye</a></li> <li><a href="/wiki/Paul_Ehrenfest" title="Paul Ehrenfest">Ehrenfest</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Hugh_Everett_III" title="Hugh Everett III">Everett</a></li> <li><a href="/wiki/Vladimir_Fock" title="Vladimir Fock">Fock</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a></li> <li><a href="/wiki/Roy_J._Glauber" title="Roy J. Glauber">Glauber</a></li> <li><a href="/wiki/Martin_Gutzwiller" title="Martin Gutzwiller">Gutzwiller</a></li> <li><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a></li> <li><a href="/wiki/Hans_Kramers" title="Hans Kramers">Kramers</a></li> <li><a href="/wiki/Willis_Lamb" title="Willis Lamb">Lamb</a></li> <li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li> <li><a href="/wiki/Max_von_Laue" title="Max von Laue">Laue</a></li> <li><a href="/wiki/Henry_Moseley" title="Henry Moseley">Moseley</a></li> <li><a href="/wiki/Robert_Andrews_Millikan" title="Robert Andrews Millikan">Millikan</a></li> <li><a href="/wiki/Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Onnes</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/Isidor_Isaac_Rabi" title="Isidor Isaac Rabi">Rabi</a></li> <li><a href="/wiki/C._V._Raman" title="C. V. Raman">Raman</a></li> <li><a href="/wiki/Johannes_Rydberg" title="Johannes Rydberg">Rydberg</a></li> <li><a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödinger</a></li> <li><a href="/wiki/Michelle_Simmons" title="Michelle Simmons">Simmons</a></li> <li><a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Sommerfeld</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Wilhelm_Wien" title="Wilhelm Wien">Wien</a></li> <li><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner</a></li> <li><a href="/wiki/Pieter_Zeeman" title="Pieter Zeeman">Zeeman</a></li> <li><a href="/wiki/Anton_Zeilinger" title="Anton Zeilinger">Zeilinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar" style="border-top:1px solid #aaa;padding-top:0.1em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_mechanics" title="Template:Quantum mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics" title="Template talk:Quantum mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics" title="Special:EditPage/Template:Quantum mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Quantum mechanics</b> is a fundamental <a href="/wiki/Scientific_theory" title="Scientific theory">theory</a> that describes the behavior of <a href="/wiki/Nature" title="Nature">nature</a> at and below the scale of <a href="/wiki/Atom" title="Atom">atoms</a>.<sup id="cite_ref-Feynman_2-0" class="reference"><a href="#cite_note-Feynman-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 1.1">: 1.1 </span></sup> It is the foundation of all <b>quantum physics</b>, which includes <a href="/wiki/Quantum_chemistry" title="Quantum chemistry">quantum chemistry</a>, <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, <a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">quantum technology</a>, and <a href="/wiki/Quantum_information_science" title="Quantum information science">quantum information science</a>. </p><p>Quantum mechanics can describe many systems that <a href="/wiki/Classical_physics" title="Classical physics">classical physics</a> cannot. Classical physics can describe many aspects of nature at an ordinary (<a href="/wiki/Macroscopic_scale" title="Macroscopic scale">macroscopic</a> and <a href="/wiki/Microscopic_scale" title="Microscopic scale">(optical) microscopic</a>) scale, but is not sufficient for describing them at very small <a href="/wiki/Submicroscopic" class="mw-redirect" title="Submicroscopic">submicroscopic</a> (atomic and <a href="/wiki/Subatomic_particle" title="Subatomic particle">subatomic</a>) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation, valid at large (macroscopic/microscopic) scale.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Quantum systems have <a href="/wiki/Bound_state" title="Bound state">bound</a> states that are <a href="/wiki/Quantization_(physics)" title="Quantization (physics)">quantized</a> to <a href="/wiki/Discrete_mathematics" title="Discrete mathematics">discrete values</a> of <a href="/wiki/Energy" title="Energy">energy</a>, <a href="/wiki/Momentum" title="Momentum">momentum</a>, <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>, and other quantities, in contrast to classical systems where these quantities can be measured continuously. Measurements of quantum systems show characteristics of both <a href="/wiki/Particle" title="Particle">particles</a> and <a href="/wiki/Wave" title="Wave">waves</a> (<a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">wave–particle duality</a>), and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a>). </p><p>Quantum mechanics <a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">arose gradually</a> from theories to explain observations that could not be reconciled with classical physics, such as <a href="/wiki/Max_Planck" title="Max Planck">Max Planck</a>'s solution in 1900 to the <a href="/wiki/Black-body_radiation" title="Black-body radiation">black-body radiation</a> problem, and the correspondence between energy and frequency in <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>'s <a href="/wiki/Annus_Mirabilis_papers#Photoelectric_effect" class="mw-redirect" title="Annus Mirabilis papers">1905 paper</a>, which explained the <a href="/wiki/Photoelectric_effect" title="Photoelectric effect">photoelectric effect</a>. These early attempts to understand microscopic phenomena, now known as the "<a href="/wiki/Old_quantum_theory" title="Old quantum theory">old quantum theory</a>", led to the full development of quantum mechanics in the mid-1920s by <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a>, <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a>, <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a>, <a href="/wiki/Max_Born" title="Max Born">Max Born</a>, <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a> and others. The modern theory is formulated in various <a href="/wiki/Mathematical_formulations_of_quantum_mechanics" class="mw-redirect" title="Mathematical formulations of quantum mechanics">specially developed mathematical formalisms</a>. In one of them, a mathematical entity called the <a href="/wiki/Wave_function" title="Wave function">wave function</a> provides information, in the form of <a href="/wiki/Probability_amplitude" title="Probability amplitude">probability amplitudes</a>, about what measurements of a particle's energy, momentum, and other physical properties may yield. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview_and_fundamental_concepts">Overview and fundamental concepts</h2></div> <p>Quantum mechanics allows the calculation of properties and behaviour of physical systems. It is typically applied to microscopic systems: molecules, atoms and sub-atomic particles. It has been demonstrated to hold for complex molecules with thousands of atoms,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> but its application to human beings raises philosophical problems, such as <a href="/wiki/Wigner%27s_friend" title="Wigner's friend">Wigner's friend</a>, and its application to the universe as a whole remains speculative.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Predictions of quantum mechanics have been verified experimentally to an extremely high degree of <a href="/wiki/Accuracy" class="mw-redirect" title="Accuracy">accuracy</a>. For example, the refinement of quantum mechanics for the interaction of light and matter, known as <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a> (QED), has been <a href="/wiki/Precision_tests_of_QED" title="Precision tests of QED">shown to agree with experiment</a> to within 1 part in 10<sup>12</sup> when predicting the magnetic properties of an electron.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>A fundamental feature of the theory is that it usually cannot predict with certainty what will happen, but only give probabilities. Mathematically, a probability is found by taking the square of the absolute value of a <a href="/wiki/Complex_number" title="Complex number">complex number</a>, known as a probability amplitude. This is known as the <a href="/wiki/Born_rule" title="Born rule">Born rule</a>, named after physicist <a href="/wiki/Max_Born" title="Max Born">Max Born</a>. For example, a quantum particle like an <a href="/wiki/Electron" title="Electron">electron</a> can be described by a wave function, which associates to each point in space a probability amplitude. Applying the Born rule to these amplitudes gives a <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> relates the collection of probability amplitudes that pertain to one moment of time to the collection of probability amplitudes that pertain to another.<sup id="cite_ref-Zwiebach2022_7-0" class="reference"><a href="#cite_note-Zwiebach2022-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 67–87">: 67–87 </span></sup> </p><p>One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between measurable quantities. The most famous form of this <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a> says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for a measurement of its position and also at the same time for a measurement of its <a href="/wiki/Momentum" title="Momentum">momentum</a>.<sup id="cite_ref-Zwiebach2022_7-1" class="reference"><a href="#cite_note-Zwiebach2022-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 427–435">: 427–435 </span></sup> </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Double-slit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Double-slit.svg/260px-Double-slit.svg.png" decoding="async" width="260" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Double-slit.svg/390px-Double-slit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Double-slit.svg/520px-Double-slit.svg.png 2x" data-file-width="512" data-file-height="245" /></a><figcaption>An illustration of the double-slit experiment in physics.</figcaption></figure> <p>Another consequence of the mathematical rules of quantum mechanics is the phenomenon of <a href="/wiki/Quantum_interference" class="mw-redirect" title="Quantum interference">quantum interference</a>, which is often illustrated with the <a href="/wiki/Double-slit_experiment" title="Double-slit experiment">double-slit experiment</a>. In the basic version of this experiment, a <a href="/wiki/Coherence_(physics)" title="Coherence (physics)">coherent light source</a>, such as a <a href="/wiki/Laser" title="Laser">laser</a> beam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate.<sup id="cite_ref-Lederman_8-0" class="reference"><a href="#cite_note-Lederman-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 102–111">: 102–111 </span></sup><sup id="cite_ref-Feynman_2-1" class="reference"><a href="#cite_note-Feynman-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 1.1–1.8">: 1.1–1.8 </span></sup> The wave nature of light causes the light waves passing through the two slits to <a href="/wiki/Interference_(wave_propagation)" class="mw-redirect" title="Interference (wave propagation)">interfere</a>, producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles.<sup id="cite_ref-Lederman_8-1" class="reference"><a href="#cite_note-Lederman-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves; the interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected <a href="/wiki/Photon" title="Photon">photon</a> passes through one slit (as would a classical particle), and not through both slits (as would a wave).<sup id="cite_ref-Lederman_8-2" class="reference"><a href="#cite_note-Lederman-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 109">: 109 </span></sup><sup id="cite_ref-Müller-Kirsten_9-0" class="reference"><a href="#cite_note-Müller-Kirsten-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Plotnitsky_10-0" class="reference"><a href="#cite_note-Plotnitsky-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> However, <a href="/wiki/Double-slit_experiment#Which_way" title="Double-slit experiment">such experiments</a> demonstrate that particles do not form the interference pattern if one detects which slit they pass through. This behavior is known as <a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">wave–particle duality</a>. In addition to light, <a href="/wiki/Electrons" class="mw-redirect" title="Electrons">electrons</a>, <a href="/wiki/Atoms" class="mw-redirect" title="Atoms">atoms</a>, and <a href="/wiki/Molecules" class="mw-redirect" title="Molecules">molecules</a> are all found to exhibit the same dual behavior when fired towards a double slit.<sup id="cite_ref-Feynman_2-2" class="reference"><a href="#cite_note-Feynman-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:QuantumTunnel.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/QuantumTunnel.jpg/260px-QuantumTunnel.jpg" decoding="async" width="260" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/QuantumTunnel.jpg/390px-QuantumTunnel.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/QuantumTunnel.jpg/520px-QuantumTunnel.jpg 2x" data-file-width="681" data-file-height="575" /></a><figcaption>A (simplified) diagram of Quantum Tunneling, a phenomenon by which a particle may move through a barrier which would be impossible under classical mechanics.</figcaption></figure> <p>Another non-classical phenomenon predicted by quantum mechanics is <a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">quantum tunnelling</a>: a particle that goes up against a <a href="/wiki/Potential_barrier" class="mw-redirect" title="Potential barrier">potential barrier</a> can cross it, even if its kinetic energy is smaller than the maximum of the potential.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling <a href="/wiki/Radioactive_decay" title="Radioactive decay">radioactive decay</a>, <a href="/wiki/Nuclear_fusion" title="Nuclear fusion">nuclear fusion</a> in stars, and applications such as <a href="/wiki/Scanning_tunnelling_microscopy" class="mw-redirect" title="Scanning tunnelling microscopy">scanning tunnelling microscopy</a>, <a href="/wiki/Tunnel_diode" title="Tunnel diode">tunnel diode</a> and <a href="/wiki/Tunnel_field-effect_transistor" title="Tunnel field-effect transistor">tunnel field-effect transistor</a>.<sup id="cite_ref-Trixler2013_12-0" class="reference"><a href="#cite_note-Trixler2013-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>When quantum systems interact, the result can be the creation of <a href="/wiki/Quantum_entanglement" title="Quantum entanglement">quantum entanglement</a>: their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "...<i>the</i> characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought".<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Quantum entanglement enables <a href="/wiki/Quantum_computing" title="Quantum computing">quantum computing</a> and is part of quantum communication protocols, such as <a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">quantum key distribution</a> and <a href="/wiki/Superdense_coding" title="Superdense coding">superdense coding</a>.<sup id="cite_ref-Caves_15-0" class="reference"><a href="#cite_note-Caves-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Contrary to popular misconception, entanglement does not allow sending signals <a href="/wiki/Faster-than-light" title="Faster-than-light">faster than light</a>, as demonstrated by the <a href="/wiki/No-communication_theorem" title="No-communication theorem">no-communication theorem</a>.<sup id="cite_ref-Caves_15-1" class="reference"><a href="#cite_note-Caves-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>Another possibility opened by entanglement is testing for "<a href="/wiki/Hidden_variable_theory" class="mw-redirect" title="Hidden variable theory">hidden variables</a>", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory provides. A collection of results, most significantly <a href="/wiki/Bell%27s_theorem" title="Bell's theorem">Bell's theorem</a>, have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of <i>local</i> hidden variables, then the results of a <a href="/wiki/Bell_test_experiments" class="mw-redirect" title="Bell test experiments">Bell test</a> will be constrained in a particular, quantifiable way. Many Bell tests have been performed and they have shown results incompatible with the constraints imposed by local hidden variables.<sup id="cite_ref-wiseman15_16-0" class="reference"><a href="#cite_note-wiseman15-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-wolchover17_17-0" class="reference"><a href="#cite_note-wolchover17-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>It is not possible to present these concepts in more than a superficial way without introducing the mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>, <a href="/wiki/Group_theory" title="Group theory">group theory</a>, and other more advanced subjects.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> Accordingly, this article will present a mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples. </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_formulation">Mathematical formulation</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Mathematical formulation of quantum mechanics</a></div> <p>In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> belonging to a (<a href="/wiki/Separable_space" title="Separable space">separable</a>) complex <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}"></span>. This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \psi ,\psi \rangle =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mo>,</mo> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi ,\psi \rangle =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d5758e7a60b4e54bc46e01b0618919c65b787a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.13ex; height:2.843ex;" alt="{\displaystyle \langle \psi ,\psi \rangle =1}"></span>, and it is well-defined up to a complex number of modulus 1 (the global phase), that is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\alpha }\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msup> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\alpha }\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7193e9b0479da798b9cf2dcc492f49166d0d7103" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.448ex; height:3.009ex;" alt="{\displaystyle e^{i\alpha }\psi }"></span> represent the same physical system. In other words, the possible states are points in the <a href="/wiki/Projective_space" title="Projective space">projective space</a> of a Hilbert space, usually called the <a href="/wiki/Complex_projective_space" title="Complex projective space">complex projective space</a>. The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex <a href="/wiki/Square-integrable" class="mw-redirect" title="Square-integrable">square-integrable</a> functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}(\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a3649087af50f42a95da2163656076529ca3139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.124ex; height:3.176ex;" alt="{\displaystyle L^{2}(\mathbb {C} )}"></span>, while the Hilbert space for the <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> of a single proton is simply the space of two-dimensional complex vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f43d6ec8a1e1fe5a85aec0dd9bdcd45ae09b06b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{2}}"></span> with the usual inner product. </p><p>Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are <a href="/wiki/Hermitian_adjoint#Hermitian_operators" title="Hermitian adjoint">Hermitian</a> (more precisely, <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">self-adjoint</a>) linear <a href="/wiki/Operator_(physics)" title="Operator (physics)">operators</a> acting on the Hilbert space. A quantum state can be an <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvector</a> of an observable, in which case it is called an <a href="/wiki/Eigenstate" class="mw-redirect" title="Eigenstate">eigenstate</a>, and the associated <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a <a href="/wiki/Quantum_superposition" title="Quantum superposition">quantum superposition</a>. When an observable is measured, the result will be one of its eigenvalues with probability given by the <a href="/wiki/Born_rule" title="Born rule">Born rule</a>: in the simplest case the eigenvalue <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> is non-degenerate and the probability is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\langle {\vec {\lambda }},\psi \rangle |^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>λ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\langle {\vec {\lambda }},\psi \rangle |^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de4b465224a805b558697f44dcf3824fb2fbfe20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.06ex; height:3.343ex;" alt="{\displaystyle |\langle {\vec {\lambda }},\psi \rangle |^{2}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>λ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8c69573b9c55ec8ea726662102b8e693bf32f01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.843ex;" alt="{\displaystyle {\vec {\lambda }}}"></span> is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \psi ,P_{\lambda }\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi ,P_{\lambda }\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ba3648378b52e2e710937b36948cd908b2d6531" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.552ex; height:2.843ex;" alt="{\displaystyle \langle \psi ,P_{\lambda }\psi \rangle }"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\lambda }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/330591f9b6fffc93ca78514576fd0d8cfac6f0c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.683ex; height:2.509ex;" alt="{\displaystyle P_{\lambda }}"></span> is the projector onto its associated eigenspace. In the continuous case, these formulas give instead the <a href="/wiki/Probability_density" class="mw-redirect" title="Probability density">probability density</a>. </p><p>After the measurement, if result <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> was obtained, the quantum state is postulated to <a href="/wiki/Collapse_of_the_wavefunction" class="mw-redirect" title="Collapse of the wavefunction">collapse</a> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\lambda }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>λ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\lambda }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8c69573b9c55ec8ea726662102b8e693bf32f01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.843ex;" alt="{\displaystyle {\vec {\lambda }}}"></span>, in the non-degenerate case, or to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle P_{\lambda }\psi {\big /}\!{\sqrt {\langle \psi ,P_{\lambda }\psi \rangle }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle P_{\lambda }\psi {\big /}\!{\sqrt {\langle \psi ,P_{\lambda }\psi \rangle }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e38d8eb9122b686c4d8f937fe59e2ec511be43ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.029ex; height:3.343ex;" alt="{\textstyle P_{\lambda }\psi {\big /}\!{\sqrt {\langle \psi ,P_{\lambda }\psi \rangle }}}"></span>, in the general case. The <a href="/wiki/Probability" title="Probability">probabilistic</a> nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous <a href="/wiki/Bohr%E2%80%93Einstein_debates" title="Bohr–Einstein debates">Bohr–Einstein debates</a>, in which the two scientists attempted to clarify these fundamental principles by way of <a href="/wiki/Thought_experiment" title="Thought experiment">thought experiments</a>. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. Newer <a href="/wiki/Interpretation_of_quantum_mechanics" class="mw-redirect" title="Interpretation of quantum mechanics">interpretations of quantum mechanics</a> have been formulated that do away with the concept of "<a href="/wiki/Collapse_of_the_wavefunction" class="mw-redirect" title="Collapse of the wavefunction">wave function collapse</a>" (see, for example, the <a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">many-worlds interpretation</a>). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become <a href="/wiki/Quantum_Entanglement" class="mw-redirect" title="Quantum Entanglement">entangled</a> so that the original quantum system ceases to exist as an independent entity (see <i><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement in quantum mechanics</a></i><sup id="cite_ref-google215_20-0" class="reference"><a href="#cite_note-google215-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup>). </p> <div class="mw-heading mw-heading3"><h3 id="Time_evolution_of_a_quantum_state">Time evolution of a quantum state</h3></div> <p>The time evolution of a quantum state is described by the Schrödinger equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi (t)=H\psi (t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi (t)=H\psi (t).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c41b5a62afa3aa83e6ea98e69d692ac0f77c2c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.236ex; height:5.509ex;" alt="{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi (t)=H\psi (t).}"></span></dd></dl> <p>Here <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> denotes the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>, the observable corresponding to the <a href="/wiki/Total_energy" class="mw-redirect" title="Total energy">total energy</a> of the system, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }"></span> is the reduced <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a>. The constant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d724d600903f701570a9f97c2a8378a6f3a80e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.109ex; height:2.176ex;" alt="{\displaystyle i\hbar }"></span> is introduced so that the Hamiltonian is reduced to the <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">classical Hamiltonian</a> in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the <a href="/wiki/Correspondence_principle" title="Correspondence principle">correspondence principle</a>. </p><p>The solution of this differential equation is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (t)=e^{-iHt/\hbar }\psi (0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>H</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (t)=e^{-iHt/\hbar }\psi (0).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b0a822b842f8d952c4e5827f312597e72a45f4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.353ex; height:3.343ex;" alt="{\displaystyle \psi (t)=e^{-iHt/\hbar }\psi (0).