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</div> </a> <ul id="toc-Waves_and_photons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantization" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Quantization"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Quantization</span> </div> </a> <ul id="toc-Quantization-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">3</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Beyond_Planck" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Beyond_Planck"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Beyond Planck</span> </div> </a> <ul id="toc-Beyond_Planck-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Beyond_Heisenberg" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Beyond_Heisenberg"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Beyond Heisenberg</span> </div> </a> <ul id="toc-Beyond_Heisenberg-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_mysteries" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_mysteries"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Further mysteries</span> </div> </a> <ul id="toc-Further_mysteries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Heisenberg_uncertainty_principle" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Heisenberg_uncertainty_principle"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Heisenberg uncertainty principle</span> </div> </a> <ul id="toc-Heisenberg_uncertainty_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uses_of_quantum_mechanics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Uses_of_quantum_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Uses of quantum mechanics</span> </div> </a> <ul id="toc-Uses_of_quantum_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Why_quantum_mechanics_is_hard_to_learn" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Why_quantum_mechanics_is_hard_to_learn"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Why quantum mechanics is hard to learn</span> </div> </a> <ul id="toc-Why_quantum_mechanics_is_hard_to_learn-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reduced_Planck's_constant" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Reduced_Planck's_constant"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Reduced Planck's constant</span> </div> </a> <ul id="toc-Reduced_Planck's_constant-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Example</span> </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_pages" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_pages"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Related pages</span> </div> </a> <ul id="toc-Related_pages-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">13</span> <span>References</span> </div> </a> <button aria-controls="toc-References-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 References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">13.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-More_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#More_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>More reading</span> </div> </a> <ul id="toc-More_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_websites" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_websites"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>Other websites</span> </div> </a> <ul id="toc-Other_websites-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" > <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 135 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-135" 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">135 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-en badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://en.wikipedia.org/wiki/Quantum_mechanics" title="Quantum mechanics – English" lang="en" hreflang="en" data-title="Quantum mechanics" data-language-autonym="English" data-language-local-name="English" class="interlanguage-link-target"><span>English</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-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-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-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 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class="mbox-text-span">This article <b>needs more <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">sources</a> for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">reliability</a>.</b><span class="hide-when-compact"> <br /><small>Please help <a class="external text" href="https://simple.wikipedia.org/w/index.php?title=Quantum_mechanics&action=edit">improve this article</a> by adding <a href="/wiki/Wikipedia:Reliable_sources" title="Wikipedia:Reliable sources">reliable sources</a>. Unsourced material may be <a href="/wiki/Template:Fact" class="mw-redirect" title="Template:Fact">challenged</a> or removed.</small><br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Quantum+mechanics%22">"Quantum mechanics"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Quantum+mechanics%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Quantum+mechanics%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Quantum+mechanics%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Quantum+mechanics%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Quantum+mechanics%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">August 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p><b><a href="/wiki/Quantum" title="Quantum">Quantum</a> mechanics</b> explains how the <a href="/wiki/Universe" title="Universe">universe</a> works with things that are even smaller than atoms. It is also called <b>quantum physics</b> or <b>quantum theory</b>. <a href="/wiki/Mechanics" title="Mechanics">Mechanics</a> is the part of physics that explains how things move and <i>quantum</i> is the <a href="/wiki/Latin" title="Latin">Latin</a> word for 'how much'. A <b>quantum</b> of energy is the least amount possible (or the least extra amount), and quantum mechanics describes how that energy moves. </p><p>Atoms were once believed to be the smallest pieces of <a href="/wiki/Matter" title="Matter">matter</a>, but modern science has shown that there are even smaller <a href="/wiki/Subatomic_particle" title="Subatomic particle">subatomic particles</a>, like <a href="/wiki/Proton" title="Proton">protons</a>, <a href="/wiki/Neutron" title="Neutron">neutrons</a> and <a href="/wiki/Electron" title="Electron">electrons</a>. <sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This claim needs references to reliable sources. (August 2023)">source?</span></a></i>]</sup> Quantum mechanics describes how the particles <a href="/wiki/Subatomic_particle" title="Subatomic particle">that make up atoms</a> work. </p><p>Quantum physics also tells us how <a href="/wiki/Electromagnetic_waves" class="mw-redirect" title="Electromagnetic waves">electromagnetic waves</a> (like <a href="/wiki/Light" title="Light">light</a>) work. <a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a> means that particles behave like waves and waves behave like particles. (They are not two kinds of things, they are something like both: this is called <i><a href="https://simple.wiktionary.org/wiki/duality" class="extiw" title="wikt:duality">duality</a></i>.) Much of modern <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Chemistry" title="Chemistry">chemistry</a> can be described and understood using the <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> rules of quantum mechanics. </p><p>The <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> used to study subatomic particles and electromagnetic waves is very complex because they act in very strange ways. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Wavelength.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Wavelength.png/250px-Wavelength.png" decoding="async" width="250" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/6/64/Wavelength.png 1.5x" data-file-width="361" data-file-height="242" /></a><figcaption>The wavelength of a wave of light</figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Waves_and_photons">Waves and photons</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_mechanics&veaction=edit&section=1" title="Change section: Waves and photons" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quantum_mechanics&action=edit&section=1" title="Edit section's source code: Waves and photons"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Photons are <a href="/wiki/Particles" class="mw-redirect" title="Particles">particles</a> that are point-sized, tinier than <a href="/wiki/Atom" title="Atom">atoms</a>. Photons are like "packets" or packages of energy. Light sources such as <a href="/wiki/Candle" title="Candle">candles</a> or <a href="/wiki/Laser" title="Laser">lasers</a> produce light in bits called <a href="/wiki/Photon" title="Photon">photons</a>. </p><p>The more photons a lamp produces, the brighter the light. Light is a form of energy that behaves like the waves in water or <a href="/wiki/Radio_wave" title="Radio wave">radio waves</a>. The distance between the top of one wave and the top of the next wave is called a '<a href="/wiki/Wavelength" title="Wavelength">wavelength</a>'. Each photon carries a certain amount, or 'quantum', of energy depending on its wavelength. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Linear_visible_spectrum.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Linear_visible_spectrum.svg/300px-Linear_visible_spectrum.svg.png" decoding="async" width="300" height="57" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Linear_visible_spectrum.svg/450px-Linear_visible_spectrum.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Linear_visible_spectrum.svg/600px-Linear_visible_spectrum.svg.png 2x" data-file-width="605" data-file-height="115" /></a><figcaption>Black at left is <b>ultraviolet</b> (high frequency); black at right is <b>infrared</b> (low frequency).</figcaption></figure> <p>A light's <a href="/wiki/Color" class="mw-redirect" title="Color">color</a> depends on its wavelength. The color <a href="/wiki/Violet_(color)" title="Violet (color)">violet</a> (the bottom or innermost color of the <a href="/wiki/Rainbow" title="Rainbow">rainbow</a>) has a wavelength of about 400 nm ("<a href="/wiki/Nanometre" title="Nanometre">nanometers</a>") which is 0.00004 centimeters or 0.000016 inches. Photons with wavelengths of 10-400 nm are called <a href="/wiki/Ultraviolet" title="Ultraviolet">ultraviolet</a> (or UV) <a href="/wiki/Light" title="Light">light</a>. Such light cannot be seen by the human eye. On the other end of the <a href="/wiki/Visual_spectrum" class="mw-redirect" title="Visual spectrum">visual spectrum</a>, red light is about 700 nm. <a href="/wiki/Infrared" title="Infrared">Infrared</a> light is about 700 nm to 300,000 nm. Human eyes are not sensitive to infrared light either. </p><p>Wavelengths are not always so small. <a href="/wiki/Radio_wave" title="Radio wave">Radio waves</a> have longer wavelengths. The wavelengths for an <a href="/wiki/FM_radio" class="mw-redirect" title="FM radio">FM radio</a> can be several meters in length (for example, stations transmitting on 99.5 FM are emitting radio energy with a wavelength of about 3 meters, which is about 10 feet). Each <a href="/wiki/Photon" title="Photon">photon</a> has a certain amount of energy related to its wavelength. The shorter the wavelength of a photon, the greater its energy. For example, an ultraviolet photon has more energy than an infrared photon. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:LowFreqVShighFreq.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/LowFreqVShighFreq.jpg/250px-LowFreqVShighFreq.jpg" decoding="async" width="250" height="108" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/LowFreqVShighFreq.jpg/375px-LowFreqVShighFreq.