Mathematical formulation of quantum mechanics

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subsection</span> </button> <ul id="toc-History_of_the_formalism-sublist" class="vector-toc-list"> <li id="toc-The_"old_quantum_theory"_and_the_need_for_new_mathematics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_"old_quantum_theory"_and_the_need_for_new_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>The "old quantum theory" and the need for new mathematics</span> </div> </a> <ul id="toc-The_"old_quantum_theory"_and_the_need_for_new_mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_"new_quantum_theory"" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_"new_quantum_theory""> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>The "new quantum theory"</span> </div> </a> <ul id="toc-The_"new_quantum_theory"-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Later_developments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Later_developments"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Later developments</span> </div> </a> <ul id="toc-Later_developments-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Postulates_of_quantum_mechanics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Postulates_of_quantum_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Postulates of quantum mechanics</span> </div> </a> <button aria-controls="toc-Postulates_of_quantum_mechanics-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 Postulates of quantum mechanics subsection</span> </button> <ul id="toc-Postulates_of_quantum_mechanics-sublist" class="vector-toc-list"> <li id="toc-Description_of_the_state_of_a_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Description_of_the_state_of_a_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Description of the state of a system</span> </div> </a> <ul id="toc-Description_of_the_state_of_a_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Measurement_on_a_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Measurement_on_a_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Measurement on a system</span> </div> </a> <ul id="toc-Measurement_on_a_system-sublist" class="vector-toc-list"> <li id="toc-Description_of_physical_quantities" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Description_of_physical_quantities"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Description of physical quantities</span> </div> </a> <ul id="toc-Description_of_physical_quantities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Results_of_measurement" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Results_of_measurement"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Results of measurement</span> </div> </a> <ul id="toc-Results_of_measurement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Effect_of_measurement_on_the_state" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Effect_of_measurement_on_the_state"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.3</span> <span>Effect of measurement on the state</span> </div> </a> <ul id="toc-Effect_of_measurement_on_the_state-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Time_evolution_of_a_system" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Time_evolution_of_a_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Time evolution of a system</span> </div> </a> <ul id="toc-Time_evolution_of_a_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_implications_of_the_postulates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_implications_of_the_postulates"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Other implications of the postulates</span> </div> </a> <ul id="toc-Other_implications_of_the_postulates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spin" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spin"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Spin</span> </div> </a> <ul id="toc-Spin-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symmetrization_postulate" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symmetrization_postulate"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Symmetrization postulate</span> </div> </a> <ul id="toc-Symmetrization_postulate-sublist" class="vector-toc-list"> <li id="toc-Exchange_Degeneracy" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Exchange_Degeneracy"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6.1</span> <span>Exchange Degeneracy</span> </div> </a> <ul id="toc-Exchange_Degeneracy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pauli_exclusion_principle" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Pauli_exclusion_principle"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6.2</span> <span>Pauli exclusion principle</span> </div> </a> <ul id="toc-Pauli_exclusion_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exceptions_for_symmetrization_postulate" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Exceptions_for_symmetrization_postulate"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6.3</span> <span>Exceptions for symmetrization postulate</span> </div> </a> <ul id="toc-Exceptions_for_symmetrization_postulate-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Mathematical_structure_of_quantum_mechanics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mathematical_structure_of_quantum_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Mathematical structure of