}"></span></dd></dl> <p>The operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(t)=e^{-iHt/\hbar }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>H</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(t)=e^{-iHt/\hbar }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17a17e78953fa7d61788cc8bc41a25ef254fc8e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.49ex; height:3.343ex;" alt="{\displaystyle U(t)=e^{-iHt/\hbar }}"></span> is known as the time-evolution operator, and has the crucial property that it is <a href="/wiki/Unitarity_(physics)" title="Unitarity (physics)">unitary</a>. This time evolution is <a href="/wiki/Determinism" title="Determinism">deterministic</a> in the sense that – given an initial quantum state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcfdf3fefa6feee8b2192815ac15a2650d8a8db2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.485ex; height:2.843ex;" alt="{\displaystyle \psi (0)}"></span> – it makes a definite prediction of what the quantum state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cf4a36b5f945be90a527b3dbe3d55d3f0439cdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.162ex; height:2.843ex;" alt="{\displaystyle \psi (t)}"></span> will be at any later time.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Atomic-orbital-clouds_spd_m0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Atomic-orbital-clouds_spd_m0.png/280px-Atomic-orbital-clouds_spd_m0.png" decoding="async" width="280" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Atomic-orbital-clouds_spd_m0.png/420px-Atomic-orbital-clouds_spd_m0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/Atomic-orbital-clouds_spd_m0.png/560px-Atomic-orbital-clouds_spd_m0.png 2x" data-file-width="1800" data-file-height="1800" /></a><figcaption>Fig. 1: <a href="/wiki/Probability_density_function" title="Probability density function">Probability densities</a> corresponding to the wave functions of an electron in a hydrogen atom possessing definite energy levels (increasing from the top of the image to the bottom: <i>n</i> = 1, 2, 3, ...) and angular momenta (increasing across from left to right: <i>s</i>, <i>p</i>, <i>d</i>, ...). Denser areas correspond to higher probability density in a position measurement. Such wave functions are directly comparable to <a href="/wiki/Chladni%27s_figures" class="mw-redirect" title="Chladni's figures">Chladni's figures</a> of <a href="/wiki/Acoustics" title="Acoustics">acoustic</a> modes of vibration in classical physics and are modes of oscillation as well, possessing a sharp <a href="/wiki/Energy" title="Energy">energy</a> and thus, a definite <a href="/wiki/Frequency" title="Frequency">frequency</a>. The <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> and energy are <a href="/wiki/Quantization_(physics)" title="Quantization (physics)">quantized</a> and take <b>only</b> discrete values like those shown. (As is the case for <a href="/wiki/Resonant_frequency" class="mw-redirect" title="Resonant frequency">resonant frequencies</a> in acoustics.)</figcaption></figure> <p>Some wave functions produce probability distributions that are independent of time, such as <a href="/wiki/Eigenstate" class="mw-redirect" title="Eigenstate">eigenstates of the Hamiltonian</a>.<sup id="cite_ref-Zwiebach2022_7-2" class="reference"><a href="#cite_note-Zwiebach2022-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 133–137">: 133–137 </span></sup> Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited <a href="/wiki/Atom" title="Atom">atom</a> is pictured classically as a particle moving in a circular trajectory around the <a href="/wiki/Atomic_nucleus" title="Atomic nucleus">atomic nucleus</a>, whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an <a href="/wiki/Atomic_orbital" title="Atomic orbital"><i>s</i> orbital</a> (<a href="/wiki/File:Atomic-orbital-clouds_spd_m0.png" title="File:Atomic-orbital-clouds spd m0.png">Fig. 1</a>). </p><p>Analytic solutions of the Schrödinger equation are known for <a href="/wiki/List_of_quantum-mechanical_systems_with_analytical_solutions" title="List of quantum-mechanical systems with analytical solutions">very few relatively simple model Hamiltonians</a> including the <a href="/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillator</a>, the <a href="/wiki/Particle_in_a_box" title="Particle in a box">particle in a box</a>, the <a href="/wiki/Dihydrogen_cation" title="Dihydrogen cation">dihydrogen cation</a>, and the <a href="/wiki/Hydrogen_atom" title="Hydrogen atom">hydrogen atom</a>. Even the <a href="/wiki/Helium" title="Helium">helium</a> atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in <a href="/wiki/Closed-form_expression" title="Closed-form expression">closed form</a>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>However, there are techniques for finding approximate solutions. One method, called <a href="/wiki/Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">perturbation theory</a>, uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a>.<sup id="cite_ref-Zwiebach2022_7-3" class="reference"><a href="#cite_note-Zwiebach2022-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 793">: 793 </span></sup> Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion.<sup id="cite_ref-Zwiebach2022_7-4" class="reference"><a href="#cite_note-Zwiebach2022-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 849">: 849 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Uncertainty_principle">Uncertainty principle</h3></div> <p>One consequence of the basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum.<sup id="cite_ref-Cohen-Tannoudji_25-0" class="reference"><a href="#cite_note-Cohen-Tannoudji-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-L&L_26-0" class="reference"><a href="#cite_note-L&L-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> Both position and momentum are observables, meaning that they are represented by <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">Hermitian operators</a>. The position operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acc59ad6d9a06d55b96b65beb0fdfc89acc1e40e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.843ex;" alt="{\displaystyle {\hat {X}}}"></span> and momentum operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a46a8cf7bc789e8c4f8de7ca0d9946a46fb8842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.812ex; height:2.843ex;" alt="{\displaystyle {\hat {P}}}"></span> do not commute, but rather satisfy the <a href="/wiki/Canonical_commutation_relation" title="Canonical commutation relation">canonical commutation relation</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\hat {X}},{\hat {P}}]=i\hbar .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\hat {X}},{\hat {P}}]=i\hbar .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/803fe39b0eeaff8d1570df480e738cf5a968cc71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.974ex; height:3.343ex;" alt="{\displaystyle [{\hat {X}},{\hat {P}}]=i\hbar .}"></span></dd></dl> <p>Given a quantum state, the Born rule lets us compute expectation values for both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>, and moreover for powers of them. Defining the uncertainty for an observable by a <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a>, we have </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{X}={\textstyle {\sqrt {\left\langle X^{2}\right\rangle -\left\langle X\right\rangle ^{2}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>⟨</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⟩</mo> </mrow> <mo>−</mo> <msup> <mrow> <mo>⟨</mo> <mi>X</mi> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{X}={\textstyle {\sqrt {\left\langle X^{2}\right\rangle -\left\langle X\right\rangle ^{2}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/457ec20972d63dfb1ecc9087e18d1f949f908c8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:21.573ex; height:4.843ex;" alt="{\displaystyle \sigma _{X}={\textstyle {\sqrt {\left\langle X^{2}\right\rangle -\left\langle X\right\rangle ^{2}}}},}"></span></dd></dl> <p>and likewise for the momentum: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{P}={\sqrt {\left\langle P^{2}\right\rangle -\left\langle P\right\rangle ^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>⟨</mo> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⟩</mo> </mrow> <mo>−</mo> <msup> <mrow> <mo>⟨</mo> <mi>P</mi> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{P}={\sqrt {\left\langle P^{2}\right\rangle -\left\langle P\right\rangle ^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63ec8f5f7b9e5957ea6d06c56068b06244acc184" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:21.383ex; height:4.843ex;" alt="{\displaystyle \sigma _{P}={\sqrt {\left\langle P^{2}\right\rangle -\left\langle P\right\rangle ^{2}}}.}"></span></dd></dl> <p>The uncertainty principle states that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{X}\sigma _{P}\geq {\frac {\hbar }{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{X}\sigma _{P}\geq {\frac {\hbar }{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/538cada7fa57155ece387e5d53d90ca366e323fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.641ex; height:5.343ex;" alt="{\displaystyle \sigma _{X}\sigma _{P}\geq {\frac {\hbar }{2}}.}"></span></dd></dl> <p>Either standard deviation can in principle be made arbitrarily small, but not both simultaneously.<sup id="cite_ref-ballentine1970_27-0" class="reference"><a href="#cite_note-ballentine1970-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> This inequality generalizes to arbitrary pairs of self-adjoint operators <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>. The <a href="/wiki/Commutator" title="Commutator">commutator</a> of these two operators is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [A,B]=AB-BA,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>A</mi> <mi>B</mi> <mo>−</mo> <mi>B</mi> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [A,B]=AB-BA,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2a47259b42e63c048c65f67d304404867841951" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.435ex; height:2.843ex;" alt="{\displaystyle [A,B]=AB-BA,}"></span></dd></dl> <p>and this provides the lower bound on the product of standard deviations: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{A}\sigma _{B}\geq {\tfrac {1}{2}}\left|{\bigl \langle }[A,B]{\bigr \rangle }\right|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">⟨</mo> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">⟩</mo> </mrow> </mrow> </mrow> <mo>|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{A}\sigma _{B}\geq {\tfrac {1}{2}}\left|{\bigl \langle }[A,B]{\bigr \rangle }\right|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0fd768b447334e150b8b98181f74b475e41ee52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.1ex; height:3.509ex;" alt="{\displaystyle \sigma _{A}\sigma _{B}\geq {\tfrac {1}{2}}\left|{\bigl \langle }[A,B]{\bigr \rangle }\right|.}"></span></dd></dl> <p>Another consequence of the canonical commutation relation is that the position and momentum operators are <a href="/wiki/Fourier_transform#Uncertainty_principle" title="Fourier transform">Fourier transforms</a> of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of the dependence in position means that the momentum operator is equivalent (up to an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i/\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i/\hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44a04e1f19b5e7bea2bfa8002a841bf8d1b4e66a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.271ex; height:2.843ex;" alt="{\displaystyle i/\hbar }"></span> factor) to taking the derivative according to the position, since in Fourier analysis <a href="/wiki/Fourier_transform#Differentiation" title="Fourier transform">differentiation corresponds to multiplication in the dual space</a>. This is why in quantum equations in position space, the momentum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bab39399bf5424f25d957cdc57c84a0622626d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}"></span> is replaced by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i\hbar {\frac {\partial }{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\hbar {\frac {\partial }{\partial x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab7fffcee704fc55eb36b137e0cc25132b5dc1bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.401ex; height:5.509ex;" alt="{\displaystyle -i\hbar {\frac {\partial }{\partial x}}}"></span>, and in particular in the <a href="/wiki/Schr%C3%B6dinger_equation#Equation" title="Schrödinger equation">non-relativistic Schrödinger equation in position space</a> the momentum-squared term is replaced with a Laplacian times <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\hbar ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\hbar ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/185934c9e69984a0905432abe63d3b41741342b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.184ex; height:2.843ex;" alt="{\displaystyle -\hbar ^{2}}"></span>.<sup id="cite_ref-Cohen-Tannoudji_25-1" class="reference"><a href="#cite_note-Cohen-Tannoudji-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Composite_systems_and_entanglement">Composite systems and entanglement</h3></div> <p>When two different quantum systems are considered together, the Hilbert space of the combined system is the <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a> of the Hilbert spaces of the two components. For example, let <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span> be two quantum systems, with Hilbert spaces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611f8542d6c72789b7e82bfcf29014d44c13aa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.429ex; height:2.509ex;" alt="{\displaystyle {\mathcal {H}}_{A}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfea1102a35aff8c6435147dc9dffc21b4cfe4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.443ex; height:2.509ex;" alt="{\displaystyle {\mathcal {H}}_{B}}"></span>, respectively. The Hilbert space of the composite system is then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}_{AB}={\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{AB}={\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54e4f6aff2be5ec9f3eb4657a133d1be17f6e2b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.134ex; height:2.509ex;" alt="{\displaystyle {\mathcal {H}}_{AB}={\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}.}"></span></dd></dl> <p>If the state for the first system is the vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a04e49109107e7765357dd92e975895840fa7700" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.978ex; height:2.509ex;" alt="{\displaystyle \psi _{A}}"></span> and the state for the second system is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ca3b90bba42c451129c4969322e175f5a1d3368" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle \psi _{B}}"></span>, then the state of the composite system is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{A}\otimes \psi _{B}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{A}\otimes \psi _{B}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/852c1e09787e91af3af1a10d1eee4eadc232d48c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.458ex; height:2.509ex;" alt="{\displaystyle \psi _{A}\otimes \psi _{B}.}"></span></dd></dl> <p>Not all states in the joint Hilbert space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}_{AB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}_{AB}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/951b54ea87d5bd62464abaf1be530e6a37a86f1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.676ex; height:2.509ex;" alt="{\displaystyle {\mathcal {H}}_{AB}}"></span> can be written in this form, however, because the superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a04e49109107e7765357dd92e975895840fa7700" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.978ex; height:2.509ex;" alt="{\displaystyle \psi _{A}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e38d84dcc21ea769a51ab502691f566c830b430" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.85ex; height:2.509ex;" alt="{\displaystyle \phi _{A}}"></span> are both possible states for system <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, and likewise <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ca3b90bba42c451129c4969322e175f5a1d3368" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle \psi _{B}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a829a7a85bb29904dec5e93605d0b976baf45755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.865ex; height:2.509ex;" alt="{\displaystyle \phi _{B}}"></span> are both possible states for system <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>, then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{\sqrt {2}}}\left(\psi _{A}\otimes \psi _{B}+\phi _{A}\otimes \phi _{B}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\sqrt {2}}}\left(\psi _{A}\otimes \psi _{B}+\phi _{A}\otimes \phi _{B}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53d226ba585a99942ec855197c23ed885b635258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.431ex; height:4.176ex;" alt="{\displaystyle {\tfrac {1}{\sqrt {2}}}\left(\psi _{A}\otimes \psi _{B}+\phi _{A}\otimes \phi _{B}\right)}"></span></dd></dl> <p>is a valid joint state that is not separable. States that are not separable are called <a href="/wiki/Quantum_entanglement" title="Quantum entanglement">entangled</a>.<sup id="cite_ref-:0_28-0" class="reference"><a href="#cite_note-:0-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_29-0" class="reference"><a href="#cite_note-:1-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p><p>If the state for a composite system is entangled, it is impossible to describe either component system <span class="texhtml mvar" style="font-style:italic;">A</span> or system <span class="texhtml mvar" style="font-style:italic;">B</span> by a state vector. One can instead define <a href="/wiki/Reduced_density_matrix" class="mw-redirect" title="Reduced density matrix">reduced density matrices</a> that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes a loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system.<sup id="cite_ref-:0_28-1" class="reference"><a href="#cite_note-:0-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_29-1" class="reference"><a href="#cite_note-:1-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> Just as density matrices specify the state of a subsystem of a larger system, analogously, <a href="/wiki/POVM" title="POVM">positive operator-valued measures</a> (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory.<sup id="cite_ref-:0_28-2" class="reference"><a href="#cite_note-:0-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-wilde_30-0" class="reference"><a href="#cite_note-wilde-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p><p>As described above, entanglement is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as <a href="/wiki/Quantum_decoherence" title="Quantum decoherence">quantum decoherence</a>. This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Equivalence_between_formulations">Equivalence between formulations</h3></div> <p>There are many mathematically equivalent formulations of quantum mechanics. One of the oldest and most common is the "<a href="/wiki/Transformation_theory_(quantum_mechanics)" title="Transformation theory (quantum mechanics)">transformation theory</a>" proposed by <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a>, which unifies and generalizes the two earliest formulations of quantum mechanics – <a href="/wiki/Matrix_mechanics" title="Matrix mechanics">matrix mechanics</a> (invented by <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a>) and wave mechanics (invented by <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a>).<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> An alternative formulation of quantum mechanics is <a href="/wiki/Feynman" class="mw-redirect" title="Feynman">Feynman</a>'s <a href="/wiki/Path_integral_formulation" title="Path integral formulation">path integral formulation</a>, in which a quantum-mechanical amplitude is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the <a href="/wiki/Action_principle" class="mw-redirect" title="Action principle">action principle</a> in classical mechanics.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Symmetries_and_conservation_laws">Symmetries and conservation laws</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a></div> <p>The Hamiltonian <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> is known as the <i>generator</i> of time evolution, since it defines a unitary time-evolution operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(t)=e^{-iHt/\hbar }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>H</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(t)=e^{-iHt/\hbar }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17a17e78953fa7d61788cc8bc41a25ef254fc8e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.49ex; height:3.343ex;" alt="{\displaystyle U(t)=e^{-iHt/\hbar }}"></span> for each value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>. From this relation between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/666c639df532e88616357c4991cabce9a57b5611" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.431ex; height:2.843ex;" alt="{\displaystyle U(t)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>, it follows that any observable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> that commutes with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> will be <i>conserved</i>: its expectation value will not change over time.<sup id="cite_ref-Zwiebach2022_7-5" class="reference"><a href="#cite_note-Zwiebach2022-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 471">: 471 </span></sup> This statement generalizes, as mathematically, any Hermitian operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> can generate a family of unitary operators parameterized by a variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>. Under the evolution generated by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, any observable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> that commutes with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> will be conserved. Moreover, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> is conserved by evolution under <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is conserved under the evolution generated by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>. This implies a quantum version of the result proven by <a href="/wiki/Emmy_Noether" title="Emmy Noether">Emmy Noether</a> in classical (<a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</a>) mechanics: for every <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable</a> <a href="/wiki/Symmetry_(physics)" title="Symmetry (physics)">symmetry</a> of a Hamiltonian, there exists a corresponding <a href="/wiki/Conservation_law" title="Conservation law">conservation law</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div> <div class="mw-heading mw-heading3"><h3 id="Free_particle">Free particle</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Free_particle" title="Free particle">Free particle</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Guassian_Dispersion.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/56/Guassian_Dispersion.gif" decoding="async" width="360" height="223" class="mw-file-element" data-file-width="360" data-file-height="223" /></a><figcaption>Position space probability density of a Gaussian <a href="/wiki/Wave_packet" title="Wave packet">wave packet</a> moving in one dimension in free space</figcaption></figure> <p>The simplest example of a quantum system with a position degree of freedom is a free particle in a single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H={\frac {1}{2m}}P^{2}=-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H={\frac {1}{2m}}P^{2}=-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/084f585ee5c6cd8a34e323a6de7943227128afc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:26.105ex; height:6.009ex;" alt="{\displaystyle H={\frac {1}{2m}}P^{2}=-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}.}"></span></dd></dl> <p>The general solution of the Schrödinger equation is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x,t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\hat {\psi }}(k,0)e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}\mathrm {d} k,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>k</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x,t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\hat {\psi }}(k,0)e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}\mathrm {d} k,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef4f021ba945856e3676808b11724109a8a74dad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:39.988ex; height:6.509ex;" alt="{\displaystyle \psi (x,t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\hat {\psi }}(k,0)e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}\mathrm {d} k,}"></span></dd></dl> <p>which is a superposition of all possible <a href="/wiki/Plane_wave" title="Plane wave">plane waves</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4cd9a9984c84a493ce547babcf58e31b04f7e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.914ex; height:4.009ex;" alt="{\displaystyle e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}}"></span>, which are eigenstates of the momentum operator with momentum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=\hbar k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=\hbar k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24fee69175538303b28ac54e907baf53d0a58dbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.875ex; height:2.509ex;" alt="{\displaystyle p=\hbar k}"></span>. The coefficients of the superposition are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\psi }}(k,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\psi }}(k,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b8323c08418da8bc376c6d78b578d4729b927ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.813ex; height:3.343ex;" alt="{\displaystyle {\hat {\psi }}(k,0)}"></span>, which is the Fourier transform of the initial quantum state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55ad442e07ca2d7986ef0787f9129fc325cde137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.849ex; height:2.843ex;" alt="{\displaystyle \psi (x,0)}"></span>. </p><p>It is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> Instead, we can consider a Gaussian <a href="/wiki/Wave_packet" title="Wave packet">wave packet</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x,0)={\frac {1}{\sqrt[{4}]{\pi a}}}e^{-{\frac {x^{2}}{2a}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mroot> <mrow> <mi>π</mi> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x,0)={\frac {1}{\sqrt[{4}]{\pi a}}}e^{-{\frac {x^{2}}{2a}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c2dae82312897d5fd4c58986c426a6009e6840" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.275ex; height:6.509ex;" alt="{\displaystyle \psi (x,0)={\frac {1}{\sqrt[{4}]{\pi a}}}e^{-{\frac {x^{2}}{2a}}}}"></span></dd></dl> <p>which has Fourier transform, and therefore momentum distribution </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\psi }}(k,0)={\sqrt[{4}]{\frac {a}{\pi }}}e^{-{\frac {ak^{2}}{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mfrac> <mi>a</mi> <mi>π</mi> </mfrac> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\psi }}(k,0)={\sqrt[{4}]{\frac {a}{\pi }}}e^{-{\frac {ak^{2}}{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4991535bba434314af8c27c16fff74f49ce367e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.682ex; height:6.343ex;" alt="{\displaystyle {\hat {\psi }}(k,0)={\sqrt[{4}]{\frac {a}{\pi }}}e^{-{\frac {ak^{2}}{2}}}.