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/5/51/LowFreqVShighFreq.jpg 2x" data-file-width="441" data-file-height="191" /></a><figcaption>Pictorial description of frequency</figcaption></figure> <p>Wavelength and <a href="/wiki/Frequency" title="Frequency">frequency</a> (the number of times the wave crests per second) are inversely proportional, which means a longer wavelength will have a lower frequency, and vice versa. If the color of the light is <a href="/wiki/Infrared" title="Infrared">infrared</a> (lower in <a href="/wiki/Frequency" title="Frequency">frequency</a> than red light), each photon can heat up what it hits. So, if a strong infrared lamp (a <a href="/wiki/Heat_lamp" title="Heat lamp">heat lamp</a>) is pointed at a person, that person will feel warm, or even hot, because of the energy stored in the many photons. The surface of the infrared lamp may even get hot enough to burn someone who may touch it. Humans cannot see infrared light, but we can feel the radiation in the form of <a href="/wiki/Heat" title="Heat">heat</a>. For example, a person walking by a brick building that has been heated by the sun will feel heat from the building without having to touch it. </p><p>The mathematical equations of quantum mechanics are abstract, which means it is impossible to know the exact physical properties of a particle (like its position or momentum) for sure. Instead, a mathematical function called the wavefunction provides information about the probability with which a particle has a given property. For example, the wavefunction can tell you what the probability is that a particle can be found in a certain location, but it can't tell you where it is for sure. Because of this uncertainty and other factors, you cannot use classical mechanics (the physics that describe how large objects move) to predict the motion of quantum particles. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:IRwarmUVpenetrates.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/IRwarmUVpenetrates.svg/350px-IRwarmUVpenetrates.svg.png" decoding="async" width="350" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/IRwarmUVpenetrates.svg/525px-IRwarmUVpenetrates.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b1/IRwarmUVpenetrates.svg/700px-IRwarmUVpenetrates.svg.png 2x" data-file-width="1374" data-file-height="413" /></a><figcaption>On the left, a <a href="/wiki/Plastic" title="Plastic">plastic</a> <a href="/wiki/Thermometer" title="Thermometer">thermometer</a> is under a bright heat lamp. This infrared radiation warms but does not damage the thermometer. On the right, another plastic thermometer gets hit by a low intensity <a href="/wiki/Ultraviolet" title="Ultraviolet">ultraviolet</a> light. This radiation damages but does not warm the thermometer.</figcaption></figure> <p><a href="/wiki/Ultraviolet" title="Ultraviolet">Ultraviolet</a> light is higher in frequency than violet light, such that it is not even in the visible light range. Each photon in the ultraviolet range has a lot of energy, enough to hurt skin cells and cause a sunburn. In fact, most forms of sunburn are not caused by heat; they are caused by the high energy of the sun's UV rays damaging your skin cells. Even higher frequencies of light (or <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a>) can penetrate deeper into the body and cause even more damage. <a href="/wiki/X-rays" class="mw-redirect" title="X-rays">X-rays</a> have so much energy that they can go deep into the human body and kill cells. Humans cannot see or feel ultraviolet light or x-rays. They may only know they have been under such high frequency light when they get a radiation burn. Areas where it is important to kill <a href="/wiki/Germs" class="mw-redirect mw-disambig" title="Germs">germs</a> often use <a href="/wiki/Ultraviolet_lamp" class="mw-redirect" title="Ultraviolet lamp">ultraviolet lamps</a> to destroy <a href="/wiki/Bacteria" title="Bacteria">bacteria</a>, <a href="/wiki/Fungi" class="mw-redirect" title="Fungi">fungi</a>, etc. X-rays are sometimes used to kill cancer cells. </p><p>Quantum mechanics started when it was discovered that if a particle has a certain frequency, it must also have a certain amount of energy. Energy is proportional to frequency (E ∝ f). The higher the frequency, the more energy a photon has, and the more damage it can do. Quantum mechanics later grew to explain the internal <a href="/wiki/Structure" title="Structure">structure</a> of atoms. Quantum mechanics also explains the way that a photon can <a href="/wiki/Interference_(physics)" class="mw-redirect" title="Interference (physics)">interfere</a> with itself, and many other things never imagined in <a href="/wiki/Classical_physics" title="Classical physics">classical physics</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Quantization">Quantization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_mechanics&veaction=edit&section=2" title="Change section: Quantization" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quantum_mechanics&action=edit&section=2" title="Edit section's source code: Quantization"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Max_Planck" title="Max Planck">Max Planck</a> discovered the relationship between frequency and energy. Nobody before had ever guessed that frequency is directly proportional to energy (this means that as one of them doubles, the other does, too). Under what are called <a href="/wiki/Natural_units" title="Natural units">natural units</a>, then the number representing the frequency of a photon would also represent its energy. The <a href="/wiki/Equation" title="Equation">equation</a> would then be: </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=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6cb4a53256d00d994355472bca039b091856752" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.153ex; height:2.509ex;" alt="{\displaystyle E=f}"></span></dd></dl> <p>meaning energy equals frequency. </p><p>But the way physics grew, there was no natural connection between the units that were used to measure energy and the units commonly used to measure time (and therefore frequency). So the formula that Planck worked out to make the numbers all come out right was: </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\times f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>h</mi> <mo>×</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=h\times f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a722c9bdbd2bab6c900f3cb56a70ea4f8a8be0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.332ex; height:2.509ex;" alt="{\displaystyle E=h\times f}"></span></dd></dl> <p>or, energy equals <i><b>h</b></i> times frequency. This <i><b>h</b></i> is a number called <a href="/wiki/Planck%27s_constant" class="mw-redirect" title="Planck's constant">Planck's constant</a> after its discoverer. </p><p>Quantum mechanics is based on the knowledge that a photon of a certain frequency means a photon of a certain amount of energy. Besides that relationship, a specific kind of atom can only give off certain frequencies of radiation, so it can also only give off photons that have certain amounts of energy. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Young%2BFringes.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Young%2BFringes.gif/150px-Young%2BFringes.gif" decoding="async" width="150" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/6/6b/Young%2BFringes.gif 1.5x" data-file-width="204" data-file-height="140" /></a><figcaption>Double-slit experiment: light goes from the light source at left to fringes (marked in the black edge) at the right.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Photoelectric_effect_in_a_solid_-_diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Photoelectric_effect_in_a_solid_-_diagram.svg/150px-Photoelectric_effect_in_a_solid_-_diagram.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Photoelectric_effect_in_a_solid_-_diagram.svg/225px-Photoelectric_effect_in_a_solid_-_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Photoelectric_effect_in_a_solid_-_diagram.svg/300px-Photoelectric_effect_in_a_solid_-_diagram.svg.png 2x" data-file-width="364" data-file-height="364" /></a><figcaption>Photoelectric effect: photons hit metal and electrons are pushed away.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_mechanics&veaction=edit&section=3" title="Change section: History" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quantum_mechanics&action=edit&section=3" title="Edit section's source code: History"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:10_Quantum_Mechanics_Masters.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/10_Quantum_Mechanics_Masters.jpg/113px-10_Quantum_Mechanics_Masters.jpg" decoding="async" width="113" height="391" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/10_Quantum_Mechanics_Masters.jpg/169px-10_Quantum_Mechanics_Masters.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/10_Quantum_Mechanics_Masters.jpg/226px-10_Quantum_Mechanics_Masters.jpg 2x" data-file-width="275" data-file-height="951" /></a><figcaption>Left to right: <a href="/wiki/Max_Planck" title="Max Planck">Max Planck</a>, <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>, <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a>, <a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">Louis de Broglie</a>, <a href="/wiki/Max_Born" title="Max Born">Max Born</a>, <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a>, <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a>, <a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Wolfgang Pauli</a>, <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a>, <a href="/wiki/Richard_Feynman" title="Richard Feynman">Richard Feynman</a>.</figcaption></figure> <p><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> thought that <a href="/wiki/Light" title="Light">light</a> was made of very small things that we would now call <a href="/wiki/Particles" class="mw-redirect" title="Particles">particles</a> (he referred to them as "Corpuscles"). <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a> thought that light was made of <a href="/wiki/Wave" class="mw-disambig" title="Wave">waves</a>. Scientists thought that a thing cannot be a particle and a wave at the same time. </p><p>Scientists did <a href="/wiki/Experiment" title="Experiment">experiments</a> to find out whether light was made of particles or waves. They found out that both ideas were right — light was somehow both waves and particles. The <a href="/wiki/Double-slit_experiment" class="mw-redirect" title="Double-slit experiment">Double-slit experiment</a> performed by <a href="/wiki/Thomas_Young" title="Thomas Young">Thomas Young</a> showed that light must act like a wave. The <a href="/wiki/Photoelectric_effect" title="Photoelectric effect">Photoelectric effect</a> discovered by Albert Einstein proved that light had to act like particles that carried specific amounts of energy, and that the energies were linked to their frequencies. This experimental result is called the "<a href="/wiki/Wave-particle_duality" class="mw-redirect" title="Wave-particle duality">wave-particle duality</a>" in quantum mechanics. Later, physicists found out that everything behaves both like a wave and like a particle, not just light. However, this effect is much smaller in large objects. </p><p>Here are some of the people who discovered the basic parts of quantum mechanics: <a href="/wiki/Max_Planck" title="Max Planck">Max Planck</a>, <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>, <a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Satyendra Nath Bose</a>, <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a>, <a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">Louis de Broglie</a>, <a href="/wiki/Max_Born" title="Max Born">Max Born</a>, <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a>, <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a>, <a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Wolfgang Pauli</a>, <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a>, <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a>, and <a href="/wiki/Richard_Feynman" title="Richard Feynman">Richard Feynman</a>. They did their work in the first half of the 20th century. </p> <div class="mw-heading mw-heading2"><h2 id="Beyond_Planck">Beyond Planck</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_mechanics&veaction=edit&section=4" title="Change section: Beyond Planck" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quantum_mechanics&action=edit&section=4" title="Edit section's source code: Beyond Planck"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Bright-line_Spectrum-Hydrogen.