quantum mechanics</span> </div> </a> <button aria-controls="toc-Mathematical_structure_of_quantum_mechanics-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 Mathematical structure of quantum mechanics subsection</span> </button> <ul id="toc-Mathematical_structure_of_quantum_mechanics-sublist" class="vector-toc-list"> <li id="toc-Pictures_of_dynamics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pictures_of_dynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Pictures of dynamics</span> </div> </a> <ul id="toc-Pictures_of_dynamics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Representations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Representations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Representations</span> </div> </a> <ul id="toc-Representations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Time_as_an_operator" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Time_as_an_operator"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Time as an operator</span> </div> </a> <ul id="toc-Time_as_an_operator-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Problem_of_measurement" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Problem_of_measurement"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Problem of measurement</span> </div> </a> <ul id="toc-Problem_of_measurement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-List_of_mathematical_tools" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#List_of_mathematical_tools"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>List of mathematical tools</span> </div> </a> <ul id="toc-List_of_mathematical_tools-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Mathematical formulation of 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 25 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-25" 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">25 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B5%D9%8A%D8%A7%D8%BA%D8%A9_%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A9_%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" title="صياغة رياضية لميكانيكا الكم – Arabic" lang="ar" hreflang="ar" data-title="صياغة رياضية لميكانيكا الكم" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</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%E0%A7%87%E0%A6%B0_%E0%A6%97%E0%A6%BE%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%B8%E0%A7%82%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%BE%E0%A6%AF%E0%A6%BC%E0%A6%A8" title="কোয়ান্টাম বলবিজ্ঞানের গাণিতিক সূত্রায়ন – Bangla" lang="bn" hreflang="bn" data-title="কোয়ান্টাম বলবিজ্ঞানের গাণিতিক সূত্রায়ন" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Postulats_de_la_mec%C3%A0nica_qu%C3%A0ntica" title="Postulats de la mecànica quàntica – Catalan" lang="ca" hreflang="ca" data-title="Postulats de la 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%C4%83%D0%BD_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%D0%B0_%D0%BD%D0%B8%D0%BA%C4%95%D1%81%C4%95%D1%81%D0%B5%D0%BC" title="Квантла механикăн математикăлла никĕсĕсем – Chuvash" lang="cv" hreflang="cv" data-title="Квантла механикăн математикăлла никĕсĕсем" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Mathematische_Formulierung_der_Quantenmechanik" title="Mathematische Formulierung der Quantenmechanik – German" lang="de" hreflang="de" data-title="Mathematische Formulierung der Quantenmechanik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Postulados_de_la_mec%C3%A1nica_cu%C3%A1ntica" title="Postulados de la mecánica cuántica – Spanish" lang="es" hreflang="es" data-title="Postulados de la 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/Aksiomoj_de_kvantuma_mekaniko" title="Aksiomoj de kvantuma mekaniko – Esperanto" lang="eo" hreflang="eo" data-title="Aksiomoj de 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-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Mekanika_kuantikoaren_formulazio_matematikoa" title="Mekanika kuantikoaren formulazio matematikoa – Basque" lang="eu" hreflang="eu" data-title="Mekanika kuantikoaren formulazio matematikoa" 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%81%D8%B1%D9%85%D9%88%D9%84%E2%80%8C%D8%A8%D9%86%D8%AF%DB%8C_%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C_%D9%85%DA%A9%D8%A7%D9%86%DB%8C%DA%A9_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%85" title="فرمول‌بندی ریاضی مکانیک کوانتم – Persian" lang="fa" hreflang="fa" data-title="فرمول‌بندی ریاضی مکانیک کوانتم" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Postulats_de_la_m%C3%A9canique_quantique" title="Postulats de la mécanique quantique – French" lang="fr" hreflang="fr" data-title="Postulats de la 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-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%96%91%EC%9E%90%EC%97%AD%ED%95%99%EC%9D%98_%EC%88%98%ED%95%99_%EA%B3%B5%EC%8B%9D%ED%99%94" title="양자역학의 수학 공식화 – Korean" lang="ko" hreflang="ko" data-title="양자역학의 수학 공식화" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Matemati%C4%8Dka_formulacija_kvantne_mehanike" title="Matematička formulacija kvantne mehanike – Croatian" lang="hr" hreflang="hr" data-title="Matematička formulacija kvantne mehanike" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Postulati_della_meccanica_quantistica" title="Postulati della meccanica quantistica – Italian" lang="it" hreflang="it" data-title="Postulati della 