}"></span></dd></dl> <p>We see that as we make <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle. </p><p>As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that the position becomes more and more uncertain. The uncertainty in momentum, however, stays constant.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Particle_in_a_box">Particle in a box</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Infinite_potential_well.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Infinite_potential_well.svg/220px-Infinite_potential_well.svg.png" decoding="async" width="220" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Infinite_potential_well.svg/330px-Infinite_potential_well.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/27/Infinite_potential_well.svg/440px-Infinite_potential_well.svg.png 2x" data-file-width="275" data-file-height="220" /></a><figcaption>1-dimensional potential energy box (or infinite potential well)</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Particle_in_a_box" title="Particle in a box">Particle in a box</a></div> <p>The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere <i>inside</i> a certain region, and therefore infinite potential energy everywhere <i>outside</i> that region.<sup id="cite_ref-Cohen-Tannoudji_25-3" class="reference"><a href="#cite_note-Cohen-Tannoudji-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 77–78">: 77–78 </span></sup> For the one-dimensional case in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> direction, the time-independent Schrödinger equation may be written </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi }{dx^{2}}}=E\psi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>E</mi> <mi>ψ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi }{dx^{2}}}=E\psi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc73e9216faf1390c3ed550b72be21fc068ec747" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.503ex; height:6.009ex;" alt="{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi }{dx^{2}}}=E\psi .}"></span></dd></dl> <p>With the differential operator defined by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {p}}_{x}=-i\hbar {\frac {d}{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {p}}_{x}=-i\hbar {\frac {d}{dx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2511885975007002c7582a9c8175689076df210a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.089ex; width:13.019ex; height:5.509ex;" alt="{\displaystyle {\hat {p}}_{x}=-i\hbar {\frac {d}{dx}}}"></span>the previous equation is evocative of the <a href="/wiki/Kinetic_energy#Kinetic_energy_of_rigid_bodies" title="Kinetic energy">classic kinetic energy analogue</a>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2m}}{\hat {p}}_{x}^{2}=E,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mi>E</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2m}}{\hat {p}}_{x}^{2}=E,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/422f57813ede91b8e348e786616c6297742b4fd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.092ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2m}}{\hat {p}}_{x}^{2}=E,}"></span></dd></dl> <p>with state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> in this case having energy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> coincident with the kinetic energy of the particle. </p><p>The general solutions of the Schrödinger equation for the particle in a box are </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)=Ae^{ikx}+Be^{-ikx}\qquad \qquad E={\frac {\hbar ^{2}k^{2}}{2m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mo>+</mo> <mi>B</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> <mspace width="2em" /> <mspace width="2em" /> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)=Ae^{ikx}+Be^{-ikx}\qquad \qquad E={\frac {\hbar ^{2}k^{2}}{2m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5fb1b2f1d5afb42edb4eb98bf89791d283c1e53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.379ex; height:5.676ex;" alt="{\displaystyle \psi (x)=Ae^{ikx}+Be^{-ikx}\qquad \qquad E={\frac {\hbar ^{2}k^{2}}{2m}}}"></span></dd></dl> <p>or, from <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)=C\sin(kx)+D\cos(kx).\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>D</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x)=C\sin(kx)+D\cos(kx).\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/229ebbbe587e0c678318cb85089a74ed289c971a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.204ex; width:30.186ex; height:2.843ex;" alt="{\displaystyle \psi (x)=C\sin(kx)+D\cos(kx).\!}"></span></dd></dl> <p>The infinite potential walls of the box determine the values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C,D,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>,</mo> <mi>D</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C,D,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e153d2c75e3492a32d4fafefec88846862c3b9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.371ex; height:2.509ex;" alt="{\displaystyle C,D,}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5fe40c588800aaab69041986b49a59664cd767a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.011ex; height:2.176ex;" alt="{\displaystyle x=L}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> must be zero. Thus, at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (0)=0=C\sin(0)+D\cos(0)=D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>=</mo> <mi>C</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>D</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (0)=0=C\sin(0)+D\cos(0)=D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34188e64cbf80486b3ad311b53b512fa36de59c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.082ex; height:2.843ex;" alt="{\displaystyle \psi (0)=0=C\sin(0)+D\cos(0)=D}"></span></dd></dl> <p>and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d375dfda80ee8df1d1d7aa8b962114044e464305" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.185ex; height:2.176ex;" alt="{\displaystyle D=0}"></span>. At <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5fe40c588800aaab69041986b49a59664cd767a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.011ex; height:2.176ex;" alt="{\displaystyle x=L}"></span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (L)=0=C\sin(kL),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>=</mo> <mi>C</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>L</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (L)=0=C\sin(kL),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7a5bc9ef330efb5b6eba5d89c2ee026bbeea6a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.524ex; height:2.843ex;" alt="{\displaystyle \psi (L)=0=C\sin(kL),}"></span></dd></dl> <p>in which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> cannot be zero as this would conflict with the postulate that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> has norm 1. Therefore, since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(kL)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>L</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(kL)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5177ccdb2057c5c1be728af20b8ef3d61f79999" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.72ex; height:2.843ex;" alt="{\displaystyle \sin(kL)=0}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle kL}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle kL}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8865c1184b2c1dff6226dae50d3be91f4f01cfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.794ex; height:2.176ex;" alt="{\displaystyle kL}"></span> must be an integer multiple of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\frac {n\pi }{L}}\qquad \qquad n=1,2,3,\ldots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>π</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mspace width="2em" /> <mspace width="2em" /> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {n\pi }{L}}\qquad \qquad n=1,2,3,\ldots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fd91af3024e1f59cf57e04884494fd2c55664f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.002ex; height:4.676ex;" alt="{\displaystyle k={\frac {n\pi }{L}}\qquad \qquad n=1,2,3,\ldots .}"></span></dd></dl> <p>This constraint on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> implies a constraint on the energy levels, yielding </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}={\frac {\hbar ^{2}\pi ^{2}n^{2}}{2mL^{2}}}={\frac {n^{2}h^{2}}{8mL^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>m</mi> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <mi>m</mi> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}={\frac {\hbar ^{2}\pi ^{2}n^{2}}{2mL^{2}}}={\frac {n^{2}h^{2}}{8mL^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe507fd1f92ebfe133aeb4f1da46ea27f569f38b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.503ex; height:6.009ex;" alt="{\displaystyle E_{n}={\frac {\hbar ^{2}\pi ^{2}n^{2}}{2mL^{2}}}={\frac {n^{2}h^{2}}{8mL^{2}}}.}"></span></dd></dl> <p>A <a href="/wiki/Finite_potential_well" title="Finite potential well">finite potential well</a> is the generalization of the infinite potential well problem to potential wells having finite depth. The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well. Another related problem is that of the <a href="/wiki/Rectangular_potential_barrier" title="Rectangular potential barrier">rectangular potential barrier</a>, which furnishes a model for the <a href="/wiki/Quantum_tunneling" class="mw-redirect" title="Quantum tunneling">quantum tunneling</a> effect that plays an important role in the performance of modern technologies such as <a href="/wiki/Flash_memory" title="Flash memory">flash memory</a> and <a href="/wiki/Scanning_tunneling_microscope" title="Scanning tunneling microscope">scanning tunneling microscopy</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Harmonic_oscillator">Harmonic oscillator</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">Quantum harmonic oscillator</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:QuantumHarmonicOscillatorAnimation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/9/90/QuantumHarmonicOscillatorAnimation.gif" decoding="async" width="300" height="373" class="mw-file-element" data-file-width="300" data-file-height="373" /></a><figcaption>Some trajectories of a <a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">harmonic oscillator</a> (i.e. a ball attached to a <a href="/wiki/Hooke%27s_law" title="Hooke's law">spring</a>) in <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> (A-B) and quantum mechanics (C-H). In quantum mechanics, the position of the ball is represented by a <a href="/wiki/Wave" title="Wave">wave</a> (called the wave function), with the <a href="/wiki/Real_part" class="mw-redirect" title="Real part">real part</a> shown in blue and the <a href="/wiki/Imaginary_part" class="mw-redirect" title="Imaginary part">imaginary part</a> shown in red. Some of the trajectories (such as C, D, E, and F) are <a href="/wiki/Standing_wave" title="Standing wave">standing waves</a> (or "<a href="/wiki/Stationary_state" title="Stationary state">stationary states</a>"). Each standing-wave frequency is proportional to a possible <a href="/wiki/Energy_level" title="Energy level">energy level</a> of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have <i>any</i> energy.</figcaption></figure> <p>As in the classical case, the potential for the quantum harmonic oscillator is given by<sup id="cite_ref-Zwiebach2022_7-6" class="reference"><a href="#cite_note-Zwiebach2022-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 234">: 234 </span></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)={\frac {1}{2}}m\omega ^{2}x^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(x)={\frac {1}{2}}m\omega ^{2}x^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9d8735e10198c090b576c765dc2778fe458b9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.594ex; height:5.176ex;" alt="{\displaystyle V(x)={\frac {1}{2}}m\omega ^{2}x^{2}.}"></span></dd></dl> <p>This problem can either be treated by directly solving the Schrödinger equation, which is not trivial, or by using the more elegant "ladder method" first proposed by Paul Dirac. The <a href="/wiki/Eigenstate" class="mw-redirect" title="Eigenstate">eigenstates</a> are given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{n}(x)={\sqrt {\frac {1}{2^{n}\,n!}}}\cdot \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\cdot e^{-{\frac {m\omega x^{2}}{2\hbar }}}\cdot H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),\qquad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>1</mn> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </msqrt> </mrow> <mo>⋅</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>ω</mi> </mrow> <mrow> <mi>π</mi> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>ω</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </mrow> </mrow> </msup> <mo>⋅</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>m</mi> <mi>ω</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </msqrt> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="2em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{n}(x)={\sqrt {\frac {1}{2^{n}\,n!}}}\cdot \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\cdot e^{-{\frac {m\omega x^{2}}{2\hbar }}}\cdot H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),\qquad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a0031b1b0c9fa4a9e4e6957718d509846522c4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:59.381ex; height:6.509ex;" alt="{\displaystyle \psi _{n}(x)={\sqrt {\frac {1}{2^{n}\,n!}}}\cdot \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\cdot e^{-{\frac {m\omega x^{2}}{2\hbar }}}\cdot H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),\qquad }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=0,1,2,\ldots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=0,1,2,\ldots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d652f9756f144ea772e4e632cb1398652d36372c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.84ex; height:2.509ex;" alt="{\displaystyle n=0,1,2,\ldots .}"></span></dd></dl> <p>where <i>H<sub>n</sub></i> are the <a href="/wiki/Hermite_polynomials" title="Hermite polynomials">Hermite polynomials</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}\left(e^{-x^{2}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}\left(e^{-x^{2}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0da8b4a24291ce8b32bcc0665f490f3c76afc7d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.636ex; height:5.509ex;" alt="{\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}\left(e^{-x^{2}}\right),}"></span></dd></dl> <p>and the corresponding energy levels are </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}=\hbar \omega \left(n+{1 \over 2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mi>ω</mi> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}=\hbar \omega \left(n+{1 \over 2}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/635fb242f4c34cd5ba947f4c8132ddd47f2872c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.86ex; height:6.176ex;" alt="{\displaystyle E_{n}=\hbar \omega \left(n+{1 \over 2}\right).}"></span></dd></dl> <p>This is another example illustrating the discretization of energy for <a href="/wiki/Bound_state" title="Bound state">bound states</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Mach–Zehnder_interferometer"><span id="Mach.E2.80.93Zehnder_interferometer"></span>Mach–Zehnder interferometer</h3></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Mach-Zehnder_interferometer.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Mach-Zehnder_interferometer.svg/290px-Mach-Zehnder_interferometer.svg.png" decoding="async" width="290" height="175" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Mach-Zehnder_interferometer.svg/435px-Mach-Zehnder_interferometer.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Mach-Zehnder_interferometer.svg/580px-Mach-Zehnder_interferometer.svg.png 2x" data-file-width="468" data-file-height="283" /></a><figcaption>Schematic of a Mach–Zehnder interferometer</figcaption></figure> <p>The <a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a> (MZI) illustrates the concepts of superposition and interference with linear algebra in dimension 2, rather than differential equations. It can be seen as a simplified version of the double-slit experiment, but it is of interest in its own right, for example in the <a href="/wiki/Delayed_choice_quantum_eraser" class="mw-redirect" title="Delayed choice quantum eraser">delayed choice quantum eraser</a>, the <a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman bomb tester</a>, and in studies of quantum entanglement.<sup id="cite_ref-Paris1999_36-0" class="reference"><a href="#cite_note-Paris1999-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Haack2010_37-0" class="reference"><a href="#cite_note-Haack2010-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p><p>We can model a photon going through the interferometer by considering that at each point it can be in a superposition of only two paths: the "lower" path which starts from the left, goes straight through both beam splitters, and ends at the top, and the "upper" path which starts from the bottom, goes straight through both beam splitters, and ends at the right. The quantum state of the photon is therefore a vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \in \mathbb {C} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \in \mathbb {C} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a51a8280039fc22cd88d90915a952f8e020f47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.086ex; height:3.009ex;" alt="{\displaystyle \psi \in \mathbb {C} ^{2}}"></span> that is a superposition of the "lower" path <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{l}={\begin{pmatrix}1\\0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{l}={\begin{pmatrix}1\\0\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feca39f1f03b4dc63be6f7a7c2060430b1217e2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.669ex; height:6.176ex;" alt="{\displaystyle \psi _{l}={\begin{pmatrix}1\\0\end{pmatrix}}}"></span> and the "upper" path <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{u}={\begin{pmatrix}0\\1\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{u}={\begin{pmatrix}0\\1\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/400e9751fcd7816718398d3892bec2ad26bb4713" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.119ex; height:6.176ex;" alt="{\displaystyle \psi _{u}={\begin{pmatrix}0\\1\end{pmatrix}}}"></span>, that is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi =\alpha \psi _{l}+\beta \psi _{u}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mi>α</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>+</mo> <mi>β</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =\alpha \psi _{l}+\beta \psi _{u}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb88364b79f3c611ca2ca1edb5356139bf4e1085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.193ex; height:2.509ex;" alt="{\displaystyle \psi =\alpha \psi _{l}+\beta \psi _{u}}"></span> for complex <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b46b57cfa0011b643037751809904d915c1b48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.854ex; height:2.509ex;" alt="{\displaystyle \alpha ,\beta }"></span>. In order to respect the postulate that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \psi ,\psi \rangle =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mo>,</mo> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi ,\psi \rangle =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d5758e7a60b4e54bc46e01b0618919c65b787a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.13ex; height:2.843ex;" alt="{\displaystyle \langle \psi ,\psi \rangle =1}"></span> we require that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha |^{2}+|\beta |^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>β</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\alpha |^{2}+|\beta |^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18cd7473cdb894839d10852890517b1fb687c73b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.617ex; height:3.343ex;" alt="{\displaystyle |\alpha |^{2}+|\beta |^{2}=1}"></span>. </p><p>Both <a href="/wiki/Beam_splitter" title="Beam splitter">beam splitters</a> are modelled as the unitary matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&i\\i&1\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&i\\i&1\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddf502efcb65d0cbac5bb8ef1a6f163ac9cf2145" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:17.617ex; height:6.509ex;" alt="{\displaystyle B={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&i\\i&1\end{pmatrix}}}"></span>, which means that when a photon meets the beam splitter it will either stay on the same path with a probability amplitude of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a0bbdb60fcb73ac67d9970a5eb0808b87fd37d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:3.176ex;" alt="{\displaystyle 1/{\sqrt {2}}}"></span>, or be reflected to the other path with a probability amplitude of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i/{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i/{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e846b6a5731f3799960a4968399d85bc0b7fb9fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.063ex; height:3.176ex;" alt="{\displaystyle i/{\sqrt {2}}}"></span>. The phase shifter on the upper arm is modelled as the unitary matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P={\begin{pmatrix}1&0\\0&e^{i\Delta \Phi }\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi mathvariant="normal">Δ</mi> <mi mathvariant="normal">Φ</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P={\begin{pmatrix}1&0\\0&e^{i\Delta \Phi }\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3df9457946dd8035c51a39e0926be9f07c7f0a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.94ex; height:6.176ex;" alt="{\displaystyle P={\begin{pmatrix}1&0\\0&e^{i\Delta \Phi }\end{pmatrix}}}"></span>, which means that if the photon is on the "upper" path it will gain a relative phase of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20cedb08e6edea3cad9b2829ef67311bbe518dd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.614ex; height:2.176ex;" alt="{\displaystyle \Delta \Phi }"></span>, and it will stay unchanged if it is in the lower path. </p><p>A photon that enters the interferometer from the left will then be acted upon with a beam splitter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>, a phase shifter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>, and another beam splitter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>, and so end up in the state </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BPB\psi _{l}=ie^{i\Delta \Phi /2}{\begin{pmatrix}-\sin(\Delta \Phi /2)\\\cos(\Delta \Phi /2)\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>P</mi> <mi>B</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mi>i</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi mathvariant="normal">Δ</mi> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BPB\psi _{l}=ie^{i\Delta \Phi /2}{\begin{pmatrix}-\sin(\Delta \Phi /2)\\\cos(\Delta \Phi /2)\end{pmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7927a94da54f5d57b8accffdb9ad456a1e3b5033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.111ex; height:6.176ex;" alt="{\displaystyle BPB\psi _{l}=ie^{i\Delta \Phi /2}{\begin{pmatrix}-\sin(\Delta \Phi /2)\\\cos(\Delta \Phi /2)\end{pmatrix}},}"></span></dd></dl> <p>and the probabilities that it will be detected at the right or at the top are given respectively by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(u)=|\langle \psi _{u},BPB\psi _{l}\rangle |^{2}=\cos ^{2}{\frac {\Delta \Phi }{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>,</mo> <mi>B</mi> <mi>P</mi> <mi>B</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(u)=|\langle \psi _{u},BPB\psi _{l}\rangle |^{2}=\cos ^{2}{\frac {\Delta \Phi }{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/110acb8ee7dc4e309de846470778f4767fe97f8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:35.63ex; height:5.343ex;" alt="{\displaystyle p(u)=|\langle \psi _{u},BPB\psi _{l}\rangle |^{2}=\cos ^{2}{\frac {\Delta \Phi }{2}},}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(l)=|\langle \psi _{l},BPB\psi _{l}\rangle |^{2}=\sin ^{2}{\frac {\Delta \Phi }{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>,</mo> <mi>B</mi> <mi>P</mi> <mi>B</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(l)=|\langle \psi _{l},BPB\psi _{l}\rangle |^{2}=\sin ^{2}{\frac {\Delta \Phi }{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86880ce53051688a4f591ebfb5183fbae013deee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:34.288ex; height:5.343ex;" alt="{\displaystyle p(l)=|\langle \psi _{l},BPB\psi _{l}\rangle |^{2}=\sin ^{2}{\frac {\Delta \Phi }{2}}.}"></span></dd></dl> <p>One can therefore use the Mach–Zehnder interferometer to estimate the <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase shift</a> by estimating these probabilities. </p><p>It is interesting to consider what would happen if the photon were definitely in either the "lower" or "upper" paths between the beam splitters. This can be accomplished by blocking one of the paths, or equivalently by removing the first beam splitter (and feeding the photon from the left or the bottom, as desired). In both cases, there will be no interference between the paths anymore, and the probabilities are given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(u)=p(l)=1/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(u)=p(l)=1/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2a9b3426984de1a64c07261c3e14b485320fee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:17.754ex; height:2.843ex;" alt="{\displaystyle p(u)=p(l)=1/2}"></span>, independently of the phase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20cedb08e6edea3cad9b2829ef67311bbe518dd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.614ex; height:2.176ex;" alt="{\displaystyle \Delta \Phi }"></span>. From this we can conclude that the photon does not take one path or another after the first beam splitter, but rather that it is in a genuine quantum superposition of the two paths.<sup id="cite_ref-vedral_38-0" class="reference"><a href="#cite_note-vedral-38"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Applications_of_quantum_mechanics" title="Applications of quantum mechanics">Applications of quantum mechanics</a></div> <p>Quantum mechanics has had enormous success in explaining many of the features of our universe, with regard to small-scale and discrete quantities and interactions which cannot be explained by <a href="/wiki/Classical_physics" title="Classical physics">classical methods</a>.