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Bright-line_Spectrum-Hydrogen.svg/358px-Bright-line_Spectrum-Hydrogen.svg.png" decoding="async" width="358" height="69" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Bright-line_Spectrum-Hydrogen.svg/537px-Bright-line_Spectrum-Hydrogen.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Bright-line_Spectrum-Hydrogen.svg/716px-Bright-line_Spectrum-Hydrogen.svg.png 2x" data-file-width="923" data-file-height="177" /></a><figcaption>Visible light given off by glowing hydrogen. (Wavelengths in nanometers.)</figcaption></figure> <p>Quantum mechanics formulae and ideas were made to explain the light that comes from glowing <a href="/wiki/Hydrogen" title="Hydrogen">hydrogen</a>. The quantum <a href="/wiki/Theory" class="mw-redirect" title="Theory">theory</a> of the <a href="/wiki/Atom" title="Atom">atom</a> also had to explain why the <a href="/wiki/Electron" title="Electron">electron</a> stays in its <a href="/wiki/Orbit" title="Orbit">orbit</a>, which other ideas were not able to explain. It followed from the older ideas that the electron would have to fall in to the center of the atom because it starts out being kept in orbit by its own energy, but it would quickly lose its energy as it revolves in its orbit. (This is because electrons and other charged particles were known to emit light and lose energy when they changed speed or turned.) </p><p>Hydrogen lamps work like neon lamps, but <a href="/wiki/Neon_lamp" title="Neon lamp">neon lamps</a> have their own unique group of colors (and frequencies) of light. Scientists learned that they could identify all elements by the light colors they produce. They just could not figure out how the frequencies were determined. </p><p>Then, a Swiss mathematician named <a href="/w/index.php?title=Johann_Balmer&action=edit&redlink=1" class="new" title="Johann Balmer (not yet started)">Johann Balmer</a> figured out an equation that told what <b>λ</b> (<b>l</b>ambda, for wave <b>l</b>ength) would be: </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 \lambda =B\left({\frac {n^{2}}{n^{2}-4}}\right)\qquad \qquad n=3,4,5,6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mi>B</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="2em" /> <mspace width="2em" /> <mi>n</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =B\left({\frac {n^{2}}{n^{2}-4}}\right)\qquad \qquad n=3,4,5,6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b934f326f434bd8f3742bad1268bfd67236c753a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.236ex; height:6.343ex;" alt="{\displaystyle \lambda =B\left({\frac {n^{2}}{n^{2}-4}}\right)\qquad \qquad n=3,4,5,6}"></span></dd></dl> <p>where <i>B</i> is a number that Balmer determined to be equal to 364.56 nm. </p><p>This equation only worked for the visible light from a hydrogen lamp. But later, the equation was made more general: </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 {1}{\lambda }}=R\left({\frac {1}{m^{2}}}-{\frac {1}{n^{2}}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </mrow> <mo>=</mo> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\lambda }}=R\left({\frac {1}{m^{2}}}-{\frac {1}{n^{2}}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f75640a81ae9d983b101e3ad86b84709ffbdbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.952ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\lambda }}=R\left({\frac {1}{m^{2}}}-{\frac {1}{n^{2}}}\right),}"></span></dd></dl> <p>where <i>R</i> is the <a href="/wiki/Rydberg_constant" title="Rydberg constant">Rydberg constant</a>, equal to 0.0110 nm<sup>−1</sup>, and <i>n</i> must be greater than <i>m</i>. </p><p>Putting in different numbers for <b>m</b> and <b>n</b>, it is easy to predict frequencies for many types of light (ultraviolet, visible, and infared). To see how this works, go to <a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/Hbase/hyde.html#c4">Hyperphysics</a> and go down past the middle of the page. (Use H = 1 for hydrogen.) </p><p>In 1908, <a href="/w/index.php?title=Walter_Ritz&action=edit&redlink=1" class="new" title="Walter Ritz (not yet started)">Walter Ritz</a> made the <a href="/w/index.php?title=Ritz_combination_principle&action=edit&redlink=1" class="new" title="Ritz combination principle (not yet started)">Ritz combination principle</a> that shows how certain gaps between frequencies keep repeating themselves. This turned out to be important to <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a> several years later. </p><p>In 1905, <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> used Planck's idea to show that a beam of light is made up of a stream of particles called <a href="/wiki/Photon" title="Photon">photons</a>. The energy of each photon depends on its <a href="/wiki/Frequency" title="Frequency">frequency</a>. Einstein's idea is the beginning of the idea in quantum mechanics that all subatomic particles like <a href="/wiki/Electron" title="Electron">electrons</a>, <a href="/wiki/Proton" title="Proton">protons</a>, <a href="/wiki/Neutrons" class="mw-redirect" title="Neutrons">neutrons</a>, and others are both waves <b>and</b> particles at the same time. (See picture of atom with the electron as waves at <a href="/wiki/Atom" title="Atom">atom</a>.) This led to a theory about subatomic particles and electromagnetic waves called <i>wave-particle duality</i>. This is where particles and waves were neither one nor the other, but had certain properties of both. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Bohr_Model.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Bohr_Model.svg/300px-Bohr_Model.svg.png" decoding="async" width="300" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Bohr_Model.svg/450px-Bohr_Model.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Bohr_Model.svg/600px-Bohr_Model.svg.png 2x" data-file-width="340" data-file-height="290" /></a><figcaption>An electron falls to lower orbit and a photon is created.</figcaption></figure> <p>In 1913, Niels Bohr came up with the idea that electrons could only take up certain orbits around the nucleus of an atom. Under Bohr's theory, the numbers called <b>m</b> and <b>n</b> in the equation above could represent orbits. Bohr's theory said electrons could begin in some orbit <b>m</b> and end up in some orbit <b>n</b>, or an electron could begin in some orbit <b>n</b> and end up in some orbit <b>m</b>. If a photon hits an electron, its energy will be absorbed, and the electron will move to a higher orbit because of that extra energy. Under Bohr's theory, if an electron falls from a higher orbit to a lower orbit, then it will have to give up energy in the form of a photon. The energy of the photon will equal the energy difference between the two orbits, and the energy of a photon makes it have a certain frequency and color. Bohr's theory provided a good explanation of many aspects of subatomic phenomena, but failed to answer why each of the colors of light produced by glowing hydrogen (and by glowing neon or any other element) has a brightness of its own, and the brightness differences are always the same for each <a href="/wiki/Chemical_element" title="Chemical element">element</a>. </p><p><span typeof="mw:File"><a href="/wiki/File:BohrEquation_to_red_light.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/BohrEquation_to_red_light.svg/900px-BohrEquation_to_red_light.svg.png" decoding="async" width="900" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/BohrEquation_to_red_light.svg/1350px-BohrEquation_to_red_light.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/BohrEquation_to_red_light.svg/1800px-BohrEquation_to_red_light.svg.png 2x" data-file-width="700" data-file-height="70" /></a></span> </p><p>By the time Niels Bohr came out with his theory, most things about the light produced by a hydrogen lamp were known, but scientists still could not explain the brightness of each of the lines produced by glowing hydrogen. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:H_plasma_spectrum.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/H_plasma_spectrum.gif/300px-H_plasma_spectrum.gif" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/H_plasma_spectrum.gif/450px-H_plasma_spectrum.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/0/0e/H_plasma_spectrum.gif 2x" data-file-width="570" data-file-height="570" /></a><figcaption>Spaced-out intensities in arbitrary units</figcaption></figure> <p>Werner Heisenberg took on the job of explaining the brightness or "intensity" of each line. He could not use any simple rule like the one Balmer had come up with. He had to use the very difficult math of classical physics that figures everything out in terms of things like the <a href="/wiki/Mass" title="Mass">mass</a> (weight) of an electron, the <a href="/wiki/Electrical_charge" class="mw-redirect" title="Electrical charge">charge</a> (static electric strength) of an electron, and other tiny quantities. Classical physics already had answers for the brightness of the bands of color that a hydrogen lamp produces, but the classical theory said that there should be a continuous rainbow, and not four separate color bands. Heisenberg's explanation is: </p><p>There is some law that says what frequencies of light glowing hydrogen will produce. It has to predict spaced-out frequencies when the electrons involved are moving between orbits close to the nucleus (center) of the atom, but it also has to predict that the frequencies will get closer and closer together as we look at what the electron does in moving between orbits farther and farther out. It will also predict that the <a href="/wiki/Intensity_(physics)" title="Intensity (physics)">intensity</a> differences between frequencies get closer and closer together as we go out. Where classical physics already gives the right answers by one set of equations the new physics has to give the same answers but by different equations. </p><p>Classical physics uses the mathematical methods of <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a> to make a math picture of the physical world, It uses collections of smooth curves that go together to make one smooth curve that gives, in this case, intensities for light of all frequencies from some light. But it is not right because that smooth curve only appears at higher frequencies. At lower frequencies, there are always isolated points and nothing connects the dots. So, to make a map of the real world, Heisenberg had to make a big change. He had to do something to pick out only the numbers that would match what was seen in nature. Sometimes people say he "guessed" these equations, but he was not making blind guesses. He found what he needed. The numbers that he calculated would put dots on a graph, but there would be no line drawn between the dots. And making one "graph" just of dots for every set of calculations would have wasted lots of paper and not have gotten anything done. Heisenberg found a way to efficiently predict the intensities for different frequencies and to organize that information in a helpful way. </p><p>Just using the empirical rule given above, the one that Balmer got started and Rydberg improved, we can see how to get one set of numbers that would help Heisenberg get the kind of picture that he wanted: </p><p>The rule says that when the electron moves from one orbit to another it either gains or loses energy, depending on whether it is getting farther from the center or nearer to it. So we can put these orbits or energy levels in as headings along the top and the side of a grid. For historical reasons the lowest orbit is called <b>n</b>, and the next orbit out is called <b>n - a</b>, then comes <b>n - b</b>, and so forth. It is confusing that they used negative numbers when the electrons were actually gaining energy, but that is just the way it is. </p><p>Since the Rydberg rule gives us frequencies, we can use that rule to put in numbers depending on where the electron goes. If the electron starts at <b>n</b> and ends up at <b>n</b>, then it has not really gone anywhere, so it did not gain energy and it did not lose energy. So the frequency is 0. If the electron starts at <b>n-a</b> and ends up at <b>n</b>, then it has fallen from a higher orbit to a lower orbit. If it does so then it loses energy, and the energy it loses shows up as a photon. The photon has a certain amount of energy, <b>e</b>, and that is related to a certain frequency <b>f</b> by the equation <b>e = h f</b>. So we know that a certain change of orbit is going to produce a certain frequency of light, <b>f</b>. If the electron starts at <b>n</b> and ends up at <b>n - a,</b> that means it has gone from a lower orbit to a higher orbit. That only happens when a photon of a certain frequency and energy comes in from the outside, is absorbed by the electron and gives it its energy, and that is what makes the electron go out to a higher orbit. So, to keep everything making sense, we write that frequency as a negative number. There was a photon with a certain frequency and now it has been taken away. </p><p>So we can make a grid like this, where f(a←b) means the frequency involved when an electron goes from energy state (orbit) <b>b</b> to energy state <b>a</b> (Again, sequences look backwards, but that is the way they were originally written.): </p><p>Grid of <b>f</b> </p> <table class="wikitable"> <tbody><tr align="center"> <th>Electron States </th> <th>n </th> <th>n-a </th> <th>n-b </th> <td>n-c </td> <th>.... </th> <th> </th></tr> <tr> <th>n </th> <td>f(n←n) </td> <td>f(n←n-a) </td> <td>f(n←n-b) </td> <td>f(n←n-c) </td> <td>..... </td> <td> </td></tr> <tr> <th>n-a </th> <td>f(n-a←n) </td> <td>f(n-a←n-a) </td> <td>f(n-a←n-b) </td> <td>f(n-a←n-c) </td> <td>..... </td> <td> </td></tr> <tr> <th>n-b </th> <td>f(n-b←n) </td> <td>f(n-b←n-a) </td> <td>f(n-b←n-b) </td> <td>f(n-b←n-c) </td> <td>..... </td> <td> </td></tr> <tr> <td>transition.... </td> <td>..... </td> <td>..... </td> <td>..... </td> <td>..... </td> <td> </td></tr></tbody></table> <p>Heisenberg did not make the grids like this. He just did the math that would let him get the intensities he was looking for. But to do that he had to multiply two amplitudes (how high a wave measures) to work out the intensity. (In classical physics, intensity equals amplitude squared.) He made an odd-looking equation to handle this problem, wrote out the rest of his paper, handed it to his boss, and went on vacation. Dr. <a href="/wiki/Max_Born" title="Max Born">Born</a> looked at his funny equation and it seemed a little crazy. He must have wondered, "Why did Heisenberg give me this strange thing? Why does he have to do it this way?" Then he realized that he was looking at a blueprint for something he already knew very well. He was used to calling the grid or table that we could write by doing, for instance, all the math for frequencies, a <b><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a></b>. And Heisenberg's weird equation was a rule for multiplying two of them together. Max Born was a very, very good mathematician. He knew that since the two matrices (grids) being multiplied represented different things (like position (x,y,z) and momentum (mv), for instance), then when you multiply the first matrix by the second you get one answer and when you multiply the second matrix by the first matrix you get another answer. Even though he did not know about matrix math, Heisenberg already saw this "different answers" problem and it had bothered him. But Dr. Born was such a good mathematician that he saw that the difference between the first matrix multiplication and the second matrix multiplication was always going to involve Planck's constant, <b>h</b>, multiplied by the square root of negative one, <b>i</b>. So within a few days of Heisenberg's discovery they already had the basic math for what Heisenberg liked to call the "indeterminacy principle." By "indeterminate" Heisenberg meant that something like an electron is just not pinned down until it gets pinned down. It is a little like a jellyfish that is always squishing around and cannot be "in one place" unless you kill it. Later, people got in the habit of calling it "<a href="/wiki/Heisenberg%27s_uncertainty_principle" title="Heisenberg's uncertainty principle">Heisenberg's uncertainty principle</a>," which made many people make the mistake of thinking that electrons and things like that are really "somewhere" but we are just uncertain about it in our own minds. That idea is wrong. It is not what Heisenberg was talking about. Having trouble measuring something is a problem, but it is not the problem Heisenberg was talking about. </p><p>Heisenberg's idea is very hard to grasp, but we can make it clearer with an example. First, we will start calling these grids "matrices," because we will soon need to talk about <a href="/wiki/Matrix_(math)" class="mw-redirect" title="Matrix (math)">matrix multiplication</a>. </p><p>Suppose that we start with two kinds of measurements, position (<b>q</b>) and momentum (<b>p</b>). In 1925, Heisenberg wrote an equation like this one: </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 Y(n,n-b)=\sum _{a}^{}\,p(n,n-a)q(n-a,n-b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </munderover> <mspace width="thinmathspace" /> <mi>p</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>a</mi> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y(n,n-b)=\sum _{a}^{}\,p(n,n-a)q(n-a,n-b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84f71143f421aefb0471384a8f213717e8e9ab77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.954ex; height:6.176ex;" alt="{\displaystyle Y(n,n-b)=\sum _{a}^{}\,p(n,n-a)q(n-a,n-b)}"></span> (Equation for the <a href="/wiki/Conjugate_variables" title="Conjugate variables">conjugate variables</a> momentum and position)</dd></dl> <p>He did not know it, but this equation gives a blueprint for writing out two matrices (grids) and for multiplying them. The rules for multiplying one matrix by another are a little messy, but here are the two matrices according to the blueprint, and then their product: </p><p>Matrix of <b>p</b> </p> <table class="wikitable"> <tbody><tr align="center"> <th>Electron States </th> <th>n-a </th> <th>n-b </th> <th>n-c </th> <th>.... </th> <th> </th></tr> <tr> <th>n </th> <td>p(n←n-a) </td> <td>p(n←n-b) </td> <td>p(n←n-c) </td> <td>..... </td> <td> </td></tr> <tr> <th>n-a </th> <td>p(n-a←n-a) </td> <td>p(n-a←n-b) </td> <td>p(n-a←n-c) </td> <td>..... </td> <td> </td></tr> <tr> <th>n-b </th> <td>p(n-b←n-a) </td> <td>p(n-b←n-b) </td> <td>p(n-b←n-c) </td> <td>..... </td> <td> </td></tr> <tr> <td>transition.... </td> <td>..... </td> <td>..... </td> <td>..... </td> <td>..... </td> <td> </td></tr></tbody></table> <p>Matrix of <b>q</b> </p> <div style="clear:both;" class=""></div> <table class="wikitable"> <tbody><tr align="center"> <th>Electron States </th> <th>n-b </th> <th>n-c </th> <th>n-d </th> <th>.... </th> <th> </th></tr> <tr> <th>n-a </th> <td>q(n-a←n-b) </td> <td>q(n-a←n-c) </td> <td>q(n-a←n-d) </td> <td>..... </td> <td> </td></tr> <tr> <th>n-b </th> <td>q(n-b←n-b) </td> <td>q(n-b←n-c) </td> <td>q(n-b←n-d) </td> <td>..... </td> <td> </td></tr> <tr> <th>n-c </th> <td>q(n-c←n-b) </td> <td>q(n-c←n-c) </td> <td>q(n-c←n-d) </td> <td>..... </td> <td> </td></tr> <tr> <td>transition.... </td> <td>..... </td> <td>..... </td> <td>..... </td> <td>..... </td> <td> </td></tr></tbody></table> <p>The matrix for the product of the above two matrices as specified by the relevant equation in Heisenberg's 1925 paper is: </p> <table class="wikitable"> <tbody><tr align="left"> <th>Electron States </th> <th>n-b </th> <th>n-c </th> <th>n-d </th> <th>..... </th></tr> <tr> <td><b>n</b> </td> <td>A </td> <td>..... </td> <td>..... </td> <td>..... </td></tr> <tr> <td><b>n-a</b> </td> <td>..... </td> <td>B </td> <td>..... </td> <td>..... </td></tr> <tr> <td><b>n-b</b> </td> <td>..... </td> <td>..... </td> <td>C </td> <td>..... </td></tr> </tbody></table> <p>Where:<br /> </p><p>A=p(n←n-a)*q(n-a←n-b)+p(n←n-b)*q(n-b←n-b)+p(n←n-c)*q(n-c←n-b)+..... </p><p>B=p(n-a←n-a)*q(n-a←n-c)+p(n-a←n-b)*q(n-b←n-c)+p(n-a←n-c)*q(n-c←n-c)+..... </p><p>C=p(n-b←n-a)*q(n-a←n-d)+p(n-b←n-b)*q(n-b←n-d)+p(n-b←n-c)*q(n-d←n-d)+..... </p><p>and so forth. </p><p>If the matrices were reversed, the following values would result: </p><p>A=q(n←n-a)*p(n-a←n-b)+q(n←n-b)*p(n-b←n-b)+q(n←n-c)*p(n-c←n-b)+.....<br /> B=q(n-a←n-a)*p(n-a←n-c)+q(n-a←n-b)*p(n-b←n-c)+q(n-a←n-c)*p(n-c←n-c)+.....<br /> C=q(n-b←n-a)*p(n-a←n-d)+q(n-b←n-b)*p(n-b←n-d)+q(n-b←n-c)*p(n-d←n-d)+..... </p><p>and so forth. </p><p>Note how changing the order of multiplication changes the numbers, step by step, that are actually multiplied. </p> <div class="mw-heading mw-heading2"><h2 id="Beyond_Heisenberg">Beyond Heisenberg</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_mechanics&veaction=edit&section=5" title="Change section: Beyond Heisenberg" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quantum_mechanics&action=edit&section=5" title="Edit section's source code: Beyond Heisenberg"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The work of Werner Heisenberg seemed to break a log jam. Very soon, many different other ways of explaining things came from people such as Louis de Broglie, Max Born, Paul Dirac, <a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Wolfgang Pauli</a>, and <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a>. </p><p>The math used by Heisenberg and earlier people is not very hard to understand, but the equations quickly grew very complicated as physicists looked more deeply into the atomic world. </p> <div class="mw-heading mw-heading2"><h2 id="Further_mysteries">Further mysteries</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_mechanics&veaction=edit&section=6" title="Change section: Further mysteries" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quantum_mechanics&action=edit&section=6" title="Edit section's source code: Further mysteries"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the early days of quantum mechanics, Albert Einstein suggested that if it were right then quantum mechanics would mean that there would be "spooky action at a distance." It turned out that quantum mechanics was right, and that what Einstein had used as a reason to reject quantum mechanics actually happened. This kind of "spooky connection" between certain quantum events is now called "<a href="/wiki/Quantum_entanglement" title="Quantum entanglement">quantum entanglement</a>". </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Superposition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Superposition.svg/300px-Superposition.svg.png" decoding="async" width="300" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Superposition.svg/450px-Superposition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Superposition.svg/600px-Superposition.svg.png 2x" data-file-width="1367" data-file-height="409" /></a><figcaption>Two entangled particles are separated: one on Earth and one taken to some distant planet. Measuring one of them forces it to "decide" which role to take, and the other one must then take the other role whenever (after that) it is measured.