<sup id="cite_ref-feynmanIII_39-0" class="reference"><a href="#cite_note-feynmanIII-39"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup> Quantum mechanics is often the only theory that can reveal the individual behaviors of the subatomic particles that make up all forms of matter (electrons, <a href="/wiki/Proton" title="Proton">protons</a>, <a href="/wiki/Neutron" title="Neutron">neutrons</a>, <a href="/wiki/Photon" title="Photon">photons</a>, and others). <a href="/wiki/Solid-state_physics" title="Solid-state physics">Solid-state physics</a> and <a href="/wiki/Materials_science" title="Materials science">materials science</a> are dependent upon quantum mechanics.<sup id="cite_ref-marvincohen2008_40-0" class="reference"><a href="#cite_note-marvincohen2008-40"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p><p>In many aspects, modern technology operates at a scale where quantum effects are significant. Important applications of quantum theory include <a href="/wiki/Quantum_chemistry" title="Quantum chemistry">quantum chemistry</a>, <a href="/wiki/Quantum_optics" title="Quantum optics">quantum optics</a>, <a href="/wiki/Quantum_computing" title="Quantum computing">quantum computing</a>, <a href="/wiki/Superconducting_magnet" title="Superconducting magnet">superconducting magnets</a>, <a href="/wiki/Light-emitting_diode" title="Light-emitting diode">light-emitting diodes</a>, the <a href="/wiki/Optical_amplifier" title="Optical amplifier">optical amplifier</a> and the laser, the <a href="/wiki/Transistor" title="Transistor">transistor</a> and <a href="/wiki/Semiconductor" title="Semiconductor">semiconductors</a> such as the <a href="/wiki/Microprocessor" title="Microprocessor">microprocessor</a>, <a href="/wiki/Medical_imaging" title="Medical imaging">medical and research imaging</a> such as <a href="/wiki/Magnetic_resonance_imaging" title="Magnetic resonance imaging">magnetic resonance imaging</a> and <a href="/wiki/Electron_microscope" title="Electron microscope">electron microscopy</a>.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule <a href="/wiki/DNA" title="DNA">DNA</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Relation_to_other_scientific_theories">Relation to other scientific theories</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a href="/wiki/Modern_physics" title="Modern physics">Modern physics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}|\psi _{n}(t)\rangle =i\hbar {\frac {d}{dt}}|\psi _{n}(t)\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}|\psi _{n}(t)\rangle =i\hbar {\frac {d}{dt}}|\psi _{n}(t)\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6227214310d46ffac904ca257878eee1fb6ce726" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.027ex; height:5.509ex;" alt="{\displaystyle {\hat {H}}|\psi _{n}(t)\rangle =i\hbar {\frac {d}{dt}}|\psi _{n}(t)\rangle }"></span> <br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <mi mathvariant="normal">Λ</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>κ</mi> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/124ab80fcb17e2733cc17ff6f93da5e52f355c77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.468ex; height:2.843ex;" alt="{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}"></span><div class="sidebar-caption" style="font-size:90%;padding-top:0.4em;font-style:italic;"><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a> and <a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></div></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Founders</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Max_Planck" title="Max Planck">Max Planck</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a></li> <li><a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a></li> <li><a href="/wiki/Max_Born" title="Max Born">Max Born</a></li> <li><a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a></li> <li><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a></li> <li><a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Pascual Jordan</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Wolfgang Pauli</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a></li> <li><a href="/wiki/Ernest_Rutherford" title="Ernest Rutherford">Ernest Rutherford</a></li> <li><a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">Louis de Broglie</a></li> <li><a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Satyendra Nath Bose</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Concepts</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Spacetime_topology" title="Spacetime topology">Topology</a></li> <li><a href="/wiki/Space" title="Space">Space</a></li> <li><a href="/wiki/Time" title="Time">Time</a></li> <li><a href="/wiki/Energy" title="Energy">Energy</a></li> <li><a href="/wiki/Matter" title="Matter">Matter</a></li> <li><a href="/wiki/Work_(physics)" title="Work (physics)">Work</a></li> <li><a href="/wiki/Randomness" title="Randomness">Randomness</a></li> <li><a href="/wiki/Information" title="Information">Information</a></li> <li><a href="/wiki/Entropy" title="Entropy">Entropy</a></li> <li><a href="/wiki/Light" title="Light">Light</a></li> <li><a href="/wiki/Particle" title="Particle">Particle</a></li> <li><a href="/wiki/Wave" title="Wave">Wave</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Branches</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Applied_physics" title="Applied physics">Applied</a></li> <li><a href="/wiki/Experimental_physics" title="Experimental physics">Experimental</a></li> <li><a href="/wiki/Theoretical_physics" title="Theoretical physics">Theoretical</a></li> <li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical</a></li> <li><a href="/wiki/Philosophy_of_physics" title="Philosophy of physics">Philosophy of physics</a></li> <li><a class="mw-selflink selflink">Quantum mechanics</a> <ul><li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_computation" class="mw-redirect" title="Quantum computation">Quantum computation</a></li></ul></li> <li><a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></li> <li><a href="/wiki/Weak_interaction" title="Weak interaction">Weak interaction</a></li> <li><a href="/wiki/Electroweak_interaction" title="Electroweak interaction">Electroweak interaction</a></li> <li><a href="/wiki/Strong_interaction" title="Strong interaction">Strong interaction</a></li> <li><a href="/wiki/Atomic_physics" title="Atomic physics">Atomic</a></li> <li><a href="/wiki/Particle_physics" title="Particle physics">Particle</a></li> <li><a href="/wiki/Nuclear_physics" title="Nuclear physics">Nuclear</a></li> <li><a href="/wiki/Atomic,_molecular,_and_optical_physics" title="Atomic, molecular, and optical physics">Atomic, molecular, and optical</a></li> <li><a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">Condensed matter</a></li> <li><a href="/wiki/Statistical_physics" class="mw-redirect" title="Statistical physics">Statistical</a></li> <li><a href="/wiki/Complex_system" title="Complex system">Complex systems</a></li> <li><a href="/wiki/Non-linear_dynamics" class="mw-redirect" title="Non-linear dynamics">Non-linear dynamics</a></li> <li><a href="/wiki/Biophysics" title="Biophysics">Biophysics</a></li> <li><a href="/wiki/Neurophysics" title="Neurophysics">Neurophysics</a></li> <li><a href="/wiki/Plasma_physics" class="mw-redirect" title="Plasma physics">Plasma physics</a></li> <li><a href="/wiki/Special_relativity" title="Special relativity">Special relativity</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General relativity</a></li> <li><a href="/wiki/Astrophysics" title="Astrophysics">Astrophysics</a></li> <li><a href="/wiki/Cosmology" title="Cosmology">Cosmology</a></li> <li><a href="/wiki/Theories_of_gravitation" class="mw-redirect" title="Theories of gravitation">Theories of gravitation</a></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Theory_of_everything" title="Theory of everything">Theory of everything</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Wilhelm_R%C3%B6ntgen" title="Wilhelm Röntgen">Röntgen</a></li> <li><a href="/wiki/Edward_Witten" title="Edward Witten">Witten</a></li> <li><a href="/wiki/Henri_Becquerel" title="Henri Becquerel">Becquerel</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/Pierre_Curie" title="Pierre Curie">Curie</a></li> <li><a href="/wiki/Wilhelm_Wien" title="Wilhelm Wien">Wien</a></li> <li><a href="/wiki/Marie_Curie" title="Marie Curie">Skłodowska-Curie</a></li> <li><a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Sommerfeld</a></li> <li><a href="/wiki/Ernest_Rutherford" title="Ernest Rutherford">Rutherford</a></li> <li><a href="/wiki/Frederick_Soddy" title="Frederick Soddy">Soddy</a></li> <li><a href="/wiki/Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Onnes</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Frank_Wilczek" title="Frank Wilczek">Wilczek</a></li> <li><a href="/wiki/Max_Born" title="Max Born">Born</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Niels_Bohr" title="Niels Bohr">Bohr</a></li> <li><a href="/wiki/Hendrik_Kramers" class="mw-redirect" title="Hendrik Kramers">Kramers</a></li> <li><a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödinger</a></li> <li><a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">de Broglie</a></li> <li><a href="/wiki/Max_von_Laue" title="Max von Laue">Laue</a></li> <li><a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Bose</a></li> <li><a href="/wiki/Arthur_Compton" title="Arthur Compton">Compton</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li> <li><a href="/wiki/Ernest_Walton" title="Ernest Walton">Walton</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/Johannes_Diderik_van_der_Waals" title="Johannes Diderik van der Waals">van der Waals</a></li> <li><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li> <li><a href="/wiki/Freeman_Dyson" title="Freeman Dyson">Dyson</a></li> <li><a href="/wiki/Pieter_Zeeman" title="Pieter Zeeman">Zeeman</a></li> <li><a href="/wiki/Henry_Moseley" title="Henry Moseley">Moseley</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel</a></li> <li><a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner</a></li> <li><a href="/wiki/Stephen_Hawking" title="Stephen Hawking">Hawking</a></li> <li><a href="/wiki/Philip_Warren_Anderson" class="mw-redirect" title="Philip Warren Anderson">P. W. Anderson</a></li> <li><a href="/wiki/Georges_Lema%C3%AEtre" title="Georges Lemaître">Lemaître</a></li> <li><a href="/wiki/Sir_George_Paget_Thomson" class="mw-redirect" title="Sir George Paget Thomson">Thomson</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler</a></li> <li><a href="/wiki/Roger_Penrose" title="Roger Penrose">Penrose</a></li> <li><a href="/wiki/Robert_A._Millikan" class="mw-redirect" title="Robert A. Millikan">Millikan</a></li> <li><a href="/wiki/Yoichiro_Nambu" title="Yoichiro Nambu">Nambu</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a></li> <li><a href="/wiki/Peter_Higgs" title="Peter Higgs">Higgs</a></li> <li><a href="/wiki/Otto_Hahn" title="Otto Hahn">Hahn</a></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a></li> <li><a href="/wiki/Yang_Chen-Ning" title="Yang Chen-Ning">Yang</a></li> <li><a href="/wiki/Tsung-Dao_Lee" title="Tsung-Dao Lee">Lee</a></li> <li><a href="/wiki/Philipp_Lenard" title="Philipp Lenard">Lenard</a></li> <li><a href="/wiki/Abdus_Salam" title="Abdus Salam">Salam</a></li> <li><a href="/wiki/Gerard_%27t_Hooft" title="Gerard 't Hooft">'t Hooft</a></li> <li><a href="/wiki/Martinus_Veltman" class="mw-redirect" title="Martinus Veltman">Veltman</a></li> <li><a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">Bell</a></li> <li><a href="/wiki/Murray_Gell-Mann" title="Murray Gell-Mann">Gell-Mann</a></li> <li><a href="/wiki/J._J._Thomson" title="J. J. Thomson">J. J. Thomson</a></li> <li><a href="/wiki/C._V._Raman" title="C. V. Raman">Raman</a></li> <li><a href="/wiki/Lawrence_Bragg" title="Lawrence Bragg">Bragg</a></li> <li><a href="/wiki/John_Bardeen" title="John Bardeen">Bardeen</a></li> <li><a href="/wiki/William_Shockley" title="William Shockley">Shockley</a></li> <li><a href="/wiki/James_Chadwick" title="James Chadwick">Chadwick</a></li> <li><a href="/wiki/Ernest_O._Lawrence" class="mw-redirect" title="Ernest O. Lawrence">Lawrence</a></li> <li><a href="/wiki/Anton_Zeilinger" title="Anton Zeilinger">Zeilinger</a></li> <li><a href="/wiki/Samuel_Goudsmit" title="Samuel Goudsmit">Goudsmit</a></li> <li><a href="/wiki/George_Uhlenbeck" title="George Uhlenbeck">Uhlenbeck</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Categories</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><div class="CategoryTreeTag" data-ct-options="{"mode":0,"hideprefix":20,"showcount":false,"namespaces":false,"notranslations":false}"><div class="CategoryTreeSection"><div class="CategoryTreeItem"><span class="CategoryTreeEmptyBullet"></span> <bdi dir="ltr"><a href="/wiki/Category:Modern_physics" title="Category:Modern physics">Modern physics</a></bdi></div><div class="CategoryTreeChildren" style="display:none"></div></div></div></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Modern_physics" title="Template:Modern physics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Modern_physics" title="Template talk:Modern physics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Modern_physics" title="Special:EditPage/Template:Modern physics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Classical_mechanics">Classical mechanics</h3></div> <p>The rules of quantum mechanics assert that the state space of a system is a Hilbert space and that observables of the system are Hermitian operators acting on vectors in that space – although they do not tell us which Hilbert space or which operators. These can be chosen appropriately in order to obtain a quantitative description of a quantum system, a necessary step in making physical predictions. An important guide for making these choices is the <a href="/wiki/Correspondence_principle" title="Correspondence principle">correspondence principle</a>, a heuristic which states that the predictions of quantum mechanics reduce to those of <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> in the regime of large <a href="/wiki/Quantum_number" title="Quantum number">quantum numbers</a>.<sup id="cite_ref-Tipler_42-0" class="reference"><a href="#cite_note-Tipler-42"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> One can also start from an established classical model of a particular system, and then try to guess the underlying quantum model that would give rise to the classical model in the correspondence limit. This approach is known as <a href="/wiki/Canonical_quantization" title="Canonical quantization">quantization</a>.<sup id="cite_ref-Peres1993_43-0" class="reference"><a href="#cite_note-Peres1993-43"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 299">: 299 </span></sup><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p><p>When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was <a href="/wiki/Theory_of_relativity" title="Theory of relativity">non-relativistic</a> classical mechanics. For instance, the well-known model of the <a href="/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillator</a> uses an explicitly non-relativistic expression for the <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> of the oscillator, and is thus a quantum version of the <a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">classical harmonic oscillator</a>.<sup id="cite_ref-Zwiebach2022_7-7" class="reference"><a href="#cite_note-Zwiebach2022-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 234">: 234 </span></sup> </p><p>Complications arise with <a href="/wiki/Chaos_theory" title="Chaos theory">chaotic systems</a>, which do not have good quantum numbers, and <a href="/wiki/Quantum_chaos" title="Quantum chaos">quantum chaos</a> studies the relationship between classical and quantum descriptions in these systems.<sup id="cite_ref-Peres1993_43-1" class="reference"><a href="#cite_note-Peres1993-43"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 353">: 353 </span></sup> </p><p><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Quantum decoherence</a> is a mechanism through which quantum systems lose <a href="/wiki/Quantum_coherence" class="mw-redirect" title="Quantum coherence">coherence</a>, and thus become incapable of displaying many typically quantum effects: <a href="/wiki/Quantum_superposition" title="Quantum superposition">quantum superpositions</a> become simply probabilistic mixtures, and quantum entanglement becomes simply classical correlations.<sup id="cite_ref-Zwiebach2022_7-8" class="reference"><a href="#cite_note-Zwiebach2022-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 687–730">: 687–730 </span></sup> Quantum coherence is not typically evident at macroscopic scales, though at temperatures approaching <a href="/wiki/Absolute_zero" title="Absolute zero">absolute zero</a> quantum behavior may manifest macroscopically.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>note 3<span class="cite-bracket">]</span></a></sup> </p><p>Many macroscopic properties of a classical system are a direct consequence of the quantum behavior of its parts. For example, the stability of bulk matter (consisting of atoms and <a href="/wiki/Molecule" title="Molecule">molecules</a> which would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic properties of matter are all results of the interaction of <a href="/wiki/Electric_charge" title="Electric charge">electric charges</a> under the rules of quantum mechanics.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Special_relativity_and_electrodynamics">Special relativity and electrodynamics</h3></div> <p>Early attempts to merge quantum mechanics with <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> involved the replacement of the Schrödinger equation with a covariant equation such as the <a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon equation</a> or the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a>. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory, <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a>, provides a fully quantum description of the <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetic interaction</a>. Quantum electrodynamics is, along with <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, one of the most accurate physical theories ever devised.<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p><p>The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been used since the inception of quantum mechanics, is to treat <a href="/wiki/Electric_charge" title="Electric charge">charged</a> particles as quantum mechanical objects being acted on by a classical <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a>. For example, the elementary quantum model of the <a href="/wiki/Hydrogen_atom" title="Hydrogen atom">hydrogen atom</a> describes the <a href="/wiki/Electric_field" title="Electric field">electric field</a> of the hydrogen atom using a classical <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle -e^{2}/(4\pi \epsilon _{_{0}}r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle -e^{2}/(4\pi \epsilon _{_{0}}r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c4876353362da0a234af137590c6379d8f8c273" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.469ex; height:3.176ex;" alt="{\displaystyle \textstyle -e^{2}/(4\pi \epsilon _{_{0}}r)}"></span> <a href="/wiki/Electric_potential" title="Electric potential">Coulomb potential</a>.<sup id="cite_ref-Zwiebach2022_7-9" class="reference"><a href="#cite_note-Zwiebach2022-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 285">: 285 </span></sup> Likewise, in a <a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach experiment</a>, a charged particle is modeled as a quantum system, while the background magnetic field is described classically.<sup id="cite_ref-Peres1993_43-2" class="reference"><a href="#cite_note-Peres1993-43"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 26">: 26 </span></sup> This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by <a href="/wiki/Charged_particle" title="Charged particle">charged particles</a>. </p><p><a href="/wiki/Field_(physics)" title="Field (physics)">Quantum field</a> theories for the <a href="/wiki/Strong_nuclear_force" class="mw-redirect" title="Strong nuclear force">strong nuclear force</a> and the <a href="/wiki/Weak_nuclear_force" class="mw-redirect" title="Weak nuclear force">weak nuclear force</a> have also been developed. The quantum field theory of the strong nuclear force is called <a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">quantum chromodynamics</a>, and describes the interactions of subnuclear particles such as <a href="/wiki/Quark" title="Quark">quarks</a> and <a href="/wiki/Gluon" title="Gluon">gluons</a>. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory (known as <a href="/wiki/Electroweak_theory" class="mw-redirect" title="Electroweak theory">electroweak theory</a>), by the physicists <a href="/wiki/Abdus_Salam" title="Abdus Salam">Abdus Salam</a>, <a href="/wiki/Sheldon_Glashow" title="Sheldon Glashow">Sheldon Glashow</a> and <a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Steven Weinberg</a>.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Relation_to_general_relativity">Relation to general relativity</h3></div> <p>Even though the predictions of both quantum theory and general relativity have been supported by rigorous and repeated <a href="/wiki/Empirical_evidence" title="Empirical evidence">empirical evidence</a>, their abstract formalisms contradict each other and they have proven extremely difficult to incorporate into one consistent, cohesive model. Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those particular applications. However, the lack of a correct theory of <a href="/wiki/Quantum_gravity" title="Quantum gravity">quantum gravity</a> is an important issue in <a href="/wiki/Physical_cosmology" title="Physical cosmology">physical cosmology</a> and the search by physicists for an elegant "<a href="/wiki/Theory_of_everything" title="Theory of everything">Theory of Everything</a>" (TOE). Consequently, resolving the inconsistencies between both theories has been a major goal of 20th- and 21st-century physics. This TOE would combine not only the models of subatomic physics but also derive the four fundamental forces of nature from a single force or phenomenon.<sup id="cite_ref-NYT-20221010_50-0" class="reference"><a href="#cite_note-NYT-20221010-50"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:String_Vibrations.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/String_Vibrations.gif/180px-String_Vibrations.gif" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/String_Vibrations.gif/270px-String_Vibrations.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/String_Vibrations.gif/360px-String_Vibrations.gif 2x" data-file-width="500" data-file-height="500" /></a><figcaption>String vibrations of particles in the quantum world of particles.</figcaption></figure> <p>One proposal for doing so is <a href="/wiki/String_theory" title="String theory">string theory</a>, which posits that the <a href="/wiki/Point_particle" title="Point particle">point-like particles</a> of <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a> are replaced by <a href="/wiki/Dimension_(mathematics_and_physics)" class="mw-redirect" title="Dimension (mathematics and physics)">one-dimensional</a> objects called <a href="/wiki/String_(physics)" title="String (physics)">strings</a>. String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its <a href="/wiki/Mass" title="Mass">mass</a>, <a href="/wiki/Charge_(physics)" title="Charge (physics)">charge</a>, and other properties determined by the <a href="/wiki/Vibration" title="Vibration">vibrational</a> state of the string. In string theory, one of the many vibrational states of the string corresponds to the <a href="/wiki/Graviton" title="Graviton">graviton</a>, a quantum mechanical particle that carries gravitational force.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> </p><p>Another popular theory is <a href="/wiki/Loop_quantum_gravity" title="Loop quantum gravity">loop quantum gravity</a> (LQG), which describes quantum properties of gravity and is thus a theory of <a href="/wiki/Quantum_spacetime" title="Quantum spacetime">quantum spacetime</a>. LQG is an attempt to merge and adapt standard quantum mechanics and standard general relativity. This theory describes space as an extremely fine fabric "woven" of finite loops called <a href="/wiki/Spin_network" title="Spin network">spin networks</a>. The evolution of a spin network over time is called a <a href="/wiki/Spin_foam" title="Spin foam">spin foam</a>. The characteristic length scale of a spin foam is the <a href="/wiki/Planck_length" class="mw-redirect" title="Planck length">Planck length</a>, approximately 1.616×10<sup>−35</sup> m, and so lengths shorter than the Planck length are not physically meaningful in LQG.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Philosophical_implications">Philosophical implications</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations of quantum mechanics</a></div> <style data-mw-deduplicate="TemplateStyles:r1233989161">.mw-parser-output .unsolved{margin:0.5em 0 1em 1em;border:#ccc solid;padding:0.35em 0.35em 0.35em 2.2em;background-color:var(--background-color-interactive-subtle);background-image:url("https://upload.wikimedia.org/wikipedia/commons/2/26/Question%2C_Web_Fundamentals.svg");background-position:top 50%left 0.35em;background-size:1.5em;background-repeat:no-repeat}@media(min-width:720px){.mw-parser-output .unsolved{clear:right;float:right;max-width:25%}}.mw-parser-output .unsolved-label{font-weight:bold}.mw-parser-output .unsolved-body{margin:0.35em;font-style:italic}.mw-parser-output .unsolved-more{font-size:smaller}</style> <div role="note" aria-labelledby="unsolved-label-physics" class="unsolved"> <div><span class="unsolved-label" id="unsolved-label-physics">Unsolved problem in physics</span>:</div> <div class="unsolved-body">Is there a preferred interpretation of quantum mechanics? How does the quantum description of reality, which includes elements such as the "<a href="/wiki/Superposition_principle" title="Superposition principle">superposition</a> of states" and "<a href="/wiki/Wave_function_collapse" title="Wave function collapse">wave function collapse</a>", give rise to the reality we perceive?</div> <div class="unsolved-more"><a href="/wiki/List_of_unsolved_problems_in_physics" title="List of unsolved problems in physics">(more unsolved problems in physics)</a></div> </div> <p>Since its inception, the many counter-intuitive aspects and results of quantum mechanics have provoked strong <a href="/wiki/Philosophy" title="Philosophy">philosophical</a> debates and many <a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">interpretations</a>. The arguments centre on the probabilistic nature of quantum mechanics, the difficulties with <a href="/wiki/Wavefunction_collapse" class="mw-redirect" title="Wavefunction collapse">wavefunction collapse</a> and the related <a href="/wiki/Measurement_problem" title="Measurement problem">measurement problem</a>, and <a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">quantum nonlocality</a>. Perhaps the only consensus that exists about these issues is that there is no consensus. <a href="/wiki/Richard_Feynman" title="Richard Feynman">Richard Feynman</a> once said, "I think I can safely say that nobody understands quantum mechanics."