</figcaption></figure> <p>When an experiment brings two things (photons, electrons, etc.) together, they must then share a common description in quantum mechanics. When they are later separated, they keep the same quantum mechanical description or "state." In the diagram, one characteristic (e.g., "up" spin) is drawn in red, and its mate (e.g., "down" spin) is drawn in blue. The purple band means that when, e.g., two electrons are put together the pair shares both characteristics. So both electrons could show either up spin or down spin. When they are later separated, one remaining on Earth and one going to some planet of the star Alpha Centauri, they still each have both spins. In other words, each one of them can "decide" to show itself as a spin-up electron or a spin-down electron. But if later on someone measures the other one, it must "decide" to show itself as having the opposite spin. </p><p>Einstein argued that over such a great distance it was crazy to think that forcing one electron to show its spin would then somehow make the other electron show an opposite characteristic. He said that the two electrons must have been spin-up or spin-down all along, but that quantum mechanics could not predict which characteristic each electron had. Being unable to predict, only being able to look at one of them with the right experiment, meant that quantum mechanics could not account for something important. Therefore, Einstein said, quantum mechanics had a big hole in it. Quantum mechanics was incomplete. </p><p>Later, it turned out that experiments showed that it was Einstein who was wrong.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Heisenberg_uncertainty_principle">Heisenberg uncertainty principle</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_mechanics&veaction=edit&section=7" title="Change section: Heisenberg uncertainty principle" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quantum_mechanics&action=edit&section=7" title="Edit section's source code: Heisenberg uncertainty principle"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1925, Werner Heisenberg described the <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty principle</a>, which says that the more we know about where a particle is, the less we can know about how fast it is going and in which direction. In other words, the more we know about the <a href="/wiki/Speed" title="Speed">speed</a> and direction of something small, the less we can know about its <a href="/wiki/Position" title="Position">position</a>. Physicists usually talk about the <a href="/wiki/Momentum" title="Momentum">momentum</a> in such discussions instead of talking about speed. Momentum is just the speed of something in a certain direction times its <a href="/wiki/Mass" title="Mass">mass</a>. </p><p>The reason behind <a href="/wiki/Heisenberg%27s_uncertainty_principle" title="Heisenberg's uncertainty principle">Heisenberg's uncertainty principle</a> says that we can never know both the location and the momentum of a particle. Because light is an abundant particle, it is used for measuring other particles. The only way to measure it is to bounce the light wave off of the particle and record the results. If a high energy, or high frequency, light beam is used, we can tell precisely where it is, but cannot tell how fast it was going. This is because the high energy photon transfers energy to the particle and changes the particle's speed. If we use a low energy photon, we can tell how fast it is going, but not where it is. This is because we are using light with a longer wavelength. The longer wavelength means the particle could be anywhere along the stretch of the wave. </p><p>The principle also says that there are many pairs of measurements for which we cannot know both of them about any <a href="/wiki/Particle" title="Particle">particle</a> (a very small thing), no matter how hard we try. The more we learn about one of such a pair, the less we can know about the other. </p><p>Even Albert Einstein had trouble accepting such a bizarre concept, and in a well-known debate said, "God does not play dice". To this, Danish physicist <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a> famously responded, "Einstein, don't tell God what to do". </p> <div class="mw-heading mw-heading2"><h2 id="Uses_of_quantum_mechanics">Uses of quantum mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_mechanics&veaction=edit&section=8" title="Change section: Uses of quantum mechanics" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quantum_mechanics&action=edit&section=8" title="Edit section's source code: Uses of quantum mechanics"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Electron" title="Electron">Electrons</a> surround every <a href="/wiki/Atom" title="Atom">atom</a>'s nucleus. <a href="/wiki/Chemical_bond" title="Chemical bond">Chemical bonds</a> link atoms to form <a href="/wiki/Molecule" title="Molecule">molecules</a>. A chemical bond links two atoms when electrons are shared between those atoms. Thus quantum mechanics is the physics of the chemical bond and of <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>. Quantum mechanics helps us understand how <a href="/wiki/Molecule" title="Molecule">molecules</a> are made, and what their properties are.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Quantum mechanics can also help us understand big things, such as <a href="/wiki/Star" title="Star">stars</a> and even the whole <a href="/wiki/Universe" title="Universe">universe</a>. Quantum mechanics is a very important part of the theory of how the <a href="/wiki/Universe" title="Universe">universe</a> began called the <a href="/wiki/Big_Bang" title="Big Bang">Big Bang</a>. </p><p>Everything made of <a href="/wiki/Matter" title="Matter">matter</a> is attracted to other matter because of a fundamental force called <a href="/wiki/Gravity" title="Gravity">gravity</a>. Einstein's theory that explains gravity is called the theory of general <a href="/wiki/Relativity" class="mw-disambig" title="Relativity">relativity</a>. A problem in modern physics is that some conclusions of quantum mechanics do not seem to <a href="/wiki/Agree" class="mw-redirect" title="Agree">agree</a> with the theory of general relativity. </p><p>Quantum mechanics is the part of physics that can explain why all <a href="/wiki/Electronics" title="Electronics">electronic technology</a> works as it does. Thus quantum mechanics explains how <a href="/wiki/Computer" title="Computer">computers</a> work, because computers are electronic machines. But the designers of the early <a href="/wiki/Computer_hardware" title="Computer hardware">computer hardware</a> of around 1950 or 1960 did not need to think about quantum mechanics. The designers of radios and televisions at that time did not think about quantum mechanics either. However, the design of the more powerful <a href="/wiki/Integrated_circuit" title="Integrated circuit">integrated circuits</a> and <a href="/wiki/Computer_memory" title="Computer memory">computer memory</a> technologies of recent years does require quantum mechanics. </p><p>Quantum mechanics has also made possible technologies such as: </p> <ul><li><a href="/wiki/Spectroscopy" title="Spectroscopy">Spectroscopy</a></li> <li><a href="/wiki/Laser" title="Laser">Lasers</a></li> <li><a href="/wiki/Magnetic_resonance_imaging" title="Magnetic resonance imaging">MRIs</a></li> <li><a href="/wiki/Compact_disc" title="Compact disc">CDs</a> and <a href="/wiki/DVD" title="DVD">DVDs</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Why_quantum_mechanics_is_hard_to_learn">Why quantum mechanics is hard to learn</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_mechanics&veaction=edit&section=9" title="Change section: Why quantum mechanics is hard to learn" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quantum_mechanics&action=edit&section=9" title="Edit section's source code: Why quantum mechanics is hard to learn"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Quantum mechanics is a <a href="https://simple.wiktionary.org/wiki/challenging" class="extiw" title="wikt:challenging">challenging</a> subject for several reasons: </p> <ul><li>Quantum mechanics explains things in very different ways from what we learn about the world when we are children.</li> <li>Understanding quantum mechanics requires more mathematics than <a href="/wiki/Algebra" title="Algebra">algebra</a> and simple <a href="/wiki/Calculus" title="Calculus">calculus</a>. It also requires <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix algebra</a>, <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, and <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a>.</li> <li>Physicists are not sure what some of the equations of quantum mechanics tell us about the real world.</li> <li>Quantum mechanics suggests that atoms and subatomic particles behave in strange ways, completely unlike anything we see in our everyday lives.</li> <li>Quantum mechanics describes things that are extremely small, so we cannot see some of them without special equipment, and we cannot see many of them at all.</li></ul> <p>Quantum mechanics describes nature in a way that is different from how we usually think about <a href="/wiki/Science" title="Science">science</a>. It tells us how <a href="/wiki/Probability" title="Probability">likely</a> to happen some things are, rather than telling us that they certainly will happen. </p><p>One example is <a href="/wiki/Young%27s_double-slit_experiment" title="Young's double-slit experiment">Young's double-slit experiment</a>. If we shoot single photons (single units of light) from a laser at a sheet of <a href="https://simple.wiktionary.org/wiki/photographic_film" class="extiw" title="wikt:photographic film">photographic film</a>, we will see a single spot of light on the developed film. If we put a sheet of metal in between, and make two very narrow slits in the sheet, when we fire many photons at the metal sheet, and they have to go through the slits, then we will see something remarkable. All the way across the sheet of developed film we will see a series of bright and dark bands. We can use mathematics to tell exactly where the bright bands will be and how bright the light was that made them, that is, we can tell ahead of time how many photons will fall on each band. But if we slow the process down and see where each photon lands on the screen we can never tell ahead of time where the next one will show up. We can know for sure that it is most likely that a photon will hit the center bright band, and that it gets less and less likely that a photon will show up at bands farther and farther from the center. So we know for sure that the bands will be brightest at the center and get dimmer and dimmer farther away. But we never know for sure which photon will go into which band. </p><p>One of the strange conclusions of quantum mechanics theory is the "<a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a>" effect. Certain properties of a particle, such as their position, speed of motion, direction of motion, and "<a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a>", cannot be talked about until something measures them (a photon bouncing off of an electron would count as a measurement of its position, for example). Before the measurement, the particle is in a "superposition of states," in which its properties have many values at the same time. Schrödinger said that quantum mechanics seemed to say that if something (such as the life or death of a cat) was determined by a quantum event, then its state would be determined by the state that resulted from the quantum event, but only at the time that somebody looked at the state of the quantum event. In the time before the state of the quantum event is looked at, perhaps "the living and dead cat (pardon the expression) [are] mixed or smeared out in equal parts."<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> <div class="mw-heading mw-heading2"><h2 id="Reduced_Planck's_constant"><span id="Reduced_Planck.27s_constant"></span>Reduced Planck's constant</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_mechanics&veaction=edit&section=10" title="Change section: Reduced Planck's constant" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quantum_mechanics&action=edit&section=10" title="Edit section's source code: Reduced Planck's constant"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>People often use the symbol <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>, which is called "h-bar." <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 ={\frac {h}{2\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar ={\frac {h}{2\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df05180652a87fe1af83af4bba3402117bd18466" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.736ex; height:5.343ex;" alt="{\displaystyle \hbar ={\frac {h}{2\pi }}}"></span> . H-bar is a unit of <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>. When this new unit is used to describe the <a href="/wiki/Atomic_orbital" title="Atomic orbital">orbits</a> of electrons in atoms, the angular momentum of any electron in orbit is always a whole number.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Example">Example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_mechanics&veaction=edit&section=11" title="Change section: Example" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quantum_mechanics&action=edit&section=11" title="Edit section's source code: Example"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The particle in a 1-dimensional well is the most simple example showing that the energy of a particle can only have specific values. The energy is said to be "quantized." The well has zero potential energy inside a range and has infinite potential energy everywhere outside that range. For the 1-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 can be written as:<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> </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>Using differential equations, we can figure out 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> can be written as </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 =Ae^{ikx}+Be^{-ikx}\;\;\;\;\;\;E={\frac {\hbar ^{2}k^{2}}{2m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <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="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <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 =Ae^{ikx}+Be^{-ikx}\;\;\;\;\;\;E={\frac {\hbar ^{2}k^{2}}{2m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be5d2a3f04ea2feebea7d7c9f8a42e0fa393c84b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.82ex; height:5.676ex;" alt="{\displaystyle \psi =Ae^{ikx}+Be^{-ikx}\;\;\;\;\;\;E={\frac {\hbar ^{2}k^{2}}{2m}}}"></span></dd></dl> <p>or as </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 =C\sin kx+D\cos kx\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mi>C</mi> <mi>sin</mi> <mo>⁡</mo> <mi>k</mi> <mi>x</mi> <mo>+</mo> <mi>D</mi> <mi>cos</mi> <mo>⁡</mo> <mi>k</mi> <mi>x</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =C\sin kx+D\cos kx\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/321c88af906c95c43a3aa4b22e2b9974c0fe17a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.385ex; height:2.509ex;" alt="{\displaystyle \psi =C\sin kx+D\cos kx\;}"></span> (by <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>).</dd></dl> <p>The walls of the box mean that the wavefunction must have a special form. The wavefunction of the particle must be zero anytime the walls are infinitely tall. At each wall: </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\;\mathrm {at} \;\;x=0,\;x=L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =0\;\mathrm {at} \;\;x=0,\;x=L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25c47164d09988cd4416a60a4cb665e51a87b563" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.057ex; height:2.509ex;" alt="{\displaystyle \psi =0\;\mathrm {at} \;\;x=0,\;x=L}"></span></dd></dl> <p>Consider <i>x</i> = 0 </p> <ul><li>sin 0 = 0, cos 0 = 1. To satisfy <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>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> </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/2061b72264512ad97942d39ac8cf8c4142b9773e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.419ex; height:2.509ex;" alt="{\displaystyle \psi =0\;}"></span> the cos term has to be removed. Hence <i>D</i> = 0</li></ul> <p>Now consider: <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 =C\sin kx\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mi>C</mi> <mi>sin</mi> <mo>⁡</mo> <mi>k</mi> <mi>x</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =C\sin kx\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30bc7b5bbbcafdc57d0a66ab3c3f52397f244e12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.194ex; height:2.509ex;" alt="{\displaystyle \psi =C\sin kx\;}"></span> </p> <ul><li>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>, <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 =C\sin kL=0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mi>C</mi> <mi>sin</mi> <mo>⁡</mo> <mi>k</mi> <mi>L</mi> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =C\sin kL=0\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3fab7ad8655a0eefe4d36589e46fddff094630" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.708ex; height:2.509ex;" alt="{\displaystyle \psi =C\sin kL=0\;}"></span></li> <li>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 C=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f750a7094a396d89a81974cdf35783db2bb287b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.027ex; height:2.176ex;" alt="{\displaystyle C=0}"></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 \psi =0\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> </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/2061b72264512ad97942d39ac8cf8c4142b9773e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.419ex; height:2.509ex;" alt="{\displaystyle \psi =0\;}"></span> for all <i>x</i>. This solution is not useful.</li> <li>therefore <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> <mi>k</mi> <mi>L</mi> <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/8d40289fc21d0863840f53a7659062edecf468ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.298ex; height:2.176ex;" alt="{\displaystyle \sin kL=0}"></span> must be true, giving us</li></ul> <dl><dd><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 kL=n\pi \;\;\;\;n=1,2,3,4,5,...\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mi>L</mi> <mo>=</mo> <mi>n</mi> <mi>π</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle kL=n\pi \;\;\;\;n=1,2,3,4,5,...\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c1704759d8c3ff88f8b305b6878975b7aa66019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.035ex; height:2.509ex;" alt="{\displaystyle kL=n\pi \;\;\;\;n=1,2,3,4,5,...\;}"></span></dd></dl></dd></dl> <p>We can see 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> must be an integer. This means that the particle can only have special energy values and cannot have the energy values in between. This is an example of energy "quantization." </p> <div class="mw-heading mw-heading2"><h2 id="Related_pages">Related pages</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_mechanics&veaction=edit&section=12" title="Change section: Related pages" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quantum_mechanics&action=edit&section=12" title="Edit section's source code: Related pages"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Electromagnetic_wave" class="mw-redirect" title="Electromagnetic wave">Electromagnetic wave</a></li> <li><a href="/wiki/Electron" title="Electron">Electron</a></li> <li><a href="/wiki/Photon" title="Photon">Photon</a></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Quantum entanglement</a></li> <li><a href="/wiki/Quantum_computer" title="Quantum computer">Quantum computer</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Quantum tunnelling</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Schrodinger_equation" class="mw-redirect" title="Schrodinger equation">Schrödinger equation</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_mechanics&veaction=edit&section=13" title="Change section: References" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quantum_mechanics&action=edit&section=13" title="Edit section's source code: References"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Feynman, Richard, 1985. <i>The Strange Theory of Light and Matter </i>. Princeton University Press.</li> <li>McEvoy, J.P. and Oscar Zarate, 1996. <i>Introducing Quantum Theory</i>. Icon Books.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_mechanics&veaction=edit&section=14" title="Change section: Notes" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quantum_mechanics&action=edit&section=14" title="Edit section's source code: Notes"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r9724317">.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-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">For an overview of the whole issue of entanglement, see J.P. McEvoy and Oscar Zarate, <i>Introducing Quantum Theory,</i> pp. 168—170.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">For a good foundation see <i>The Nature of the Chemical Bond,</i> by Linus Pauling.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Schrödinger: "The Present Situation in Quantum Mechanics," p. 8 of 22.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><i>Scientific American Reader</i>, Simon and Schuster, 1953, p. 117.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Derivation of particle in a box, <a rel="nofollow" class="external text" href="http://chemistry.tidalswan.com/index.php?title=Quantum_Mechanics">chemistry.tidalswan.com</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="More_reading">More reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_mechanics&veaction=edit&section=15" title="Change section: More reading" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Quantum_mechanics&action=edit&section=15" title="Edit section's source code: More reading"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r9724332">.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="CITEREFCox&_Forshaw2011" class="citation book cs1">Cox, Brian; & Forshaw, Jeff (2011). <a rel="nofollow" class="external text" href="https://archive.org/details/quantumuniversee0000coxb_j8t1"><i>The Quantum Universe: Everything That Can Happen Does Happen</i></a>. 