<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> According to <a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Steven Weinberg</a>, "There is now in my opinion no entirely satisfactory interpretation of quantum mechanics."<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> </p><p>The views of <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a>, Werner Heisenberg and other physicists are often grouped together as the "<a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen interpretation</a>".<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> According to these views, the probabilistic nature of quantum mechanics is not a <i>temporary</i> feature which will eventually be replaced by a deterministic theory, but is instead a <i>final</i> renunciation of the classical idea of "causality". Bohr in particular emphasized that any well-defined application of the quantum mechanical formalism must always make reference to the experimental arrangement, due to the <a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)">complementary</a> nature of evidence obtained under different experimental situations. Copenhagen-type interpretations were adopted by Nobel laureates in quantum physics, including Bohr,<sup id="cite_ref-BohrComo_58-0" class="reference"><a href="#cite_note-BohrComo-58"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> Heisenberg,<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> Schrödinger,<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> Feynman,<sup id="cite_ref-Feynman_2-3" class="reference"><a href="#cite_note-Feynman-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and Zeilinger<sup id="cite_ref-MaKoflerZeilinger_61-0" class="reference"><a href="#cite_note-MaKoflerZeilinger-61"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> as well as 21st-century researchers in quantum foundations.<sup id="cite_ref-:25_62-0" class="reference"><a href="#cite_note-:25-62"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>, himself one of the founders of <a href="/wiki/Old_quantum_theory" title="Old quantum theory">quantum theory</a>, was troubled by its apparent failure to respect some cherished metaphysical principles, such as <a href="/wiki/Determinism" title="Determinism">determinism</a> and <a href="/wiki/Principle_of_locality" title="Principle of locality">locality</a>. Einstein's long-running exchanges with Bohr about the meaning and status of quantum mechanics are now known as the <a href="/wiki/Bohr%E2%80%93Einstein_debates" title="Bohr–Einstein debates">Bohr–Einstein debates</a>. Einstein believed that underlying quantum mechanics must be a theory that explicitly forbids <a href="/wiki/Action_at_a_distance" title="Action at a distance">action at a distance</a>. He argued that quantum mechanics was incomplete, a theory that was valid but not fundamental, analogous to how <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a> is valid, but the fundamental theory behind it is <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>. In 1935, Einstein and his collaborators <a href="/wiki/Boris_Podolsky" title="Boris Podolsky">Boris Podolsky</a> and <a href="/wiki/Nathan_Rosen" title="Nathan Rosen">Nathan Rosen</a> published an argument that the principle of locality implies the incompleteness of quantum mechanics, a <a href="/wiki/Thought_experiment" title="Thought experiment">thought experiment</a> later termed the <a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">Einstein–Podolsky–Rosen paradox</a>.<sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>note 4<span class="cite-bracket">]</span></a></sup> In 1964, <a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">John Bell</a> showed that EPR's principle of locality, together with determinism, was actually incompatible with quantum mechanics: they implied constraints on the correlations produced by distance systems, now known as <a href="/wiki/Bell_inequalities" class="mw-redirect" title="Bell inequalities">Bell inequalities</a>, that can be violated by entangled particles.<sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> Since then <a href="/wiki/Bell_test" title="Bell test">several experiments</a> have been performed to obtain these correlations, with the result that they do in fact violate Bell inequalities, and thus falsify the conjunction of locality with determinism.<sup id="cite_ref-wiseman15_16-1" class="reference"><a href="#cite_note-wiseman15-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-wolchover17_17-1" class="reference"><a href="#cite_note-wolchover17-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Bohmian_mechanics" class="mw-redirect" title="Bohmian mechanics">Bohmian mechanics</a> shows that it is possible to reformulate quantum mechanics to make it deterministic, at the price of making it explicitly nonlocal. It attributes not only a wave function to a physical system, but in addition a real position, that evolves deterministically under a nonlocal guiding equation. The evolution of a physical system is given at all times by the Schrödinger equation together with the guiding equation; there is never a collapse of the wave function. This solves the measurement problem.<sup id="cite_ref-69" class="reference"><a href="#cite_note-69"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Schroedingers_cat_film.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Schroedingers_cat_film.svg/220px-Schroedingers_cat_film.svg.png" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Schroedingers_cat_film.svg/330px-Schroedingers_cat_film.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Schroedingers_cat_film.svg/440px-Schroedingers_cat_film.svg.png 2x" data-file-width="600" data-file-height="400" /></a><figcaption><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a> in the many-worlds interpretation of quantum mechanics, where a branching of the universe occurs through a superposition of two quantum mechanical states.</figcaption></figure> <p>Everett's <a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">many-worlds interpretation</a>, formulated in 1956, holds that <i>all</i> the possibilities described by quantum theory <i>simultaneously</i> occur in a multiverse composed of mostly independent parallel universes.<sup id="cite_ref-70" class="reference"><a href="#cite_note-70"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> This is a consequence of removing the axiom of the collapse of the wave packet. All possible states of the measured system and the measuring apparatus, together with the observer, are present in a real physical quantum superposition. While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we do not observe the multiverse as a whole, but only one parallel universe at a time. Exactly how this is supposed to work has been the subject of much debate. Several attempts have been made to make sense of this and derive the Born rule,<sup id="cite_ref-dewitt73_71-0" class="reference"><a href="#cite_note-dewitt73-71"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-wallace2003_72-0" class="reference"><a href="#cite_note-wallace2003-72"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> with no consensus on whether they have been successful.<sup id="cite_ref-ballentine1973_73-0" class="reference"><a href="#cite_note-ballentine1973-73"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-74" class="reference"><a href="#cite_note-74"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-kent2009_75-0" class="reference"><a href="#cite_note-kent2009-75"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational quantum mechanics</a> appeared in the late 1990s as a modern derivative of Copenhagen-type ideas,<sup id="cite_ref-76" class="reference"><a href="#cite_note-76"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/QBism" class="mw-redirect" title="QBism">QBism</a> was developed some years later.<sup id="cite_ref-:23_77-0" class="reference"><a href="#cite_note-:23-77"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History of quantum mechanics</a> and <a href="/wiki/Atomic_theory" class="mw-redirect" title="Atomic theory">Atomic theory</a></div> <p>Quantum mechanics was developed in the early decades of the 20th century, driven by the need to explain phenomena that, in some cases, had been observed in earlier times. Scientific inquiry into the wave nature of light began in the 17th and 18th centuries, when scientists such as <a href="/wiki/Robert_Hooke" title="Robert Hooke">Robert Hooke</a>, <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a> and <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> proposed a wave theory of light based on experimental observations.<sup id="cite_ref-Born_&_Wolf_78-0" class="reference"><a href="#cite_note-Born_&_Wolf-78"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup> In 1803 English <a href="/wiki/Polymath" title="Polymath">polymath</a> <a href="/wiki/Thomas_Young_(scientist)" title="Thomas Young (scientist)">Thomas Young</a> described the famous <a href="/wiki/Young%27s_interference_experiment" title="Young's interference experiment">double-slit experiment</a>.<sup id="cite_ref-79" class="reference"><a href="#cite_note-79"><span class="cite-bracket">[</span>75<span class="cite-bracket">]</span></a></sup> This experiment played a major role in the general acceptance of the <a href="/wiki/Wave_theory_of_light" class="mw-redirect" title="Wave theory of light">wave theory of light</a>. </p><p>During the early 19th century, <a href="/wiki/Chemistry" title="Chemistry">chemical</a> research by <a href="/wiki/John_Dalton" title="John Dalton">John Dalton</a> and <a href="/wiki/Amedeo_Avogadro" title="Amedeo Avogadro">Amedeo Avogadro</a> lent weight to the <a href="/wiki/Atomic_theory" class="mw-redirect" title="Atomic theory">atomic theory</a> of matter, an idea that <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a>, <a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</a> and others built upon to establish the <a href="/wiki/Kinetic_theory_of_gases" title="Kinetic theory of gases">kinetic theory of gases</a>. The successes of kinetic theory gave further credence to the idea that matter is composed of atoms, yet the theory also had shortcomings that would only be resolved by the development of quantum mechanics.<sup id="cite_ref-Feynman-kinetic-theory_80-0" class="reference"><a href="#cite_note-Feynman-kinetic-theory-80"><span class="cite-bracket">[</span>76<span class="cite-bracket">]</span></a></sup> While the early conception of atoms from <a href="/wiki/Ancient_Greek_philosophy" title="Ancient Greek philosophy">Greek philosophy</a> had been that they were indivisible units – the word "atom" deriving from the <a href="/wiki/Greek_language" title="Greek language">Greek</a> for "uncuttable" –  the 19th century saw the formulation of hypotheses about subatomic structure. One important discovery in that regard was <a href="/wiki/Michael_Faraday" title="Michael Faraday">Michael Faraday</a>'s 1838 observation of a glow caused by an electrical discharge inside a glass tube containing gas at low pressure. <a href="/wiki/Julius_Pl%C3%BCcker" title="Julius Plücker">Julius Plücker</a>, <a href="/wiki/Johann_Wilhelm_Hittorf" title="Johann Wilhelm Hittorf">Johann Wilhelm Hittorf</a> and <a href="/wiki/Eugen_Goldstein" title="Eugen Goldstein">Eugen Goldstein</a> carried on and improved upon Faraday's work, leading to the identification of <a href="/wiki/Cathode_rays" class="mw-redirect" title="Cathode rays">cathode rays</a>, which <a href="/wiki/J._J._Thomson" title="J. J. Thomson">J. J. Thomson</a> found to consist of subatomic particles that would be called electrons.<sup id="cite_ref-81" class="reference"><a href="#cite_note-81"><span class="cite-bracket">[</span>77<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-82" class="reference"><a href="#cite_note-82"><span class="cite-bracket">[</span>78<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Max_Planck_(1858-1947).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Max_Planck_%281858-1947%29.jpg/170px-Max_Planck_%281858-1947%29.jpg" decoding="async" width="170" height="239" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Max_Planck_%281858-1947%29.jpg/255px-Max_Planck_%281858-1947%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/Max_Planck_%281858-1947%29.jpg/340px-Max_Planck_%281858-1947%29.jpg 2x" data-file-width="1810" data-file-height="2542" /></a><figcaption><a href="/wiki/Max_Planck" title="Max Planck">Max Planck</a> is considered the father of the quantum theory.</figcaption></figure> <p>The <a href="/wiki/Black-body_radiation" title="Black-body radiation">black-body radiation</a> problem was discovered by <a href="/wiki/Gustav_Kirchhoff" title="Gustav Kirchhoff">Gustav Kirchhoff</a> in 1859. In 1900, Max Planck proposed the hypothesis that energy is radiated and absorbed in discrete "quanta" (or energy packets), yielding a calculation that precisely matched the observed patterns of black-body radiation.<sup id="cite_ref-83" class="reference"><a href="#cite_note-83"><span class="cite-bracket">[</span>79<span class="cite-bracket">]</span></a></sup> The word <i>quantum</i> derives from the <a href="/wiki/Latin_language" class="mw-redirect" title="Latin language">Latin</a>, meaning "how great" or "how much".<sup id="cite_ref-84" class="reference"><a href="#cite_note-84"><span class="cite-bracket">[</span>80<span class="cite-bracket">]</span></a></sup> According to Planck, quantities of energy could be thought of as divided into "elements" whose size (<i>E</i>) would be proportional to their <a href="/wiki/Frequency" title="Frequency">frequency</a> (<i>ν</i>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=h\nu \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>h</mi> <mi>ν</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=h\nu \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24767663bb931daecf1eb628be4fea46f32e3622" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.026ex; height:2.176ex;" alt="{\displaystyle E=h\nu \ }"></span>,</dd></dl> <p>where <i>h</i> is the <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a>. Planck cautiously insisted that this was only an aspect of the processes of absorption and emission of radiation and was not the <i>physical reality</i> of the radiation.<sup id="cite_ref-85" class="reference"><a href="#cite_note-85"><span class="cite-bracket">[</span>81<span class="cite-bracket">]</span></a></sup> In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery.<sup id="cite_ref-Kragh_86-0" class="reference"><a href="#cite_note-Kragh-86"><span class="cite-bracket">[</span>82<span class="cite-bracket">]</span></a></sup> However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis <a href="/wiki/Local_realism" class="mw-redirect" title="Local realism">realistically</a> and used it to explain the <a href="/wiki/Photoelectric_effect" title="Photoelectric effect">photoelectric effect</a>, in which shining light on certain materials can eject electrons from the material. Niels Bohr then developed Planck's ideas about radiation into a <a href="/wiki/Bohr_model" title="Bohr model">model of the hydrogen atom</a> that successfully predicted the <a href="/wiki/Spectral_line" title="Spectral line">spectral lines</a> of hydrogen.<sup id="cite_ref-87" class="reference"><a href="#cite_note-87"><span class="cite-bracket">[</span>83<span class="cite-bracket">]</span></a></sup> Einstein further developed this idea to show that an <a href="/wiki/Electromagnetic_wave" class="mw-redirect" title="Electromagnetic wave">electromagnetic wave</a> such as light could also be described as a particle (later called the photon), with a discrete amount of energy that depends on its frequency.<sup id="cite_ref-88" class="reference"><a href="#cite_note-88"><span class="cite-bracket">[</span>84<span class="cite-bracket">]</span></a></sup> In his paper "On the Quantum Theory of Radiation", Einstein expanded on the interaction between energy and matter to explain the absorption and emission of energy by atoms. Although overshadowed at the time by his general theory of relativity, this paper articulated the mechanism underlying the stimulated emission of radiation,<sup id="cite_ref-89" class="reference"><a href="#cite_note-89"><span class="cite-bracket">[</span>85<span class="cite-bracket">]</span></a></sup> which became the basis of the laser.<sup id="cite_ref-90" class="reference"><a href="#cite_note-90"><span class="cite-bracket">[</span>86<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Solvay_conference_1927.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Solvay_conference_1927.jpg/310px-Solvay_conference_1927.jpg" decoding="async" width="310" height="224" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Solvay_conference_1927.jpg/465px-Solvay_conference_1927.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Solvay_conference_1927.jpg/620px-Solvay_conference_1927.jpg 2x" data-file-width="3000" data-file-height="2171" /></a><figcaption>The 1927 <a href="/wiki/Solvay_Conference" title="Solvay Conference">Solvay Conference</a> in <a href="/wiki/Brussels" title="Brussels">Brussels</a> was the fifth world physics conference.</figcaption></figure> <p>This phase is known as the <a href="/wiki/Old_quantum_theory" title="Old quantum theory">old quantum theory</a>. Never complete or self-consistent, the old quantum theory was rather a set of <a href="/wiki/Heuristic" title="Heuristic">heuristic</a> corrections to classical mechanics.<sup id="cite_ref-terHaar_91-0" class="reference"><a href="#cite_note-terHaar-91"><span class="cite-bracket">[</span>87<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-92" class="reference"><a href="#cite_note-92"><span class="cite-bracket">[</span>88<span class="cite-bracket">]</span></a></sup> The theory is now understood as a <a href="/wiki/WKB_approximation#Application_to_the_Schr.C3.B6dinger_equation" title="WKB approximation">semi-classical approximation</a> to modern quantum mechanics.<sup id="cite_ref-93" class="reference"><a href="#cite_note-93"><span class="cite-bracket">[</span>89<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-94" class="reference"><a href="#cite_note-94"><span class="cite-bracket">[</span>90<span class="cite-bracket">]</span></a></sup> Notable results from this period include, in addition to the work of Planck, Einstein and Bohr mentioned above, Einstein and <a href="/wiki/Peter_Debye" title="Peter Debye">Peter Debye</a>'s work on the <a href="/wiki/Specific_heat" class="mw-redirect" title="Specific heat">specific heat</a> of solids, Bohr and <a href="/wiki/Hendrika_Johanna_van_Leeuwen" title="Hendrika Johanna van Leeuwen">Hendrika Johanna van Leeuwen</a>'s <a href="/wiki/Bohr%E2%80%93Van_Leeuwen_theorem" title="Bohr–Van Leeuwen theorem">proof</a> that classical physics cannot account for <a href="/wiki/Diamagnetism" title="Diamagnetism">diamagnetism</a>, and <a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Arnold Sommerfeld</a>'s extension of the Bohr model to include special-relativistic effects.<sup id="cite_ref-terHaar_91-1" class="reference"><a href="#cite_note-terHaar-91"><span class="cite-bracket">[</span>87<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Aharoni_95-0" class="reference"><a href="#cite_note-Aharoni-95"><span class="cite-bracket">[</span>91<span class="cite-bracket">]</span></a></sup> </p><p>In the mid-1920s quantum mechanics was developed to become the standard formulation for atomic physics. In 1923, the French physicist <a href="/wiki/Louis-Victor_de_Broglie" class="mw-redirect" title="Louis-Victor de Broglie">Louis de Broglie</a> put forward his theory of matter waves by stating that particles can exhibit wave characteristics and vice versa. Building on de Broglie's approach, modern quantum mechanics was born in 1925, when the German physicists Werner Heisenberg, Max Born, and <a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Pascual Jordan</a><sup id="cite_ref-Edwards79_96-0" class="reference"><a href="#cite_note-Edwards79-96"><span class="cite-bracket">[</span>92<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Edwards81_97-0" class="reference"><a href="#cite_note-Edwards81-97"><span class="cite-bracket">[</span>93<span class="cite-bracket">]</span></a></sup> developed <a href="/wiki/Matrix_mechanics" title="Matrix mechanics">matrix mechanics</a> and the Austrian physicist Erwin Schrödinger invented <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">wave mechanics</a>. Born introduced the probabilistic interpretation of Schrödinger's wave function in July 1926.<sup id="cite_ref-98" class="reference"><a href="#cite_note-98"><span class="cite-bracket">[</span>94<span class="cite-bracket">]</span></a></sup> Thus, the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth <a href="/wiki/Solvay_Conference" title="Solvay Conference">Solvay Conference</a> in 1927.<sup id="cite_ref-pais1997_99-0" class="reference"><a href="#cite_note-pais1997-99"><span class="cite-bracket">[</span>95<span class="cite-bracket">]</span></a></sup> </p><p>By 1930, quantum mechanics had been further unified and formalized by <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>, Paul Dirac and <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a><sup id="cite_ref-100" class="reference"><a href="#cite_note-100"><span class="cite-bracket">[</span>96<span class="cite-bracket">]</span></a></sup> with greater emphasis on <a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">measurement</a>, the statistical nature of our knowledge of reality, and <a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">philosophical speculation about the 'observer'</a>. It has since permeated many disciplines, including quantum chemistry, <a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">quantum electronics</a>, <a href="/wiki/Quantum_optics" title="Quantum optics">quantum optics</a>, and <a href="/wiki/Quantum_information_science" title="Quantum information science">quantum information science</a>. It also provides a useful framework for many features of the modern <a href="/wiki/Periodic_table" title="Periodic table">periodic table of elements</a>, and describes the behaviors of <a href="/wiki/Atoms" class="mw-redirect" title="Atoms">atoms</a> during <a href="/wiki/Chemical_bond" title="Chemical bond">chemical bonding</a> and the flow of electrons in computer <a href="/wiki/Semiconductor" title="Semiconductor">semiconductors</a>, and therefore plays a crucial role in many modern technologies. While quantum mechanics was constructed to describe the world of the very small, it is also needed to explain some <a href="/wiki/Macroscopic" class="mw-redirect" title="Macroscopic">macroscopic</a> phenomena such as <a href="/wiki/Superconductivity" title="Superconductivity">superconductors</a><sup id="cite_ref-feynman2015_101-0" class="reference"><a href="#cite_note-feynman2015-101"><span class="cite-bracket">[</span>97<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Superfluid" class="mw-redirect" title="Superfluid">superfluids</a>.<sup id="cite_ref-102" class="reference"><a href="#cite_note-102"><span class="cite-bracket">[</span>98<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 35em;"> <ul><li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li> <li><a href="/wiki/Einstein%27s_thought_experiments" title="Einstein's thought experiments">Einstein's thought experiments</a></li> <li><a href="/wiki/List_of_textbooks_on_classical_and_quantum_mechanics" class="mw-redirect" title="List of textbooks on classical and quantum mechanics">List of textbooks on classical and quantum mechanics</a></li> <li><a href="/wiki/Macroscopic_quantum_phenomena" title="Macroscopic quantum phenomena">Macroscopic quantum phenomena</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase-space formulation</a></li> <li><a href="/wiki/Regularization_(physics)" title="Regularization (physics)">Regularization (physics)</a></li> <li><a href="/wiki/Two-state_quantum_system" title="Two-state quantum system">Two-state quantum system</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Explanatory_notes">Explanatory notes</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text">A momentum eigenstate would be a perfectly monochromatic wave of infinite extent, which is not square-integrable. Likewise, a position eigenstate would be a <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta distribution</a>, not square-integrable and technically not a function at all. Consequently, neither can belong to the particle's Hilbert space. Physicists sometimes introduce fictitious "bases" for a Hilbert space comprising elements outside that space. These are invented for calculational convenience and do not represent physical states.<sup id="cite_ref-Cohen-Tannoudji_25-2" class="reference"><a href="#cite_note-Cohen-Tannoudji-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 100–105">: 100–105 </span></sup></span> </li> <li id="cite_note-feynmanIII-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-feynmanIII_39-0">^</a></b></span> <span class="reference-text">See, for example, <a href="/wiki/The_Feynman_Lectures_on_Physics" title="The Feynman Lectures on Physics">the Feynman Lectures on Physics</a> for some of the technological applications which use quantum mechanics, e.g., <a href="/wiki/Transistor" title="Transistor">transistors</a> (vol <b>III</b>, pp. 14–11 ff), <a href="/wiki/Integrated_circuit" title="Integrated circuit">integrated circuits</a>, which are follow-on technology in solid-state physics (vol <b>II</b>, pp. 8–6), and <a href="/wiki/Laser" title="Laser">lasers</a> (vol <b>III</b>, pp. 9–13).</span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text">See <i><a href="/wiki/Macroscopic_quantum_phenomena" title="Macroscopic quantum phenomena">Macroscopic quantum phenomena</a></i>, <i><a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensate</a></i>, and <i><a href="/wiki/Quantum_machine" title="Quantum machine">Quantum machine</a></i></span> </li> <li id="cite_note-67"><span class="mw-cite-backlink"><b><a href="#cite_ref-67">^</a></b></span> <span class="reference-text">The published form of the EPR argument was due to Podolsky, and Einstein himself was not satisfied with it. In his own publications and correspondence, Einstein used a different argument to insist that quantum mechanics is an incomplete theory.<sup id="cite_ref-spekkens_63-0" class="reference"><a href="#cite_note-spekkens-63"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-howard_64-0" class="reference"><a href="#cite_note-howard-64"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Born1926-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Born1926_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBorn1926" class="citation journal cs1"><a href="/wiki/Max_Born" title="Max Born">Born, M.</a> (1926). "Zur Quantenmechanik der Stoßvorgänge" [On the Quantum Mechanics of Collision Processes]. <i>Zeitschrift für Physik</i>. <b>37</b> (12): <span class="nowrap">863–</span>867. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1926ZPhy...37..863B">1926ZPhy...37..863B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01397477">10.1007/BF01397477</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1434-6001">1434-6001</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119896026">119896026</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zeitschrift+f%C3%BCr+Physik&rft.atitle=Zur+Quantenmechanik+der+Sto%C3%9Fvorg%C3%A4nge&rft.volume=37&rft.issue=12&rft.pages=%3Cspan+class%3D%22nowrap%22%3E863-%3C%2Fspan%3E867&rft.date=1926&rft_id=info%3Adoi%2F10.1007%2FBF01397477&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119896026%23id-name%3DS2CID&rft.issn=1434-6001&rft_id=info%3Abibcode%2F1926ZPhy...37..863B&rft.aulast=Born&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-Feynman-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Feynman_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Feynman_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Feynman_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Feynman_2-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynmanLeightonSands1964" class="citation book cs1">Feynman, Richard; Leighton, Robert; Sands, Matthew (1964). <a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/III_01.html"><i>The Feynman Lectures on Physics</i></a>. Vol. 3. California Institute of Technology. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-50064-6" title="Special:BookSources/978-0-201-50064-6"><bdi>978-0-201-50064-6</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">19 December</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Feynman+Lectures+on+Physics&rft.pub=California+Institute+of+Technology&rft.date=1964&rft.isbn=978-0-201-50064-6&rft.aulast=Feynman&rft.aufirst=Richard&rft.au=Leighton%2C+Robert&rft.au=Sands%2C+Matthew&rft_id=https%3A%2F%2Ffeynmanlectures.caltech.edu%2FIII_01.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJaeger2014" class="citation journal cs1">Jaeger, Gregg (September 2014). 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US: Prometheus Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-61614-281-0" title="Special:BookSources/978-1-61614-281-0"><bdi>978-1-61614-281-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Physics+for+Poets&rft.place=US&rft.pub=Prometheus+Books&rft.date=2011&rft.isbn=978-1-61614-281-0&rft.aulast=Lederman&rft.aufirst=Leon+M.&rft.au=Hill%2C+Christopher+T.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqY_yOwHg_WYC%26pg%3DPA102&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-Müller-Kirsten-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Müller-Kirsten_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMüller-Kirsten2006" class="citation book cs1">Müller-Kirsten, H. J. W. (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=p1_Z81Le58MC&pg=PA14"><i>Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral</i></a>. US: World Scientific. p. 14. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-256-691-1" title="Special:BookSources/978-981-256-691-1"><bdi>978-981-256-691-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Quantum+Mechanics%3A+Schr%C3%B6dinger+Equation+and+Path+Integral&rft.place=US&rft.pages=14&rft.pub=World+Scientific&rft.date=2006&rft.isbn=978-981-256-691-1&rft.aulast=M%C3%BCller-Kirsten&rft.aufirst=H.+J.+W.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dp1_Z81Le58MC%26pg%3DPA14&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-Plotnitsky-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Plotnitsky_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlotnitsky2012" class="citation book cs1">Plotnitsky, Arkady (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=dmdUp97S4AYC&pg=PA75"><i>Niels Bohr and Complementarity: An Introduction</i></a>. US: Springer. pp. <span class="nowrap">75–</span>76. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4614-4517-3" title="Special:BookSources/978-1-4614-4517-3"><bdi>978-1-4614-4517-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Niels+Bohr+and+Complementarity%3A+An+Introduction&rft.place=US&rft.pages=%3Cspan+class%3D%22nowrap%22%3E75-%3C%2Fspan%3E76&rft.pub=Springer&rft.date=2012&rft.isbn=978-1-4614-4517-3&rft.aulast=Plotnitsky&rft.aufirst=Arkady&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DdmdUp97S4AYC%26pg%3DPA75&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffiths1995" class="citation book cs1"><a href="/wiki/David_J._Griffiths" title="David J. 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Retrieved <span class="nowrap">2024-06-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Notre+Dame+News&rft.atitle=Developing+more+energy-efficient+transistors+through+quantum+tunneling&rft.date=2012-03-27&rft.aulast=Phifer&rft.aufirst=Arnold&rft_id=https%3A%2F%2Fnews.nd.edu%2Fnews%2Fdeveloping-more-energy-efficient-transistors-through-quantum-tunneling%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBub2019" class="citation book cs1"><a href="/wiki/Jeffrey_Bub" title="Jeffrey Bub">Bub, Jeffrey</a> (2019). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/qt-entangle/">"Quantum entanglement"</a>. In Zalta, Edward N. (ed.). <a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy"><i>Stanford Encyclopedia of Philosophy</i></a>. 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Wiseman">Wiseman, Howard</a> (October 2015). <a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnature15631">"Death by experiment for local realism"</a>. <i><a href="/wiki/Nature_(journal)" title="Nature (journal)">Nature</a></i>. <b>526</b> (7575): <span class="nowrap">649–</span>650. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnature15631">10.1038/nature15631</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0028-0836">0028-0836</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26503054">26503054</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Death+by+experiment+for+local+realism&rft.volume=526&rft.issue=7575&rft.pages=%3Cspan+class%3D%22nowrap%22%3E649-%3C%2Fspan%3E650&rft.date=2015-10&rft.issn=0028-0836&rft_id=info%3Apmid%2F26503054&rft_id=info%3Adoi%2F10.1038%2Fnature15631&rft.aulast=Wiseman&rft.aufirst=Howard&rft_id=https%3A%2F%2Fdoi.org%2F10.1038%252Fnature15631&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-wolchover17-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-wolchover17_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-wolchover17_17-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWolchover2017" class="citation web cs1"><a href="/wiki/Natalie_Wolchover" title="Natalie Wolchover">Wolchover, Natalie</a> (7 February 2017). <a rel="nofollow" class="external text" href="https://www.quantamagazine.org/20170207-bell-test-quantum-loophole/">"Experiment Reaffirms Quantum Weirdness"</a>. <i><a href="/wiki/Quanta_Magazine" title="Quanta Magazine">Quanta Magazine</a></i><span class="reference-accessdate">. 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Baez">Baez, John C.</a> (20 March 2020). <a rel="nofollow" class="external text" href="https://math.ucr.edu/home/baez/books.html">"How to Learn Math and Physics"</a>. <i>University of California, Riverside</i><span class="reference-accessdate">. Retrieved <span class="nowrap">19 December</span> 2020</span>. <q>there's no way to understand the interpretation of quantum mechanics without also being able to <i>solve quantum mechanics problems</i> – to understand the theory, you need to be able to use it (and vice versa)</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=University+of+California%2C+Riverside&rft.atitle=How+to+Learn+Math+and+Physics&rft.date=2020-03-20&rft.aulast=Baez&rft.aufirst=John+C.&rft_id=https%3A%2F%2Fmath.ucr.edu%2Fhome%2Fbaez%2Fbooks.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSagan1996" class="citation book cs1"><a href="/wiki/Carl_Sagan" title="Carl Sagan">Sagan, Carl</a> (1996). <a href="/wiki/The_Demon-Haunted_World" title="The Demon-Haunted World"><i>The Demon-Haunted World: Science as a Candle in the Dark</i></a>. Ballantine Books. p. 249. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-345-40946-9" title="Special:BookSources/0-345-40946-9"><bdi>0-345-40946-9</bdi></a>. <q><span class="cs1-kern-left"></span>"For most physics students, (the "mathematical underpinning" of quantum mechanics) might occupy them from, say, third grade to early graduate school – roughly 15 years. [...] The job of the popularizer of science, trying to get across some idea of quantum mechanics to a general audience that has not gone through these initiation rites, is daunting. Indeed, there are no successful popularizations of quantum mechanics in my opinion – partly for this reason.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Demon-Haunted+World%3A+Science+as+a+Candle+in+the+Dark&rft.pages=249&rft.pub=Ballantine+Books&rft.date=1996&rft.isbn=0-345-40946-9&rft.aulast=Sagan&rft.aufirst=Carl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-google215-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-google215_20-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreensteinZajonc2006" class="citation book cs1">Greenstein, George; Zajonc, Arthur (2006). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230102102134/https://books.google.com/books?id=5t0tm0FB1CsC&pg=PA215">"8 Measurement"</a>. <i>The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics</i> (2nd ed.). Jones and Bartlett. p. 215. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7637-2470-2" title="Special:BookSources/978-0-7637-2470-2"><bdi>978-0-7637-2470-2</bdi></a>. Archived from <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5t0tm0FB1CsC&pg=PA215">the original</a> on 2023-01-02.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=8+Measurement&rft.btitle=The+Quantum+Challenge%3A+Modern+Research+on+the+Foundations+of+Quantum+Mechanics&rft.pages=215&rft.edition=2nd&rft.pub=Jones+and+Bartlett&rft.date=2006&rft.isbn=978-0-7637-2470-2&rft.aulast=Greenstein&rft.aufirst=George&rft.au=Zajonc%2C+Arthur&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5t0tm0FB1CsC%26pg%3DPA215&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeinberg2010" class="citation book cs1">Weinberg, Steven (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OLrZkgPsZR0C"><i>Dreams Of A Final Theory: The Search for The Fundamental Laws of Nature</i></a>. Random House. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OLrZkgPsZR0C&pg=PT82">82</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4070-6396-6" title="Special:BookSources/978-1-4070-6396-6"><bdi>978-1-4070-6396-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dreams+Of+A+Final+Theory%3A+The+Search+for+The+Fundamental+Laws+of+Nature&rft.pages=82&rft.pub=Random+House&rft.date=2010&rft.isbn=978-1-4070-6396-6&rft.aulast=Weinberg&rft.aufirst=Steven&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOLrZkgPsZR0C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZhangDeng1993" class="citation journal cs1">Zhang, Ruiqin; Deng, Conghao (1993-01-01). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevA.47.71">"Exact solutions of the Schrödinger equation for some quantum-mechanical many-body systems"</a>. <i>Physical Review A</i>. <b>47</b> (1): <span class="nowrap">71–</span>77. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993PhRvA..47...71Z">1993PhRvA..47...71Z</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevA.47.71">10.1103/PhysRevA.47.71</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1050-2947">1050-2947</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9908895">9908895</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+A&rft.atitle=Exact+solutions+of+the+Schr%C3%B6dinger+equation+for+some+quantum-mechanical+many-body+systems&rft.volume=47&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E71-%3C%2Fspan%3E77&rft.date=1993-01-01&rft_id=info%3Adoi%2F10.1103%2FPhysRevA.47.71&rft.issn=1050-2947&rft_id=info%3Apmid%2F9908895&rft_id=info%3Abibcode%2F1993PhRvA..47...71Z&rft.aulast=Zhang&rft.aufirst=Ruiqin&rft.au=Deng%2C+Conghao&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRevA.47.71&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiDrummondSchuckOlevano2019" class="citation journal cs1">Li, Jing; Drummond, N. D.; Schuck, Peter; Olevano, Valerio (2019-04-01). <a rel="nofollow" class="external text" href="https://doi.org/10.21468%2FSciPostPhys.6.4.040">"Comparing many-body approaches against the helium atom exact solution"</a>. <i>SciPost Physics</i>. <b>6</b> (4): 040. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1801.09977">1801.09977</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2019ScPP....6...40L">2019ScPP....6...40L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.21468%2FSciPostPhys.6.4.040">10.21468/SciPostPhys.6.4.040</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2542-4653">2542-4653</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SciPost+Physics&rft.atitle=Comparing+many-body+approaches+against+the+helium+atom+exact+solution&rft.volume=6&rft.issue=4&rft.pages=040&rft.date=2019-04-01&rft_id=info%3Aarxiv%2F1801.09977&rft.issn=2542-4653&rft_id=info%3Adoi%2F10.21468%2FSciPostPhys.6.4.040&rft_id=info%3Abibcode%2F2019ScPP....6...40L&rft.aulast=Li&rft.aufirst=Jing&rft.au=Drummond%2C+N.+D.&rft.au=Schuck%2C+Peter&rft.au=Olevano%2C+Valerio&rft_id=https%3A%2F%2Fdoi.org%2F10.21468%252FSciPostPhys.6.4.040&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDrake2023" class="citation book cs1">Drake, Gordon W. F. (2023). "High Precision Calculations for Helium". In Drake, Gordon W. F. (ed.). <i>Springer Handbook of Atomic, Molecular, and Optical Physics</i>. Springer Handbooks. Cham: Springer International Publishing. pp. <span class="nowrap">199–</span>216. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-030-73893-8_12">10.1007/978-3-030-73893-8_12</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-73892-1" title="Special:BookSources/978-3-030-73892-1"><bdi>978-3-030-73892-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=High+Precision+Calculations+for+Helium&rft.btitle=Springer+Handbook+of+Atomic%2C+Molecular%2C+and+Optical+Physics&rft.place=Cham&rft.series=Springer+Handbooks&rft.pages=%3Cspan+class%3D%22nowrap%22%3E199-%3C%2Fspan%3E216&rft.pub=Springer+International+Publishing&rft.date=2023&rft_id=info%3Adoi%2F10.1007%2F978-3-030-73893-8_12&rft.isbn=978-3-030-73892-1&rft.aulast=Drake&rft.aufirst=Gordon+W.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-Cohen-Tannoudji-25"><span class="mw-cite-backlink">^ <a href="#cite_ref-Cohen-Tannoudji_25-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Cohen-Tannoudji_25-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Cohen-Tannoudji_25-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Cohen-Tannoudji_25-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohen-TannoudjiDiuLaloë2005" class="citation book cs1"><a href="/wiki/Claude_Cohen-Tannoudji" title="Claude Cohen-Tannoudji">Cohen-Tannoudji, Claude</a>; Diu, Bernard; Laloë, Franck (2005). <i>Quantum Mechanics</i>. Translated by Hemley, Susan Reid; Ostrowsky, Nicole; Ostrowsky, Dan. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-16433-X" title="Special:BookSources/0-471-16433-X"><bdi>0-471-16433-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics&rft.pub=John+Wiley+%26+Sons&rft.date=2005&rft.isbn=0-471-16433-X&rft.aulast=Cohen-Tannoudji&rft.aufirst=Claude&rft.au=Diu%2C+Bernard&rft.au=Lalo%C3%AB%2C+Franck&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-L&L-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-L&L_26-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandauLifschitz1977" class="citation book cs1"><a href="/wiki/Lev_Landau" title="Lev Landau">Landau, Lev D.</a>; <a href="/wiki/Evgeny_Lifshitz" title="Evgeny Lifshitz">Lifschitz, Evgeny M.</a> (1977). <a rel="nofollow" class="external text" href="https://archive.org/details/QuantumMechanics_104"><i>Quantum Mechanics: Non-Relativistic Theory</i></a>. Vol. 3 (3rd ed.). <a href="/wiki/Pergamon_Press" title="Pergamon Press">Pergamon Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-020940-1" title="Special:BookSources/978-0-08-020940-1"><bdi>978-0-08-020940-1</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/2284121">2284121</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics%3A+Non-Relativistic+Theory&rft.edition=3rd&rft.pub=Pergamon+Press&rft.date=1977&rft_id=info%3Aoclcnum%2F2284121&rft.isbn=978-0-08-020940-1&rft.aulast=Landau&rft.aufirst=Lev+D.&rft.au=Lifschitz%2C+Evgeny+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2FQuantumMechanics_104&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-ballentine1970-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-ballentine1970_27-0">^</a></b></span> <span class="reference-text">Section 3.2 of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBallentine1970" class="citation cs2">Ballentine, Leslie E. (1970), "The Statistical Interpretation of Quantum Mechanics", <i>Reviews of Modern Physics</i>, <b>42</b> (4): <span class="nowrap">358–</span>381, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1970RvMP...42..358B">1970RvMP...42..358B</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.42.358">10.1103/RevModPhys.42.358</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120024263">120024263</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=The+Statistical+Interpretation+of+Quantum+Mechanics&rft.volume=42&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E358-%3C%2Fspan%3E381&rft.date=1970&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120024263%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.42.358&rft_id=info%3Abibcode%2F1970RvMP...42..358B&rft.aulast=Ballentine&rft.aufirst=Leslie+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span>. This fact is experimentally well-known for example in quantum optics; see e.g. chap. 2 and Fig. 2.1 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeonhardt1997" class="citation cs2">Leonhardt, Ulf (1997), <a rel="nofollow" class="external text" href="https://archive.org/details/measuringquantum0000leon"><i>Measuring the Quantum State of Light</i></a>, Cambridge: Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-49730-2" title="Special:BookSources/0-521-49730-2"><bdi>0-521-49730-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Measuring+the+Quantum+State+of+Light&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=0-521-49730-2&rft.aulast=Leonhardt&rft.aufirst=Ulf&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmeasuringquantum0000leon&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span>.</span> </li> <li id="cite_note-:0-28"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_28-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_28-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_28-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNielsenChuang2010" class="citation book cs1"><a href="/wiki/Michael_Nielsen" title="Michael Nielsen">Nielsen, Michael A.</a>; <a href="/wiki/Isaac_Chuang" title="Isaac Chuang">Chuang, Isaac L.</a> (2010). <i><a href="/wiki/Quantum_Computation_and_Quantum_Information" title="Quantum Computation and Quantum Information">Quantum Computation and Quantum Information</a></i> (2nd ed.). Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-00217-3" title="Special:BookSources/978-1-107-00217-3"><bdi>978-1-107-00217-3</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/844974180">844974180</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Computation+and+Quantum+Information&rft.place=Cambridge&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2010&rft_id=info%3Aoclcnum%2F844974180&rft.isbn=978-1-107-00217-3&rft.aulast=Nielsen&rft.aufirst=Michael+A.&rft.au=Chuang%2C+Isaac+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-:1-29"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_29-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_29-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRieffelPolak2011" class="citation book cs1"><a href="/wiki/Eleanor_Rieffel" title="Eleanor Rieffel">Rieffel, Eleanor G.</a>; Polak, Wolfgang H. (2011). <a href="/wiki/Quantum_Computing:_A_Gentle_Introduction" title="Quantum Computing: A Gentle Introduction"><i>Quantum Computing: A Gentle Introduction</i></a>. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-01506-6" title="Special:BookSources/978-0-262-01506-6"><bdi>978-0-262-01506-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Computing%3A+A+Gentle+Introduction&rft.pub=MIT+Press&rft.date=2011&rft.isbn=978-0-262-01506-6&rft.aulast=Rieffel&rft.aufirst=Eleanor+G.&rft.au=Polak%2C+Wolfgang+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-wilde-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-wilde_30-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilde2017" class="citation book cs1">Wilde, Mark M. (2017). <i>Quantum Information Theory</i> (2nd ed.). Cambridge University Press. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1106.1445">1106.1445</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2F9781316809976.001">10.1017/9781316809976.001</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-17616-4" title="Special:BookSources/978-1-107-17616-4"><bdi>978-1-107-17616-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/973404322">973404322</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2515538">2515538</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Information+Theory&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2017&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2515538%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2F9781316809976.001&rft_id=info%3Aoclcnum%2F973404322&rft_id=info%3Aarxiv%2F1106.1445&rft.isbn=978-1-107-17616-4&rft.aulast=Wilde&rft.aufirst=Mark+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchlosshauer2019" class="citation journal cs1">Schlosshauer, Maximilian (October 2019). 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What do you get? Just maybe, a holographic cosmos"</a>. <i><a href="/wiki/The_New_York_Times" title="The New York Times">The New York Times</a></i><span class="reference-accessdate">. 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Phys</i>. <b>34</b> (3): <span class="nowrap">415–</span>438. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0303050">quant-ph/0303050</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003SHPMP..34..415W">2003SHPMP..34..415W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS1355-2198%2803%2900036-4">10.1016/S1355-2198(03)00036-4</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1921913">1921913</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Stud.+Hist.+Phil.+Mod.+Phys.&rft.atitle=Everettian+Rationality%3A+defending+Deutsch%27s+approach+to+probability+in+the+Everett+interpretation&rft.volume=34&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E415-%3C%2Fspan%3E438&rft.date=2003&rft_id=info%3Aarxiv%2Fquant-ph%2F0303050&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A1921913%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2FS1355-2198%2803%2900036-4&rft_id=info%3Abibcode%2F2003SHPMP..34..415W&rft.aulast=Wallace&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-ballentine1973-73"><span class="mw-cite-backlink"><b><a href="#cite_ref-ballentine1973_73-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBallentine1973" class="citation journal cs1">Ballentine, L. E. (1973). "Can the statistical postulate of quantum theory be derived? – A critique of the many-universes interpretation". <i>Foundations of Physics</i>. <b>3</b> (2): <span class="nowrap">229–</span>240. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1973FoPh....3..229B">1973FoPh....3..229B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00708440">10.1007/BF00708440</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121747282">121747282</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Foundations+of+Physics&rft.atitle=Can+the+statistical+postulate+of+quantum+theory+be+derived%3F+%E2%80%93+A+critique+of+the+many-universes+interpretation&rft.volume=3&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E229-%3C%2Fspan%3E240&rft.date=1973&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121747282%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF00708440&rft_id=info%3Abibcode%2F1973FoPh....3..229B&rft.aulast=Ballentine&rft.aufirst=L.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-74"><span class="mw-cite-backlink"><b><a href="#cite_ref-74">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandsman2008" class="citation book cs1">Landsman, N. 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Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-70622-9" title="Special:BookSources/978-3-540-70622-9"><bdi>978-3-540-70622-9</bdi></a>. <q>The conclusion seems to be that no generally accepted derivation of the Born rule has been given to date, but this does not imply that such a derivation is impossible in principle.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Born+rule+and+its+interpretation&rft.btitle=Compendium+of+Quantum+Physics&rft.pub=Springer&rft.date=2008&rft.isbn=978-3-540-70622-9&rft.aulast=Landsman&rft.aufirst=N.+P.&rft_id=http%3A%2F%2Fwww.math.ru.nl%2F~landsman%2FBorn.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-kent2009-75"><span class="mw-cite-backlink"><b><a href="#cite_ref-kent2009_75-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKent2010" class="citation book cs1"><a href="/wiki/Adrian_Kent" title="Adrian Kent">Kent, Adrian</a> (2010). 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Oxford University Press. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0905.0624">0905.0624</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009arXiv0905.0624K">2009arXiv0905.0624K</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=One+world+versus+many%3A+The+inadequacy+of+Everettian+accounts+of+evolution%2C+probability%2C+and+scientific+confirmation&rft.btitle=Many+Worlds%3F+Everett%2C+Quantum+Theory+and+Reality&rft.pub=Oxford+University+Press&rft.date=2010&rft_id=info%3Aarxiv%2F0905.0624&rft_id=info%3Abibcode%2F2009arXiv0905.0624K&rft.aulast=Kent&rft.aufirst=Adrian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-76"><span class="mw-cite-backlink"><b><a href="#cite_ref-76">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVan_Fraassen2010" class="citation journal cs1"><a href="/wiki/Bas_van_Fraassen" title="Bas van Fraassen">Van Fraassen, Bas C.</a> (April 2010). <a rel="nofollow" class="external text" href="http://link.springer.com/10.1007/s10701-009-9326-5">"Rovelli's World"</a>. <i><a href="/wiki/Foundations_of_Physics" title="Foundations of Physics">Foundations of Physics</a></i>. <b>40</b> (4): <span class="nowrap">390–</span>417. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010FoPh...40..390V">2010FoPh...40..390V</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10701-009-9326-5">10.1007/s10701-009-9326-5</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0015-9018">0015-9018</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17217776">17217776</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Foundations+of+Physics&rft.atitle=Rovelli%27s+World&rft.volume=40&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E390-%3C%2Fspan%3E417&rft.date=2010-04&rft_id=info%3Adoi%2F10.1007%2Fs10701-009-9326-5&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17217776%23id-name%3DS2CID&rft.issn=0015-9018&rft_id=info%3Abibcode%2F2010FoPh...40..390V&rft.aulast=Van+Fraassen&rft.aufirst=Bas+C.