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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/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="/w/index.php?title=Introduction_to_quantum_mechanics&action=edit&redlink=1" class="new" title="Introduction to quantum mechanics (not yet started)">Introduction</a></li> <li><a href="/w/index.php?title=History_of_quantum_mechanics&action=edit&redlink=1" class="new" title="History of quantum mechanics (not yet started)">History</a> <ul><li><a href="/w/index.php?title=Timeline_of_quantum_mechanics&action=edit&redlink=1" class="new" title="Timeline of quantum mechanics (not yet started)">Timeline</a></li></ul></li> <li><a href="/w/index.php?title=Glossary_of_elementary_quantum_mechanics&action=edit&redlink=1" class="new" title="Glossary of elementary quantum mechanics (not yet started)">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" class="mw-redirect" title="Born rule">Born rule</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/Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li> <li><a href="/w/index.php?title=Degenerate_energy_levels&action=edit&redlink=1" class="new" title="Degenerate energy levels (not yet started)">Degenerate levels</a></li></ul></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Quantum_superposition" class="mw-redirect" title="Quantum superposition">Superposition</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%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li> <li><a href="/w/index.php?title=Wave_function_collapse&action=edit&redlink=1" class="new" title="Wave function collapse (not yet started)">Collapse</a></li></ul></li> <li><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&redlink=1" class="new" title="Bra–ket notation (not yet 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started)">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/w/index.php?title=Scattering_theory&action=edit&redlink=1" class="new" title="Scattering theory (not yet started)">Scattering theory</a></li> <li><a href="/w/index.php?title=Symmetry_in_quantum_mechanics&action=edit&redlink=1" class="new" title="Symmetry in quantum mechanics (not yet started)">Symmetry in quantum mechanics</a></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="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&redlink=1" class="new" title="Mathematical formulation of quantum mechanics (not yet started)">Formulations</a></li> <li><a href="/w/index.php?title=Heisenberg_picture&action=edit&redlink=1" class="new" title="Heisenberg picture (not yet started)">Heisenberg</a></li> <li><a href="/w/index.php?title=Interaction_picture&action=edit&redlink=1" class="new" title="Interaction picture (not yet started)">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li> <li><a href="/w/index.php?title=Schr%C3%B6dinger_picture&action=edit&redlink=1" class="new" title="Schrödinger picture (not yet started)">Schrödinger</a></li> <li><a href="/w/index.php?title=Path_integral_formulation&action=edit&redlink=1" class="new" title="Path integral formulation (not yet started)">Path integral formulation</a></li> <li><a href="/w/index.php?title=Phase-space_formulation&action=edit&redlink=1" class="new" title="Phase-space formulation (not yet started)">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/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/w/index.php?title=Klein%E2%80%93Gordon_equation&action=edit&redlink=1" class="new" title="Klein–Gordon equation (not yet started)">Klein–Gordon</a></li> <li><a href="/w/index.php?title=Pauli_equation&action=edit&redlink=1" class="new" title="Pauli equation (not yet started)">Pauli</a></li> <li><a href="/w/index.php?title=Rydberg_formula&action=edit&redlink=1" class="new" title="Rydberg formula (not yet started)">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/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/w/index.php?title=Quantum_Bayesianism&action=edit&redlink=1" class="new" title="Quantum Bayesianism (not yet started)">Bayesian</a></li> <li><a href="/w/index.php?title=Consistent_histories&action=edit&redlink=1" class="new" title="Consistent histories (not yet started)">Consistent histories</a></li> <li><a href="/w/index.php?title=De_Broglie%E2%80%93Bohm_theory&action=edit&redlink=1" class="new" title="De Broglie–Bohm theory (not yet started)">de Broglie–Bohm</a></li> <li><a href="/w/index.php?title=Ensemble_interpretation&action=edit&redlink=1" class="new" title="Ensemble interpretation (not yet started)">Ensemble</a></li> <li><a href="/w/index.php?title=Hidden-variable_theory&action=edit&redlink=1" class="new" title="Hidden-variable theory (not yet started)">Hidden-variable</a> <ul><li><a href="/w/index.php?title=Local_hidden-variable_theory&action=edit&redlink=1" class="new" title="Local hidden-variable theory (not yet started)">Local</a> <ul><li><a href="/w/index.php?title=Superdeterminism&action=edit&redlink=1" class="new" title="Superdeterminism (not yet started)">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/w/index.php?title=Objective-collapse_theory&action=edit&redlink=1" class="new" title="Objective-collapse theory (not yet started)">Objective collapse</a></li> <li><a href="/w/index.php?title=Quantum_logic&action=edit&redlink=1" class="new" title="Quantum logic (not yet started)">Quantum logic</a></li> <li><a href="/w/index.php?title=Relational_quantum_mechanics&action=edit&redlink=1" class="new" title="Relational quantum mechanics (not yet started)">Relational</a></li> <li><a href="/w/index.php?title=Transactional_interpretation&action=edit&redlink=1" class="new" title="Transactional interpretation (not yet started)">Transactional</a></li> <li><a href="/w/index.php?title=Von_Neumann%E2%80%93Wigner_interpretation&action=edit&redlink=1" class="new" title="Von Neumann–Wigner interpretation (not yet started)">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="/w/index.php?title=Bell_test&action=edit&redlink=1" class="new" title="Bell test (not yet started)">Bell test</a></li> <li><a href="/w/index.php?title=Davisson%E2%80%93Germer_experiment&action=edit&redlink=1" class="new" title="Davisson–Germer experiment (not yet started)">Davisson–Germer</a></li> <li><a href="/w/index.php?title=Delayed-choice_quantum_eraser&action=edit&redlink=1" class="new" title="Delayed-choice quantum eraser (not yet started)">Delayed-choice quantum eraser</a></li> <li><a href="/wiki/Double-slit_experiment" class="mw-redirect" title="Double-slit experiment">Double-slit</a></li> <li><a href="/w/index.php?title=Franck%E2%80%93Hertz_experiment&action=edit&redlink=1" class="new" title="Franck–Hertz experiment (not yet started)">Franck–Hertz</a></li> <li><a href="/w/index.php?title=Mach%E2%80%93Zehnder_interferometer&action=edit&redlink=1" class="new" title="Mach–Zehnder interferometer (not yet started)">Mach–Zehnder interferometer</a></li> <li><a href="/w/index.php?title=Elitzur%E2%80%93Vaidman_bomb_tester&action=edit&redlink=1" class="new" title="Elitzur–Vaidman bomb tester (not yet started)">Elitzur–Vaidman</a></li> <li><a href="/w/index.php?title=Popper%27s_experiment&action=edit&redlink=1" class="new" title="Popper's experiment (not yet started)">Popper</a></li> <li><a href="/w/index.php?title=Quantum_eraser_experiment&action=edit&redlink=1" class="new" title="Quantum eraser experiment (not yet started)">Quantum eraser</a></li> <li><a href="/w/index.php?title=Stern%E2%80%93Gerlach_experiment&action=edit&redlink=1" class="new" title="Stern–Gerlach experiment (not yet started)">Stern–Gerlach</a></li> <li><a href="/w/index.php?title=Wheeler%27s_delayed-choice_experiment&action=edit&redlink=1" class="new" title="Wheeler's delayed-choice experiment (not yet started)">Wheeler's delayed choice</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/w/index.php?title=Quantum_nanoscience&action=edit&redlink=1" class="new" title="Quantum nanoscience (not yet started)">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="/w/index.php?title=Quantum_biology&action=edit&redlink=1" class="new" title="Quantum biology (not yet started)">Quantum biology</a></li> <li><a href="/w/index.php?title=Quantum_chemistry&action=edit&redlink=1" class="new" title="Quantum chemistry (not yet started)">Quantum chemistry</a></li> <li><a href="/w/index.php?title=Quantum_chaos&action=edit&redlink=1" class="new" title="Quantum chaos (not yet started)">Quantum chaos</a></li> <li><a href="/w/index.php?title=Quantum_cosmology&action=edit&redlink=1" class="new" title="Quantum cosmology (not yet started)">Quantum cosmology</a></li> <li><a href="/w/index.php?title=Quantum_differential_calculus&action=edit&redlink=1" class="new" title="Quantum differential calculus (not yet started)">Quantum differential calculus</a></li> <li><a href="/w/index.php?title=Quantum_dynamics&action=edit&redlink=1" class="new" title="Quantum dynamics (not yet started)">Quantum dynamics</a></li> <li><a href="/w/index.php?title=Quantum_geometry&action=edit&redlink=1" class="new" title="Quantum geometry (not yet started)">Quantum geometry</a></li> <li><a href="/w/index.php?title=Measurement_problem&action=edit&redlink=1" class="new" title="Measurement problem (not yet started)">Quantum measurement problem</a></li> <li><a href="/w/index.php?title=Quantum_mind&action=edit&redlink=1" class="new" title="Quantum mind (not yet started)">Quantum mind</a></li> <li><a href="/w/index.php?title=Quantum_stochastic_calculus&action=edit&redlink=1" class="new" title="Quantum stochastic calculus (not yet started)">Quantum stochastic calculus</a></li> <li><a href="/w/index.php?title=Quantum_spacetime&action=edit&redlink=1" class="new" title="Quantum spacetime (not yet started)">Quantum spacetime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/w/index.php?title=Quantum_technology&action=edit&redlink=1" class="new" title="Quantum technology (not yet started)">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="/w/index.php?title=Quantum_algorithm&action=edit&redlink=1" class="new" title="Quantum algorithm (not yet started)">Quantum algorithms</a></li> <li><a href="/w/index.php?title=Quantum_amplifier&action=edit&redlink=1" class="new" title="Quantum amplifier (not yet started)">Quantum amplifier</a></li> <li><a href="/w/index.php?title=Quantum_bus&action=edit&redlink=1" class="new" title="Quantum bus (not yet started)">Quantum bus</a></li> <li><a href="/w/index.php?title=Quantum_cellular_automaton&action=edit&redlink=1" class="new" title="Quantum cellular automaton (not yet started)">Quantum cellular automata</a> <ul><li><a href="/w/index.php?title=Quantum_finite_automaton&action=edit&redlink=1" class="new" title="Quantum finite automaton (not yet started)">Quantum finite automata</a></li></ul></li> <li><a href="/w/index.php?title=Quantum_channel&action=edit&redlink=1" class="new" title="Quantum channel (not yet started)">Quantum channel</a></li> <li><a href="/w/index.php?title=Quantum_circuit&action=edit&redlink=1" class="new" title="Quantum circuit (not yet started)">Quantum circuit</a></li> <li><a href="/w/index.php?title=Quantum_complexity_theory&action=edit&redlink=1" class="new" title="Quantum complexity theory (not yet started)">Quantum complexity theory</a></li> <li><a href="/wiki/Quantum_computing" class="mw-redirect" title="Quantum computing">Quantum computing</a> <ul><li><a href="/w/index.php?title=Timeline_of_quantum_computing_and_communication&action=edit&redlink=1" class="new" title="Timeline of quantum computing and communication (not yet started)">Timeline</a></li></ul></li> <li><a href="/w/index.php?title=Quantum_cryptography&action=edit&redlink=1" class="new" title="Quantum cryptography (not yet started)">Quantum cryptography</a></li> <li><a href="/w/index.php?title=Quantum_electronics&action=edit&redlink=1" class="new" title="Quantum electronics (not yet started)">Quantum electronics</a></li> <li><a href="/w/index.php?title=Quantum_error_correction&action=edit&redlink=1" class="new" title="Quantum error correction (not yet started)">Quantum error correction</a></li> <li><a href="/w/index.php?title=Quantum_imaging&action=edit&redlink=1" class="new" title="Quantum imaging (not yet started)">Quantum imaging</a></li> <li><a href="/w/index.php?title=Quantum_image_processing&action=edit&redlink=1" class="new" title="Quantum image processing (not yet started)">Quantum image processing</a></li> <li><a href="/w/index.php?title=Quantum_information&action=edit&redlink=1" class="new" title="Quantum information (not yet started)">Quantum information</a></li> <li><a href="/w/index.php?title=Quantum_key_distribution&action=edit&redlink=1" class="new" title="Quantum key distribution (not yet started)">Quantum key distribution</a></li> <li><a href="/w/index.php?title=Quantum_logic&action=edit&redlink=1" class="new" title="Quantum logic (not yet started)">Quantum logic</a></li> <li><a href="/w/index.php?title=Quantum_logic_gate&action=edit&redlink=1" class="new" title="Quantum logic gate (not yet started)">Quantum logic gates</a></li> <li><a href="/w/index.php?title=Quantum_machine&action=edit&redlink=1" class="new" title="Quantum machine (not yet started)">Quantum machine</a></li> <li><a href="/w/index.php?title=Quantum_machine_learning&action=edit&redlink=1" class="new" title="Quantum machine learning (not yet started)">Quantum machine learning</a></li> <li><a href="/w/index.php?title=Quantum_metamaterial&action=edit&redlink=1" class="new" title="Quantum metamaterial (not yet started)">Quantum metamaterial</a></li> <li><a href="/w/index.php?title=Quantum_metrology&action=edit&redlink=1" class="new" title="Quantum metrology (not yet started)">Quantum metrology</a></li> <li><a href="/w/index.php?title=Quantum_network&action=edit&redlink=1" class="new" title="Quantum network (not yet started)">Quantum network</a></li> <li><a href="/w/index.php?title=Quantum_neural_network&action=edit&redlink=1" class="new" title="Quantum neural network (not yet started)">Quantum neural network</a></li> <li><a href="/w/index.php?title=Quantum_optics&action=edit&redlink=1" class="new" title="Quantum optics (not yet started)">Quantum optics</a></li> <li><a href="/w/index.php?title=Quantum_programming&action=edit&redlink=1" class="new" title="Quantum programming (not yet started)">Quantum programming</a></li> <li><a href="/w/index.php?title=Quantum_sensor&action=edit&redlink=1" class="new" title="Quantum sensor (not yet started)">Quantum sensing</a></li> <li><a href="/w/index.php?title=Quantum_simulator&action=edit&redlink=1" class="new" title="Quantum simulator (not yet started)">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="/w/index.php?title=Quantum_statistical_mechanics&action=edit&redlink=1" class="new" title="Quantum statistical mechanics (not yet started)">Quantum statistical mechanics</a></li> <li><a href="/w/index.php?title=Quantum_field_theory&action=edit&redlink=1" class="new" title="Quantum field theory (not yet started)">Quantum field theory</a> <ul><li><a href="/w/index.php?title=History_of_quantum_field_theory&action=edit&redlink=1" class="new" title="History of quantum field theory (not yet started)">History</a></li></ul></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/w/index.php?title=Relativistic_quantum_mechanics&action=edit&redlink=1" class="new" title="Relativistic quantum mechanics (not yet started)">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="/w/index.php?title=Schr%C3%B6dinger%27s_cat_in_popular_culture&action=edit&redlink=1" class="new" title="Schrödinger's cat in popular culture (not yet started)">in popular culture</a></li></ul></li> <li><a href="/w/index.php?title=Wigner%27s_friend&action=edit&redlink=1" class="new" title="Wigner's friend (not yet started)">Wigner's friend</a></li> <li><a href="/w/index.php?title=Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox&action=edit&redlink=1" class="new" title="Einstein–Podolsky–Rosen paradox (not yet started)">EPR paradox</a></li> <li><a href="/w/index.php?title=Quantum_mysticism&action=edit&redlink=1" class="new" title="Quantum mysticism (not yet started)">Quantum mysticism</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles nomobile"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r9122492"></div><div role="navigation" class="navbox" aria-labelledby="Branches_of_physics" 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:r9122447"><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" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">v</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:Branches_of_physics&action=edit&redlink=1" class="new" title="Template talk:Branches of physics (not yet started)"><abbr title="Discuss this template" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">t</abbr></a></li><li class="nv-edit"><a class="external text" href="https://simple.wikipedia.org/w/index.php?title=Template:Branches_of_physics&action=edit"><abbr title="Edit this template" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">e</abbr></a></li></ul></div><div id="Branches_of_physics" style="font-size:114%;margin:0 4em"><a href="/w/index.php?title=Branches_of_physics&action=edit&redlink=1" class="new" title="Branches of physics (not yet started)">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="/w/index.php?title=Basic_research&action=edit&redlink=1" class="new" title="Basic research (not yet started)">Pure</a></li> <li><a href="/wiki/Applied_physics" title="Applied physics">Applied</a> <ul><li><a href="/w/index.php?title=Engineering_physics&action=edit&redlink=1" class="new" title="Engineering physics (not yet started)">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" class="mw-redirect" title="Theoretical physics">Theoretical</a> <ul><li><a href="/w/index.php?title=Computational_physics&action=edit&redlink=1" class="new" title="Computational physics (not yet started)">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="/w/index.php?title=Analytical_mechanics&action=edit&redlink=1" class="new" title="Analytical mechanics (not yet started)">Analytical</a></li> <li><a href="/wiki/Celestial_mechanics" title="Celestial mechanics">Celestial</a></li> <li><a href="/w/index.php?title=Continuum_mechanics&action=edit&redlink=1" class="new" title="Continuum mechanics (not yet started)">Continuum</a></li></ul></li> <li><a href="/wiki/Acoustics" title="Acoustics">Acoustics</a></li> <li><a href="/w/index.php?title=Classical_electromagnetism&action=edit&redlink=1" class="new" title="Classical electromagnetism (not yet started)">Classical electromagnetism</a></li> <li><a href="/w/index.php?title=Classical_optics&action=edit&redlink=1" class="new" title="Classical optics (not yet started)">Classical optics</a> <ul><li><a href="/w/index.php?title=Geometrical_optics&action=edit&redlink=1" class="new" title="Geometrical optics (not yet started)">Ray</a></li> <li><a href="/w/index.php?title=Physical_optics&action=edit&redlink=1" class="new" title="Physical optics (not yet started)">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="/w/index.php?title=Modern_physics&action=edit&redlink=1" class="new" title="Modern physics (not yet started)">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="/w/index.php?title=Relativistic_mechanics&action=edit&redlink=1" class="new" title="Relativistic mechanics (not yet started)">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 class="mw-selflink selflink">Quantum mechanics</a></li> <li><a href="/wiki/Particle_physics" title="Particle physics">Particle physics</a></li> <li><a href="/w/index.php?title=Atomic,_molecular,_and_optical_physics&action=edit&redlink=1" class="new" title="Atomic, molecular, and optical physics (not yet started)">Atomic, molecular, and optical physics</a> <ul><li><a href="/wiki/Atomic_physics" title="Atomic physics">Atomic</a></li> <li><a href="/w/index.php?title=Molecular_physics&action=edit&redlink=1" class="new" title="Molecular physics (not yet started)">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></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Interdisciplinary</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="/w/index.php?title=Atmospheric_physics&action=edit&redlink=1" class="new" title="Atmospheric physics (not yet started)">Atmospheric physics</a></li> <li><a href="/wiki/Biophysics" title="Biophysics">Biophysics</a></li> <li><a href="/w/index.php?title=Chemical_physics&action=edit&redlink=1" class="new" title="Chemical physics (not yet started)">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="/w/index.php?title=Physical_oceanography&action=edit&redlink=1" class="new" title="Physical oceanography (not yet started)">Ocean physics</a></li> <li><a href="/w/index.php?title=Quantum_information_science&action=edit&redlink=1" class="new" title="Quantum information science (not yet started)">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="/w/index.php?title=History_of_physics&action=edit&redlink=1" class="new" title="History of physics (not yet started)">History of physics</a></li> <li><a href="/wiki/Nobel_Prize_in_Physics" class="mw-redirect" title="Nobel Prize in Physics">Nobel Prize in Physics</a></li> <li><a href="/w/index.php?title=Physics_education&action=edit&redlink=1" class="new" title="Physics education (not yet started)">Physics education</a></li> <li><a href="/w/index.php?title=Timeline_of_fundamental_physics_discoveries&action=edit&redlink=1" class="new" title="Timeline of fundamental physics discoveries (not yet started)">Timeline of physics discoveries</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles nomobile"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r9122492"></div><div role="navigation" class="navbox" aria-labelledby="Chemical_bonding_theory" 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" 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href="/wiki/Chemical_bond" title="Chemical bond">Chemical bonding</a> theory</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div id="*Atomic_orbital_*Quantum_mechanics"> <ul><li><a href="/wiki/Atomic_orbital" title="Atomic orbital">Atomic orbital</a></li> <li><a class="mw-selflink selflink">Quantum mechanics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of bonds</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">By symmetry</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/Sigma_bond" title="Sigma bond">Sigma (σ)</a></li> <li><a href="/wiki/Pi_bond" title="Pi bond">Pi (π)</a></li> <li><a href="/w/index.php?title=Delta_bond&action=edit&redlink=1" class="new" title="Delta bond (not yet started)">Delta (δ)</a></li> <li><a href="/w/index.php?title=Phi_bond&action=edit&redlink=1" class="new" title="Phi bond (not yet started)">Phi (φ)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By <a href="/w/index.php?title=Bond_order&action=edit&redlink=1" class="new" title="Bond order (not yet started)">multiplicity</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/Double_bond" title="Double bond">2 (double)</a></li> <li>3 (triple) (see /<a href="/wiki/Triple_bond" class="mw-redirect" title="Triple bond">triple bond</a>)</li> <li><a href="/w/index.php?title=Single_bond&action=edit&redlink=1" class="new" title="Single bond (not yet started)">1 (single)</a></li> <li><a href="/w/index.php?title=Quadruple_bond&action=edit&redlink=1" class="new" title="Quadruple bond (not yet started)">4 (quadruple)</a></li> <li><a href="/w/index.php?title=Quintuple_bond&action=edit&redlink=1" class="new" title="Quintuple bond (not yet started)">5 (quintuple)</a></li> <li><a href="/w/index.php?title=Sextuple_bond&action=edit&redlink=1" class="new" title="Sextuple bond (not yet started)">6 (sextuple)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</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="/w/index.php?title=Triplet_state&action=edit&redlink=1" class="new" title="Triplet state (not yet started)">Triplet</a></li> <li><a href="/w/index.php?title=Singlet_state&action=edit&redlink=1" class="new" title="Singlet state (not yet started)">Singlet</a></li> <li><a href="/w/index.php?title=Exchange_interaction&action=edit&redlink=1" class="new" title="Exchange interaction (not yet started)">Exchange-coupled</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/w/index.php?title=Valence_bond_theory&action=edit&redlink=1" class="new" title="Valence bond theory (not yet started)">Valence bond theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</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/Lewis_structure" title="Lewis structure">Lewis structure</a></li> <li><a href="/w/index.php?title=Orbital_hybridisation&action=edit&redlink=1" class="new" title="Orbital hybridisation (not yet started)">Hybrid orbital</a></li> <li><a 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