&rft_id=http%3A%2F%2Flink.springer.com%2F10.1007%2Fs10701-009-9326-5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-:23-77"><span class="mw-cite-backlink"><b><a href="#cite_ref-:23_77-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHealey2016" class="citation book cs1">Healey, Richard (2016). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/quantum-bayesian/">"Quantum-Bayesian and Pragmatist Views of Quantum Theory"</a>. In Zalta, Edward N. (ed.). <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>. Metaphysics Research Lab, Stanford University.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Quantum-Bayesian+and+Pragmatist+Views+of+Quantum+Theory&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.pub=Metaphysics+Research+Lab%2C+Stanford+University&rft.date=2016&rft.aulast=Healey&rft.aufirst=Richard&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fquantum-bayesian%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-Born_&_Wolf-78"><span class="mw-cite-backlink"><b><a href="#cite_ref-Born_&_Wolf_78-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBornWolf1999" class="citation book cs1"><a href="/wiki/Max_Born" title="Max Born">Born, Max</a>; <a href="/wiki/Emil_Wolf" title="Emil Wolf">Wolf, Emil</a> (1999). <a href="/wiki/Principles_of_Optics" title="Principles of Optics"><i>Principles of Optics</i></a>. 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Retrieved <span class="nowrap">30 September</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Feynman+Lectures+on+Physics&rft.pub=California+Institute+of+Technology&rft.date=1964&rft.isbn=978-0-201-50064-6&rft.aulast=Feynman&rft.aufirst=Richard&rft.au=Leighton%2C+Robert&rft.au=Sands%2C+Matthew&rft_id=https%3A%2F%2Ffeynmanlectures.caltech.edu%2FI_40.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-81"><span class="mw-cite-backlink"><b><a href="#cite_ref-81">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartin1986" class="citation cs2">Martin, Andre (1986), "Cathode Ray Tubes for Industrial and Military Applications", in Hawkes, Peter (ed.), <i>Advances in Electronics and Electron Physics, Volume 67</i>, Academic Press, p. 183, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-057733-3" title="Special:BookSources/978-0-08-057733-3"><bdi>978-0-08-057733-3</bdi></a>, <q>Evidence for the existence of "cathode-rays" was first found by Plücker and Hittorf ...</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Cathode+Ray+Tubes+for+Industrial+and+Military+Applications&rft.btitle=Advances+in+Electronics+and+Electron+Physics%2C+Volume+67&rft.pages=183&rft.pub=Academic+Press&rft.date=1986&rft.isbn=978-0-08-057733-3&rft.aulast=Martin&rft.aufirst=Andre&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-82"><span class="mw-cite-backlink"><b><a href="#cite_ref-82">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDahl1997" class="citation book cs1">Dahl, Per F. (1997). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xUzaWGocMdMC"><i>Flash of the Cathode Rays: A History of J J Thomson's Electron</i></a>. CRC Press. pp. <span class="nowrap">47–</span>57. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7503-0453-5" title="Special:BookSources/978-0-7503-0453-5"><bdi>978-0-7503-0453-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Flash+of+the+Cathode+Rays%3A+A+History+of+J+J+Thomson%27s+Electron&rft.pages=%3Cspan+class%3D%22nowrap%22%3E47-%3C%2Fspan%3E57&rft.pub=CRC+Press&rft.date=1997&rft.isbn=978-0-7503-0453-5&rft.aulast=Dahl&rft.aufirst=Per+F.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxUzaWGocMdMC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-83"><span class="mw-cite-backlink"><b><a href="#cite_ref-83">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMehraRechenberg1982" class="citation book cs1"><a href="/wiki/Jagdish_Mehra" title="Jagdish Mehra">Mehra, J.</a>; Rechenberg, H. (1982). <i>The Historical Development of Quantum Theory, Vol. 1: The Quantum Theory of Planck, Einstein, Bohr and Sommerfeld. Its Foundation and the Rise of Its Difficulties (1900–1925)</i>. New York: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90642-3" title="Special:BookSources/978-0-387-90642-3"><bdi>978-0-387-90642-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Historical+Development+of+Quantum+Theory%2C+Vol.+1%3A+The+Quantum+Theory+of+Planck%2C+Einstein%2C+Bohr+and+Sommerfeld.+Its+Foundation+and+the+Rise+of+Its+Difficulties+%281900%E2%80%931925%29&rft.place=New+York&rft.pub=Springer-Verlag&rft.date=1982&rft.isbn=978-0-387-90642-3&rft.aulast=Mehra&rft.aufirst=J.&rft.au=Rechenberg%2C+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-84"><span class="mw-cite-backlink"><b><a href="#cite_ref-84">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.merriam-webster.com/dictionary/quantum">"Quantum – Definition and More"</a>. Merriam-Webster Dictionary. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121026104456/http://www.merriam-webster.com/dictionary/quantum">Archived</a> from the original on Oct 26, 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">18 August</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Quantum+%E2%80%93+Definition+and+More&rft.pub=Merriam-Webster+Dictionary&rft_id=http%3A%2F%2Fwww.merriam-webster.com%2Fdictionary%2Fquantum&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-85"><span class="mw-cite-backlink"><b><a href="#cite_ref-85">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKuhn1978" class="citation book cs1"><a href="/wiki/Thomas_Samuel_Kuhn" class="mw-redirect" title="Thomas Samuel Kuhn">Kuhn, T. S.</a> (1978). <i>Black-body theory and the quantum discontinuity 1894–1912</i>. Oxford: Clarendon Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-502383-1" title="Special:BookSources/978-0-19-502383-1"><bdi>978-0-19-502383-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Black-body+theory+and+the+quantum+discontinuity+1894%E2%80%931912&rft.place=Oxford&rft.pub=Clarendon+Press&rft.date=1978&rft.isbn=978-0-19-502383-1&rft.aulast=Kuhn&rft.aufirst=T.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-Kragh-86"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kragh_86-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKragh2000" class="citation web cs1"><a href="/wiki/Helge_Kragh" title="Helge Kragh">Kragh, Helge</a> (1 December 2000). <a rel="nofollow" class="external text" href="https://physicsworld.com/a/max-planck-the-reluctant-revolutionary/">"Max Planck: the reluctant revolutionary"</a>. <i><a href="/wiki/Physics_World" title="Physics World">Physics World</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">12 December</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Physics+World&rft.atitle=Max+Planck%3A+the+reluctant+revolutionary&rft.date=2000-12-01&rft.aulast=Kragh&rft.aufirst=Helge&rft_id=https%3A%2F%2Fphysicsworld.com%2Fa%2Fmax-planck-the-reluctant-revolutionary%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-87"><span class="mw-cite-backlink"><b><a href="#cite_ref-87">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStachel2009" class="citation book cs1"><a href="/wiki/John_Stachel" title="John Stachel">Stachel, John</a> (2009). "Bohr and the Photon". <i>Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle</i>. The Western Ontario Series in Philosophy of Science. Vol. 73. Dordrecht: Springer. pp. <span class="nowrap">69–</span>83. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4020-9107-0_5">10.1007/978-1-4020-9107-0_5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-9106-3" title="Special:BookSources/978-1-4020-9106-3"><bdi>978-1-4020-9106-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bohr+and+the+Photon&rft.btitle=Quantum+Reality%2C+Relativistic+Causality%2C+and+Closing+the+Epistemic+Circle&rft.place=Dordrecht&rft.series=The+Western+Ontario+Series+in+Philosophy+of+Science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E69-%3C%2Fspan%3E83&rft.pub=Springer&rft.date=2009&rft_id=info%3Adoi%2F10.1007%2F978-1-4020-9107-0_5&rft.isbn=978-1-4020-9106-3&rft.aulast=Stachel&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-88"><span class="mw-cite-backlink"><b><a href="#cite_ref-88">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEinstein1905" class="citation journal cs1">Einstein, Albert (1905). <a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19053220607">"Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt"</a> [On a heuristic point of view concerning the production and transformation of light]. <i><a href="/wiki/Annalen_der_Physik" title="Annalen der Physik">Annalen der Physik</a></i>. <b>17</b> (6): <span class="nowrap">132–</span>148. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1905AnP...322..132E">1905AnP...322..132E</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19053220607">10.1002/andp.19053220607</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annalen+der+Physik&rft.atitle=%C3%9Cber+einen+die+Erzeugung+und+Verwandlung+des+Lichtes+betreffenden+heuristischen+Gesichtspunkt&rft.volume=17&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E132-%3C%2Fspan%3E148&rft.date=1905&rft_id=info%3Adoi%2F10.1002%2Fandp.19053220607&rft_id=info%3Abibcode%2F1905AnP...322..132E&rft.aulast=Einstein&rft.aufirst=Albert&rft_id=https%3A%2F%2Fdoi.org%2F10.1002%252Fandp.19053220607&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span> Reprinted in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStachel1989" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/John_Stachel" title="John Stachel">Stachel, John</a>, ed. (1989). <i>The Collected Papers of Albert Einstein</i> (in German). Vol. 2. Princeton University Press. pp. <span class="nowrap">149–</span>166.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Collected+Papers+of+Albert+Einstein&rft.pages=%3Cspan+class%3D%22nowrap%22%3E149-%3C%2Fspan%3E166&rft.pub=Princeton+University+Press&rft.date=1989&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span> See also "Einstein's early work on the quantum hypothesis", ibid. pp. 134–148.</span> </li> <li id="cite_note-89"><span class="mw-cite-backlink"><b><a href="#cite_ref-89">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEinstein1917" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein, Albert</a> (1917). "Zur Quantentheorie der Strahlung" [On the Quantum Theory of Radiation]. <i><a href="/wiki/Physikalische_Zeitschrift" title="Physikalische Zeitschrift">Physikalische Zeitschrift</a></i> (in German). <b>18</b>: <span class="nowrap">121–</span>128. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1917PhyZ...18..121E">1917PhyZ...18..121E</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physikalische+Zeitschrift&rft.atitle=Zur+Quantentheorie+der+Strahlung&rft.volume=18&rft.pages=%3Cspan+class%3D%22nowrap%22%3E121-%3C%2Fspan%3E128&rft.date=1917&rft_id=info%3Abibcode%2F1917PhyZ...18..121E&rft.aulast=Einstein&rft.aufirst=Albert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span> Translated in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEinstein1967" class="citation book cs1">Einstein, A. (1967). "On the Quantum Theory of Radiation". <i>The Old Quantum Theory</i>. Elsevier. pp. <span class="nowrap">167–</span>183. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fb978-0-08-012102-4.50018-8">10.1016/b978-0-08-012102-4.50018-8</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-012102-4" title="Special:BookSources/978-0-08-012102-4"><bdi>978-0-08-012102-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=On+the+Quantum+Theory+of+Radiation&rft.btitle=The+Old+Quantum+Theory&rft.pages=%3Cspan+class%3D%22nowrap%22%3E167-%3C%2Fspan%3E183&rft.pub=Elsevier&rft.date=1967&rft_id=info%3Adoi%2F10.1016%2Fb978-0-08-012102-4.50018-8&rft.isbn=978-0-08-012102-4&rft.aulast=Einstein&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-90"><span class="mw-cite-backlink"><b><a href="#cite_ref-90">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBall2017" class="citation web cs1"><a href="/wiki/Philip_Ball" title="Philip Ball">Ball, Philip</a> (2017-08-31). <a rel="nofollow" class="external text" href="https://physicsworld.com/a/a-century-ago-einstein-sparked-the-notion-of-the-laser/">"A century ago Einstein sparked the notion of the laser"</a>. <i><a href="/wiki/Physics_World" title="Physics World">Physics World</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-03-23</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Physics+World&rft.atitle=A+century+ago+Einstein+sparked+the+notion+of+the+laser&rft.date=2017-08-31&rft.aulast=Ball&rft.aufirst=Philip&rft_id=https%3A%2F%2Fphysicsworld.com%2Fa%2Fa-century-ago-einstein-sparked-the-notion-of-the-laser%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-terHaar-91"><span class="mw-cite-backlink">^ <a href="#cite_ref-terHaar_91-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-terHaar_91-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFter_Haar1967" class="citation book cs1">ter Haar, D. (1967). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/oldquantumtheory0000haar"><i>The Old Quantum Theory</i></a></span>. Pergamon Press. pp. <span class="nowrap">3–</span>75. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-012101-7" title="Special:BookSources/978-0-08-012101-7"><bdi>978-0-08-012101-7</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/66-29628">66-29628</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Old+Quantum+Theory&rft.pages=%3Cspan+class%3D%22nowrap%22%3E3-%3C%2Fspan%3E75&rft.pub=Pergamon+Press&rft.date=1967&rft_id=info%3Alccn%2F66-29628&rft.isbn=978-0-08-012101-7&rft.aulast=ter+Haar&rft.aufirst=D.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Foldquantumtheory0000haar&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-92"><span class="mw-cite-backlink"><b><a href="#cite_ref-92">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBokulichBokulich2020" class="citation encyclopaedia cs1">Bokulich, Alisa; Bokulich, Peter (2020-08-13). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/bohr-correspondence/">"Bohr's Correspondence Principle"</a>. In <a href="/wiki/Edward_N._Zalta" title="Edward N. Zalta">Zalta, Edward N.</a> (ed.). <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bohr%27s+Correspondence+Principle&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.date=2020-08-13&rft.aulast=Bokulich&rft.aufirst=Alisa&rft.au=Bokulich%2C+Peter&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fbohr-correspondence%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-93"><span class="mw-cite-backlink"><b><a href="#cite_ref-93">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Semi-classical_approximation">"Semi-classical approximation"</a>. <i>Encyclopedia of Mathematics</i><span class="reference-accessdate">. Retrieved <span class="nowrap">1 February</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Semi-classical+approximation&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DSemi-classical_approximation&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-94"><span class="mw-cite-backlink"><b><a href="#cite_ref-94">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSakuraiNapolitano2014" class="citation book cs1"><a href="/wiki/J._J._Sakurai" title="J. J. Sakurai">Sakurai, J. J.</a>; Napolitano, J. (2014). "Quantum Dynamics". <a href="/wiki/Modern_Quantum_Mechanics" title="Modern Quantum Mechanics"><i>Modern Quantum Mechanics</i></a>. Pearson. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-292-02410-3" title="Special:BookSources/978-1-292-02410-3"><bdi>978-1-292-02410-3</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/929609283">929609283</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Quantum+Dynamics&rft.btitle=Modern+Quantum+Mechanics&rft.pub=Pearson&rft.date=2014&rft_id=info%3Aoclcnum%2F929609283&rft.isbn=978-1-292-02410-3&rft.aulast=Sakurai&rft.aufirst=J.+J.&rft.au=Napolitano%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-Aharoni-95"><span class="mw-cite-backlink"><b><a href="#cite_ref-Aharoni_95-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAharoni1996" class="citation book cs1"><a href="/wiki/Amikam_Aharoni" title="Amikam Aharoni">Aharoni, Amikam</a> (1996). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoth00ahar/page/6"><i>Introduction to the Theory of Ferromagnetism</i></a>. <a href="/wiki/Clarendon_Press" class="mw-redirect" title="Clarendon Press">Clarendon Press</a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoth00ahar/page/6">6–7</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-851791-2" title="Special:BookSources/0-19-851791-2"><bdi>0-19-851791-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+Theory+of+Ferromagnetism&rft.pages=6-7&rft.pub=Clarendon+Press&rft.date=1996&rft.isbn=0-19-851791-2&rft.aulast=Aharoni&rft.aufirst=Amikam&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoth00ahar%2Fpage%2F6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-Edwards79-96"><span class="mw-cite-backlink"><b><a href="#cite_ref-Edwards79_96-0">^</a></b></span> <span class="reference-text">David Edwards, "The Mathematical Foundations of Quantum Mechanics", <i>Synthese</i>, Volume 42, Number 1/September, 1979, pp. 1–70.</span> </li> <li id="cite_note-Edwards81-97"><span class="mw-cite-backlink"><b><a href="#cite_ref-Edwards81_97-0">^</a></b></span> <span class="reference-text">David Edwards, "The Mathematical Foundations of Quantum Field Theory: Fermions, Gauge Fields, and Super-symmetry, Part I: Lattice Field Theories", <i>International Journal of Theoretical Physics</i>, Vol. 20, No. 7 (1981).</span> </li> <li id="cite_note-98"><span class="mw-cite-backlink"><b><a href="#cite_ref-98">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernstein2005" class="citation journal cs1"><a href="/wiki/Jeremy_Bernstein" title="Jeremy Bernstein">Bernstein, Jeremy</a> (November 2005). <a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.2060717">"Max Born and the quantum theory"</a>. <i><a href="/wiki/American_Journal_of_Physics" title="American Journal of Physics">American Journal of Physics</a></i>. <b>73</b> (11): <span class="nowrap">999–</span>1008. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005AmJPh..73..999B">2005AmJPh..73..999B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.2060717">10.1119/1.2060717</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9505">0002-9505</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=Max+Born+and+the+quantum+theory&rft.volume=73&rft.issue=11&rft.pages=%3Cspan+class%3D%22nowrap%22%3E999-%3C%2Fspan%3E1008&rft.date=2005-11&rft.issn=0002-9505&rft_id=info%3Adoi%2F10.1119%2F1.2060717&rft_id=info%3Abibcode%2F2005AmJPh..73..999B&rft.aulast=Bernstein&rft.aufirst=Jeremy&rft_id=https%3A%2F%2Fdoi.org%2F10.1119%252F1.2060717&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-pais1997-99"><span class="mw-cite-backlink"><b><a href="#cite_ref-pais1997_99-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPais1997" class="citation book cs1"><a href="/wiki/Abraham_Pais" title="Abraham Pais">Pais, Abraham</a> (1997). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/taleoftwocontine00pais"><i>A Tale of Two Continents: A Physicist's Life in a Turbulent World</i></a></span>. Princeton, New Jersey: Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-01243-1" title="Special:BookSources/0-691-01243-1"><bdi>0-691-01243-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Tale+of+Two+Continents%3A+A+Physicist%27s+Life+in+a+Turbulent+World&rft.place=Princeton%2C+New+Jersey&rft.pub=Princeton+University+Press&rft.date=1997&rft.isbn=0-691-01243-1&rft.aulast=Pais&rft.aufirst=Abraham&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftaleoftwocontine00pais&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-100"><span class="mw-cite-backlink"><b><a href="#cite_ref-100">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVan_Hove1958" class="citation journal cs1">Van Hove, Leon (1958). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10206-2/S0002-9904-1958-10206-2.pdf">"Von Neumann's contributions to quantum mechanics"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a></i>. <b>64</b> (3): Part 2:95–99. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9904-1958-10206-2">10.1090/s0002-9904-1958-10206-2</a></span>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20240120073106/https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10206-2/S0002-9904-1958-10206-2.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on Jan 20, 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=Von+Neumann%27s+contributions+to+quantum+mechanics&rft.volume=64&rft.issue=3&rft.pages=Part+2%3A95-99&rft.date=1958&rft_id=info%3Adoi%2F10.1090%2Fs0002-9904-1958-10206-2&rft.aulast=Van+Hove&rft.aufirst=Leon&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fbull%2F1958-64-03%2FS0002-9904-1958-10206-2%2FS0002-9904-1958-10206-2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-feynman2015-101"><span class="mw-cite-backlink"><b><a href="#cite_ref-feynman2015_101-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynman" class="citation web cs1"><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman, Richard</a>. <a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/III_21.html#Ch21-S5">"The Feynman Lectures on Physics Vol. III Ch. 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity, 21-4"</a>. <a href="/wiki/California_Institute_of_Technology" title="California Institute of Technology">California Institute of Technology</a>. <a rel="nofollow" class="external text" href="https://archive.today/20161215225248/http://www.feynmanlectures.caltech.edu/III_21.html%23Ch21-S5">Archived</a> from the original on 15 Dec 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">24 November</span> 2015</span>. <q>...it was long believed that the wave function of the Schrödinger equation would never have a macroscopic representation analogous to the macroscopic representation of the amplitude for photons. On the other hand, it is now realized that the phenomena of superconductivity presents us with just this situation.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Feynman+Lectures+on+Physics+Vol.+III+Ch.+21%3A+The+Schr%C3%B6dinger+Equation+in+a+Classical+Context%3A+A+Seminar+on+Superconductivity%2C+21-4&rft.pub=California+Institute+of+Technology&rft.aulast=Feynman&rft.aufirst=Richard&rft_id=https%3A%2F%2Ffeynmanlectures.caltech.edu%2FIII_21.html%23Ch21-S5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> <li id="cite_note-102"><span class="mw-cite-backlink"><b><a href="#cite_ref-102">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPackard2006" class="citation web cs1">Packard, Richard (2006). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151125112132/http://research.physics.berkeley.edu/packard/publications/Articles/LT24_Berk_expts_on_macro_sup_effects.pdf">"Berkeley Experiments on Superfluid Macroscopic Quantum Effects"</a> <span class="cs1-format">(PDF)</span>. Physics Department, University of California, Berkeley. Archived from <a rel="nofollow" class="external text" href="http://physics.berkeley.edu/sites/default/files/_/lt24_berk_expts_on_macro_sup_effects.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 25 November 2015<span class="reference-accessdate">. Retrieved <span class="nowrap">24 November</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Berkeley+Experiments+on+Superfluid+Macroscopic+Quantum+Effects&rft.pub=Physics+Department%2C+University+of+California%2C+Berkeley&rft.date=2006&rft.aulast=Packard&rft.aufirst=Richard&rft_id=http%3A%2F%2Fphysics.berkeley.edu%2Fsites%2Fdefault%2Ffiles%2F_%2Flt24_berk_expts_on_macro_sup_effects.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <p>The following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a minimum of technical apparatus. </p> <ul><li><a href="/wiki/Marvin_Chester" title="Marvin Chester">Chester, Marvin</a> (1987). <i>Primer of Quantum Mechanics</i>. John Wiley. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-42878-8" title="Special:BookSources/0-486-42878-8">0-486-42878-8</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxForshaw2011" class="citation book cs1"><a href="/wiki/Brian_Cox_(physicist)" title="Brian Cox (physicist)">Cox, Brian</a>; <a href="/wiki/Jeff_Forshaw" title="Jeff Forshaw">Forshaw, Jeff</a> (2011). <a href="/wiki/The_Quantum_Universe" title="The Quantum Universe"><i>The Quantum Universe: Everything That Can Happen Does Happen</i></a>. Allen Lane. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84614-432-5" title="Special:BookSources/978-1-84614-432-5"><bdi>978-1-84614-432-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Quantum+Universe%3A+Everything+That+Can+Happen+Does+Happen&rft.pub=Allen+Lane&rft.date=2011&rft.isbn=978-1-84614-432-5&rft.aulast=Cox&rft.aufirst=Brian&rft.au=Forshaw%2C+Jeff&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Richard Feynman</a>, 1985. <i><a href="/wiki/QED:_The_Strange_Theory_of_Light_and_Matter" title="QED: The Strange Theory of Light and Matter">QED: The Strange Theory of Light and Matter</a></i>, Princeton University Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-08388-6" title="Special:BookSources/0-691-08388-6">0-691-08388-6</a>. Four elementary lectures on quantum electrodynamics and <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, yet containing many insights for the expert.</li> <li><a href="/wiki/Giancarlo_Ghirardi" title="Giancarlo Ghirardi">Ghirardi, GianCarlo</a>, 2004. <i>Sneaking a Look at God's Cards</i>, Gerald Malsbary, trans. Princeton Univ. Press. The most technical of the works cited here. Passages using <a href="/wiki/Algebra" title="Algebra">algebra</a>, <a href="/wiki/Trigonometry" title="Trigonometry">trigonometry</a>, and <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">bra–ket notation</a> can be passed over on a first reading.</li> <li><a href="/wiki/N._David_Mermin" title="N. David Mermin">N. David Mermin</a>, 1990, "Spooky actions at a distance: mysteries of the QT" in his <i>Boojums All the Way Through</i>. Cambridge University Press: 110–76.</li> <li><a href="/wiki/Victor_Stenger" class="mw-redirect" title="Victor Stenger">Victor Stenger</a>, 2000. <i>Timeless Reality: Symmetry, Simplicity, and Multiple Universes</i>. Buffalo, NY: Prometheus Books. Chpts. 5–8. Includes <a href="/wiki/Cosmological" class="mw-redirect" title="Cosmological">cosmological</a> and <a href="/wiki/Philosophical" class="mw-redirect" title="Philosophical">philosophical</a> considerations.</li></ul> <p>More technical: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernstein,_Jeremy2009" class="citation book cs1"><a href="/wiki/Jeremy_Bernstein" title="Jeremy Bernstein">Bernstein, Jeremy</a> (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=j0Me3brYOL0C"><i>Quantum Leaps</i></a>. Cambridge, Massachusetts: Belknap Press of Harvard University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-674-03541-6" title="Special:BookSources/978-0-674-03541-6"><bdi>978-0-674-03541-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Leaps&rft.place=Cambridge%2C+Massachusetts&rft.pub=Belknap+Press+of+Harvard+University+Press&rft.date=2009&rft.isbn=978-0-674-03541-6&rft.au=Bernstein%2C+Jeremy&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dj0Me3brYOL0C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBohm,_David1989" class="citation book cs1"><a href="/wiki/David_Bohm" title="David Bohm">Bohm, David</a> (1989). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/quantumtheory0000bohm"><i>Quantum Theory</i></a></span>. Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-65969-5" title="Special:BookSources/978-0-486-65969-5"><bdi>978-0-486-65969-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Theory&rft.pub=Dover+Publications&rft.date=1989&rft.isbn=978-0-486-65969-5&rft.au=Bohm%2C+David&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumtheory0000bohm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBinney,_JamesSkinner,_David2008" class="citation book cs1"><a href="/wiki/James_Binney" title="James Binney">Binney, James</a>; Skinner, David (2008). <i>The Physics of Quantum Mechanics</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-968857-9" title="Special:BookSources/978-0-19-968857-9"><bdi>978-0-19-968857-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Physics+of+Quantum+Mechanics&rft.pub=Oxford+University+Press&rft.date=2008&rft.isbn=978-0-19-968857-9&rft.au=Binney%2C+James&rft.au=Skinner%2C+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEisberg,_RobertResnick,_Robert1985" class="citation book cs1">Eisberg, Robert; <a href="/wiki/Robert_Resnick" title="Robert Resnick">Resnick, Robert</a> (1985). <a rel="nofollow" class="external text" href="https://archive.org/details/quantumphysicsof00eisb"><i>Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles</i></a> (2nd ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-87373-0" title="Special:BookSources/978-0-471-87373-0"><bdi>978-0-471-87373-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Physics+of+Atoms%2C+Molecules%2C+Solids%2C+Nuclei%2C+and+Particles&rft.edition=2nd&rft.pub=Wiley&rft.date=1985&rft.isbn=978-0-471-87373-0&rft.au=Eisberg%2C+Robert&rft.au=Resnick%2C+Robert&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumphysicsof00eisb&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></li> <li><a href="/wiki/Bryce_DeWitt" title="Bryce DeWitt">Bryce DeWitt</a>, R. Neill Graham, eds., 1973. <i>The Many-Worlds Interpretation of Quantum Mechanics</i>, Princeton Series in Physics, Princeton University Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-08131-X" title="Special:BookSources/0-691-08131-X">0-691-08131-X</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEverett1957" class="citation journal cs1"><a href="/wiki/Hugh_Everett" class="mw-redirect" title="Hugh Everett">Everett, Hugh</a> (1957). "Relative State Formulation of Quantum Mechanics". <i>Reviews of Modern Physics</i>. <b>29</b> (3): <span class="nowrap">454–</span>462. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1957RvMP...29..454E">1957RvMP...29..454E</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.29.454">10.1103/RevModPhys.29.454</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17178479">17178479</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=Relative+State+Formulation+of+Quantum+Mechanics&rft.volume=29&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E454-%3C%2Fspan%3E462&rft.date=1957&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17178479%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.29.454&rft_id=info%3Abibcode%2F1957RvMP...29..454E&rft.aulast=Everett&rft.aufirst=Hugh&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynmanLeightonSands1965" class="citation book cs1"><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman, Richard P.</a>; <a href="/wiki/Robert_B._Leighton" title="Robert B. Leighton">Leighton, Robert B.</a>; Sands, Matthew (1965). <a href="/wiki/The_Feynman_Lectures_on_Physics" title="The Feynman Lectures on Physics"><i>The Feynman Lectures on Physics</i></a>. Vol. <span class="nowrap">1–</span>3. Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7382-0008-8" title="Special:BookSources/978-0-7382-0008-8"><bdi>978-0-7382-0008-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Feynman+Lectures+on+Physics&rft.pub=Addison-Wesley&rft.date=1965&rft.isbn=978-0-7382-0008-8&rft.aulast=Feynman&rft.aufirst=Richard+P.&rft.au=Leighton%2C+Robert+B.&rft.au=Sands%2C+Matthew&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></li> <li><a href="/wiki/Daniel_Greenberger" title="Daniel Greenberger">D. Greenberger</a>, <a href="/wiki/Klaus_Hentschel" title="Klaus Hentschel">K. Hentschel</a>, F. Weinert, eds., 2009. <i>Compendium of quantum physics, Concepts, experiments, history and philosophy</i>, Springer-Verlag, Berlin, Heidelberg. Short articles on many QM topics.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffiths,_David_J.2004" class="citation book cs1"><a href="/wiki/David_J._Griffiths" title="David J. Griffiths">Griffiths, David J.</a> (2004). <i><a href="/wiki/Introduction_to_Quantum_Mechanics_(book)" title="Introduction to Quantum Mechanics (book)">Introduction to Quantum Mechanics</a></i> (2nd ed.). Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-111892-8" title="Special:BookSources/978-0-13-111892-8"><bdi>978-0-13-111892-8</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/40251748">40251748</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Quantum+Mechanics&rft.edition=2nd&rft.pub=Prentice+Hall&rft.date=2004&rft_id=info%3Aoclcnum%2F40251748&rft.isbn=978-0-13-111892-8&rft.au=Griffiths%2C+David+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span> A standard undergraduate text.</li> <li><a href="/wiki/Max_Jammer" title="Max Jammer">Max Jammer</a>, 1966. <i>The Conceptual Development of Quantum Mechanics</i>. McGraw Hill.</li> <li><a href="/wiki/Hagen_Kleinert" title="Hagen Kleinert">Hagen Kleinert</a>, 2004. <i>Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets</i>, 3rd ed. Singapore: World Scientific. <a rel="nofollow" class="external text" href="http://www.physik.fu-berlin.de/~kleinert/b5">Draft of 4th edition.</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080615134934/http://www.physik.fu-berlin.de/~kleinert/b5">Archived</a> 2008-06-15 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFL.D._Landau,_E.M._Lifshitz1977" class="citation book cs1"><a href="/wiki/Lev_Landau" title="Lev Landau">L.D. Landau</a>, <a href="/wiki/Evgeny_Lifshitz" title="Evgeny Lifshitz">E.M. Lifshitz</a> (1977). <i>Quantum Mechanics: Non-Relativistic Theory</i>. Vol. 3 (3rd ed.). <a href="/wiki/Pergamon_Press" title="Pergamon Press">Pergamon Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-020940-1" title="Special:BookSources/978-0-08-020940-1"><bdi>978-0-08-020940-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics%3A+Non-Relativistic+Theory&rft.edition=3rd&rft.pub=Pergamon+Press&rft.date=1977&rft.isbn=978-0-08-020940-1&rft.au=L.D.+Landau%2C+E.M.+Lifshitz&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://archive.org/details/QuantumMechanics_104">Online copy</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiboff,_Richard_L.2002" class="citation book cs1"><a href="/wiki/Liboff,_Richard" class="mw-redirect" title="Liboff, Richard">Liboff, Richard L.</a> (2002). <i>Introductory Quantum Mechanics</i>. Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8053-8714-8" title="Special:BookSources/978-0-8053-8714-8"><bdi>978-0-8053-8714-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introductory+Quantum+Mechanics&rft.pub=Addison-Wesley&rft.date=2002&rft.isbn=978-0-8053-8714-8&rft.au=Liboff%2C+Richard+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></li> <li>Gunther Ludwig, 1968. <i>Wave Mechanics</i>. London: Pergamon Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-08-203204-1" title="Special:BookSources/0-08-203204-1">0-08-203204-1</a></li> <li><a href="/wiki/George_Mackey" title="George Mackey">George Mackey</a> (2004). <i>The mathematical foundations of quantum mechanics</i>. Dover Publications. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-43517-2" title="Special:BookSources/0-486-43517-2">0-486-43517-2</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMerzbacher,_Eugen1998" class="citation book cs1"><a href="/wiki/Eugen_Merzbacher" title="Eugen Merzbacher">Merzbacher, Eugen</a> (1998). <i>Quantum Mechanics</i>. Wiley, John & Sons, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-88702-7" title="Special:BookSources/978-0-471-88702-7"><bdi>978-0-471-88702-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics&rft.pub=Wiley%2C+John+%26+Sons%2C+Inc&rft.date=1998&rft.isbn=978-0-471-88702-7&rft.au=Merzbacher%2C+Eugen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></li> <li><a href="/wiki/Albert_Messiah" title="Albert Messiah">Albert Messiah</a>, 1966. <i>Quantum Mechanics</i> (Vol. I), English translation from French by G.M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III. <a rel="nofollow" class="external text" href="https://archive.org/details/QuantumMechanicsVolumeI">online</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOmnès,_Roland1999" class="citation book cs1"><a href="/wiki/Roland_Omn%C3%A8s" title="Roland Omnès">Omnès, Roland</a> (1999). <a rel="nofollow" class="external text" href="https://archive.org/details/understandingqua00omne"><i>Understanding Quantum Mechanics</i></a>. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-00435-8" title="Special:BookSources/978-0-691-00435-8"><bdi>978-0-691-00435-8</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/39849482">39849482</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Understanding+Quantum+Mechanics&rft.pub=Princeton+University+Press&rft.date=1999&rft_id=info%3Aoclcnum%2F39849482&rft.isbn=978-0-691-00435-8&rft.au=Omn%C3%A8s%2C+Roland&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Funderstandingqua00omne&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFScerri2006" class="citation book cs1"><a href="/wiki/Eric_R._Scerri" class="mw-redirect" title="Eric R. Scerri">Scerri, Eric. R.</a> (2006). <i>The Periodic Table: Its Story and Its Significance</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-530573-6" title="Special:BookSources/0-19-530573-6"><bdi>0-19-530573-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Periodic+Table%3A+Its+Story+and+Its+Significance&rft.pub=Oxford+University+Press&rft.date=2006&rft.isbn=0-19-530573-6&rft.aulast=Scerri&rft.aufirst=Eric.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span> Considers the extent to which chemistry and the periodic system have been reduced to quantum mechanics.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchiff1955" class="citation book cs1"><a href="/wiki/Leonard_I._Schiff" title="Leonard I. Schiff">Schiff, Leonard I.</a> (1955). <i>Quantum Mechanics</i>. Mc-Graw Hill, Inc.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics&rft.pub=Mc-Graw+Hill%2C+Inc.&rft.date=1955&rft.aulast=Schiff&rft.aufirst=Leonard+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShankar,_R.1994" class="citation book cs1"><a href="/wiki/Ramamurti_Shankar" title="Ramamurti Shankar">Shankar, R.</a> (1994). <i>Principles of Quantum Mechanics</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-306-44790-7" title="Special:BookSources/978-0-306-44790-7"><bdi>978-0-306-44790-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Quantum+Mechanics&rft.pub=Springer&rft.date=1994&rft.isbn=978-0-306-44790-7&rft.au=Shankar%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStone,_A._Douglas2013" class="citation book cs1"><a href="/wiki/A._Douglas_Stone" title="A. Douglas Stone">Stone, A. Douglas</a> (2013). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/einsteinquantumq0000ston"><i>Einstein and the Quantum</i></a></span>. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-13968-5" title="Special:BookSources/978-0-691-13968-5"><bdi>978-0-691-13968-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Einstein+and+the+Quantum&rft.pub=Princeton+University+Press&rft.date=2013&rft.isbn=978-0-691-13968-5&rft.au=Stone%2C+A.+Douglas&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Feinsteinquantumq0000ston&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTransnational_College_of_Lex1996" class="citation book cs1"><a href="/wiki/Transnational_College_of_Lex" class="mw-redirect" title="Transnational College of Lex">Transnational College of Lex</a> (1996). <i>What is Quantum Mechanics? A Physics Adventure</i>. Language Research Foundation, Boston. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-9643504-1-0" title="Special:BookSources/978-0-9643504-1-0"><bdi>978-0-9643504-1-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/34661512">34661512</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=What+is+Quantum+Mechanics%3F+A+Physics+Adventure&rft.pub=Language+Research+Foundation%2C+Boston&rft.date=1996&rft_id=info%3Aoclcnum%2F34661512&rft.isbn=978-0-9643504-1-0&rft.au=Transnational+College+of+Lex&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></li> <li><a href="/wiki/Martinus_J._G._Veltman" title="Martinus J. G. Veltman">Veltman, Martinus J.G.</a> (2003), <i>Facts and Mysteries in Elementary Particle Physics</i>.</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output 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/></span></span></span><span class="sister-link"><a href="https://en.wikibooks.org/wiki/Quantum_Mechanics" class="extiw" title="b:Quantum Mechanics">Textbooks</a> from Wikibooks</span></li><li><span class="sister-logo"><span class="mw-valign-middle" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/27px-Wikiversity_logo_2017.svg.png" decoding="async" width="27" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/41px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/54px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></span></span></span><span class="sister-link"><a href="https://en.wikiversity.org/wiki/Quantum_mechanics" class="extiw" title="v:Quantum mechanics">Resources</a> from Wikiversity</span></li></ul></div></div> </div> <ul><li>J. O'Connor and E. F. Robertson: <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/The_Quantum_age_begins.html">A history of quantum mechanics.</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080913201312/http://www.quantiki.org/wiki/index.php/Introduction_to_Quantum_Theory">Introduction to Quantum Theory at Quantiki.</a></li> <li><a rel="nofollow" class="external text" href="http://bethe.cornell.edu/">Quantum Physics Made Relatively Simple</a>: three video lectures by <a href="/wiki/Hans_Bethe" title="Hans Bethe">Hans Bethe</a>.</li></ul> <dl><dt>Course material</dt> <dd></dd></dl> <ul><li><a rel="nofollow" class="external text" href="http://oyc.yale.edu/sites/default/files/notes_quantum_cookbook.pdf">Quantum Cook Book</a> and <a rel="nofollow" class="external text" href="http://oyc.yale.edu/physics/phys-201#sessions">PHYS 201: Fundamentals of Physics II</a> by <a href="/wiki/Ramamurti_Shankar" title="Ramamurti Shankar">Ramamurti Shankar</a>, Yale OpenCourseware.</li> <li><i><a rel="nofollow" class="external text" href="http://www.lightandmatter.com/mod/">Modern Physics: With waves, thermodynamics, and optics</a></i> – an online textbook.</li> <li><a href="/wiki/MIT_OpenCourseWare" title="MIT OpenCourseWare">MIT OpenCourseWare</a>: <a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/chemistry/">Chemistry</a> and <a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/physics/">Physics</a>. See <a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/">8.04</a>, <a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/index.htm">8.05</a> and <a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/physics/8-06-quantum-physics-iii-spring-2018/index.htm">8.06</a>.</li> <li><a rel="nofollow" class="external text" href="http://physics.csbsju.edu/QM/"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠5<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> Examples in Quantum Mechanics</a>.</li></ul> <dl><dt>Philosophy</dt> <dd></dd></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIsmael" class="citation encyclopaedia cs1">Ismael, Jenann. <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/qm/">"Quantum Mechanics"</a>. In <a href="/wiki/Edward_N._Zalta" title="Edward N. Zalta">Zalta, Edward N.</a> (ed.). <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Quantum+Mechanics&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.aulast=Ismael&rft.aufirst=Jenann&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fqm%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZalta" class="citation encyclopaedia cs1"><a href="/wiki/Edward_N._Zalta" title="Edward N. Zalta">Zalta, Edward N.</a> (ed.). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/qt-issues/">"Philosophical Issues in Quantum Theory"</a>. <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Philosophical+Issues+in+Quantum+Theory&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fqt-issues%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+mechanics" class="Z3988"></span></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 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aria-labelledby="Quantum_mechanics332" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_mechanics_topics" title="Template:Quantum mechanics topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics_topics" title="Template talk:Quantum mechanics topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics_topics" title="Special:EditPage/Template:Quantum mechanics topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_mechanics332" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Quantum mechanics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a> <ul><li><a href="/wiki/Timeline_of_quantum_mechanics" title="Timeline of quantum mechanics">Timeline</a></li></ul></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamentals</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Born_rule" title="Born rule">Born rule</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li> <li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)"> Complementarity</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a> <ul><li><a href="/wiki/Ground_state" title="Ground state">Ground state</a></li> <li><a href="/wiki/Excited_state" title="Excited state">Excited state</a></li> <li><a href="/wiki/Degenerate_energy_levels" title="Degenerate energy levels">Degenerate levels</a></li> <li><a href="/wiki/Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li></ul></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Formulations</a></li> <li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Equations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Weyl_equation" title="Weyl equation">Weyl</a></li> <li><a href="/wiki/Majorana_equation" title="Majorana equation">Majorana</a></li> <li><a href="/wiki/Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell test</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li> <li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed choice</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_nanoscience" class="mw-redirect" title="Quantum nanoscience">Science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_biology" title="Quantum biology">Quantum biology</a></li> <li><a href="/wiki/Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li> <li><a href="/wiki/Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li> <li><a href="/wiki/Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li> <li><a href="/wiki/Quantum_mind" title="Quantum mind">Quantum mind</a></li> <li><a href="/wiki/Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li> <li><a href="/wiki/Quantum_spacetime" title="Quantum spacetime">Quantum spacetime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li> <li><a href="/wiki/Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">Quantum bus</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a> <ul><li><a href="/wiki/Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li> <li><a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">Quantum electronics</a></li> <li><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li> <li><a href="/wiki/Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li> <li><a href="/wiki/Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li> <li><a href="/wiki/Quantum_machine" title="Quantum machine">Quantum machine</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li> <li><a href="/wiki/Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li> <li><a href="/wiki/Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">Quantum network</a></li> <li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li> <li><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Extensions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li> <li><a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a> <ul><li><a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></li></ul></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat_in_popular_culture" title="Schrödinger's cat in popular culture">in popular culture</a></li></ul></li> <li><a href="/wiki/Wigner%27s_friend" title="Wigner's friend">Wigner's friend</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Quantum_mysticism" title="Quantum mysticism">Quantum mysticism</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Quantum_mechanics" title="Category:Quantum mechanics">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Major_branches_of_physics48" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Branches_of_physics" title="Template:Branches of physics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Branches_of_physics" title="Template talk:Branches of physics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Branches_of_physics" title="Special:EditPage/Template:Branches of physics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_branches_of_physics48" style="font-size:114%;margin:0 4em">Major <a href="/wiki/Branches_of_physics" title="Branches of physics">branches of physics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisions</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Basic_research" title="Basic research">Pure</a></li> <li><a href="/wiki/Applied_physics" title="Applied physics">Applied</a> <ul><li><a href="/wiki/Engineering_physics" title="Engineering physics">Engineering</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Approaches</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Experimental_physics" title="Experimental physics">Experimental</a></li> <li><a href="/wiki/Theoretical_physics" title="Theoretical physics">Theoretical</a> <ul><li><a href="/wiki/Computational_physics" title="Computational physics">Computational</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Classical_physics" title="Classical physics">Classical</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a> <ul><li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newtonian</a></li> <li><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical</a></li> <li><a href="/wiki/Celestial_mechanics" title="Celestial mechanics">Celestial</a></li> <li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum</a></li></ul></li> <li><a href="/wiki/Acoustics" title="Acoustics">Acoustics</a></li> <li><a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">Classical electromagnetism</a></li> <li><a href="/wiki/Classical_optics" class="mw-redirect" title="Classical optics">Classical optics</a> <ul><li><a href="/wiki/Geometrical_optics" title="Geometrical optics">Ray</a></li> <li><a href="/wiki/Physical_optics" title="Physical optics">Wave</a></li></ul></li> <li><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a> <ul><li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical</a></li> <li><a href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">Non-equilibrium</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Modern_physics" title="Modern physics">Modern</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Relativistic_mechanics" title="Relativistic mechanics">Relativistic mechanics</a> <ul><li><a href="/wiki/Special_relativity" title="Special relativity">Special</a></li> <li><a href="/wiki/General_relativity" title="General relativity">General</a></li></ul></li> <li><a href="/wiki/Nuclear_physics" title="Nuclear physics">Nuclear physics</a></li> <li><a href="/wiki/Particle_physics" title="Particle physics">Particle physics</a></li> <li><a class="mw-selflink selflink">Quantum mechanics</a></li> <li><a href="/wiki/Atomic,_molecular,_and_optical_physics" title="Atomic, molecular, and optical physics">Atomic, molecular, and optical physics</a> <ul><li><a href="/wiki/Atomic_physics" title="Atomic physics">Atomic</a></li> <li><a href="/wiki/Molecular_physics" title="Molecular physics">Molecular</a></li> <li><a href="/wiki/Optics#Modern_optics" title="Optics">Modern optics</a></li></ul></li> <li><a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">Condensed matter physics</a> <ul><li><a href="/wiki/Solid-state_physics" title="Solid-state physics">Solid-state physics</a></li> <li><a href="/wiki/Crystallography" title="Crystallography">Crystallography</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Applied_and_interdisciplinary_physics" title="Category:Applied and interdisciplinary physics">Interdisciplinary</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Astrophysics" title="Astrophysics">Astrophysics</a></li> <li><a href="/wiki/Atmospheric_physics" title="Atmospheric physics">Atmospheric physics</a></li> <li><a href="/wiki/Biophysics" title="Biophysics">Biophysics</a></li> <li><a href="/wiki/Chemical_physics" title="Chemical physics">Chemical physics</a></li> <li><a href="/wiki/Geophysics" title="Geophysics">Geophysics</a></li> <li><a href="/wiki/Materials_science" title="Materials science">Materials science</a></li> <li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li> <li><a href="/wiki/Medical_physics" title="Medical physics">Medical physics</a></li> <li><a href="/wiki/Physical_oceanography" title="Physical oceanography">Ocean physics</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_physics" title="History of physics">History of physics</a></li> <li><a href="/wiki/Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a></li> <li><a href="/wiki/Philosophy_of_physics" title="Philosophy of physics">Philosophy of physics</a></li> <li><a href="/wiki/Physics_education" title="Physics education">Physics education</a> <ul><li><a href="/wiki/Physics_education_research" title="Physics education research">research</a></li></ul></li> <li><a href="/wiki/Timeline_of_fundamental_physics_discoveries" title="Timeline of fundamental physics discoveries">Timeline of physics discoveries</a></li></ul> </div></td></tr></tbody></table></div> <style data-mw-deduplicate="TemplateStyles:r1130092004">.mw-parser-output .portal-bar{font-size:88%;font-weight:bold;display:flex;justify-content:center;align-items:baseline}.mw-parser-output .portal-bar-bordered{padding:0 2em;background-color:#fdfdfd;border:1px solid #a2a9b1;clear:both;margin:1em auto 0}.mw-parser-output .portal-bar-related{font-size:100%;justify-content:flex-start}.mw-parser-output .portal-bar-unbordered{padding:0 1.7em;margin-left:0}.mw-parser-output .portal-bar-header{margin:0 1em 0 0.5em;flex:0 0 auto;min-height:24px}.mw-parser-output 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