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%94%D7%A4%D7%95%D7%A1%D7%98%D7%95%D7%9C%D7%98%D7%99%D7%9D_%D7%A9%D7%9C_%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%A7%D7%95%D7%95%D7%A0%D7%98%D7%99%D7%9D" title="הפוסטולטים של תורת הקוונטים – Hebrew" lang="he" hreflang="he" data-title="הפוסטולטים של תורת הקוונטים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Leges_motus_quanticae" title="Leges motus quanticae – Latin" lang="la" hreflang="la" data-title="Leges motus quanticae" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wiskundige_structuur_van_de_kwantummechanica" title="Wiskundige structuur van de kwantummechanica – Dutch" lang="nl" hreflang="nl" data-title="Wiskundige structuur van de kwantummechanica" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</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%E3%81%AE%E6%95%B0%E5%AD%A6%E7%9A%84%E5%AE%9A%E5%BC%8F%E5%8C%96" title="量子力学の数学的定式化 – Japanese" lang="ja" hreflang="ja" data-title="量子力学の数学的定式化" 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-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Postulats_de_la_fisica_quantica" title="Postulats de la fisica quantica – Occitan" lang="oc" hreflang="oc" data-title="Postulats de la fisica quantica" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a 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.sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist nowraplinks" style="width:19.0em;"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1799e4a910c7d26396922a20ef5ceec25ca1871c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.882ex; height:5.509ex;" alt="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }"></span><div class="sidebar-caption" style="font-size:90%;padding-top:0.4em;font-style:italic;"><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a></div></td></tr><tr><td class="sidebar-above hlist nowrap" style="display:block;margin-bottom:0.4em;"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a></li></ul></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Background</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)">Complementarity</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_number" title="Quantum number">Quantum number</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">State</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li></ul></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Experiments</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell's inequality</a></li> <li><a href="/wiki/CHSH_inequality" title="CHSH inequality">CHSH inequality</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Leggett_inequality" title="Leggett inequality">Leggett inequality</a></li> <li><a href="/wiki/Leggett%E2%80%93Garg_inequality" title="Leggett–Garg inequality">Leggett–Garg inequality</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li></ul> </div> <ul><li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a> <ul><li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice</a></li></ul></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed-choice</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Formulations</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a class="mw-selflink selflink">Overview</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase-space</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Sum-over-histories (path integral)</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective-collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Advanced topics</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Yakir_Aharonov" title="Yakir Aharonov">Aharonov</a></li> <li><a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">Bell</a></li> <li><a href="/wiki/Hans_Bethe" title="Hans Bethe">Bethe</a></li> <li><a href="/wiki/Patrick_Blackett" title="Patrick Blackett">Blackett</a></li> <li><a href="/wiki/Felix_Bloch" title="Felix Bloch">Bloch</a></li> <li><a href="/wiki/David_Bohm" title="David Bohm">Bohm</a></li> <li><a href="/wiki/Niels_Bohr" title="Niels Bohr">Bohr</a></li> <li><a href="/wiki/Max_Born" title="Max Born">Born</a></li> <li><a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Bose</a></li> <li><a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">de Broglie</a></li> <li><a href="/wiki/Arthur_Compton" title="Arthur Compton">Compton</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Clinton_Davisson" title="Clinton Davisson">Davisson</a></li> <li><a href="/wiki/Peter_Debye" title="Peter Debye">Debye</a></li> <li><a href="/wiki/Paul_Ehrenfest" title="Paul Ehrenfest">Ehrenfest</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Hugh_Everett_III" title="Hugh Everett III">Everett</a></li> <li><a href="/wiki/Vladimir_Fock" title="Vladimir Fock">Fock</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a></li> <li><a href="/wiki/Roy_J._Glauber" title="Roy J. Glauber">Glauber</a></li> <li><a href="/wiki/Martin_Gutzwiller" title="Martin Gutzwiller">Gutzwiller</a></li> <li><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a></li> <li><a href="/wiki/Hans_Kramers" title="Hans Kramers">Kramers</a></li> <li><a href="/wiki/Willis_Lamb" title="Willis Lamb">Lamb</a></li> <li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li> <li><a href="/wiki/Max_von_Laue" title="Max von Laue">Laue</a></li> <li><a href="/wiki/Henry_Moseley" title="Henry Moseley">Moseley</a></li> <li><a href="/wiki/Robert_Andrews_Millikan" title="Robert Andrews Millikan">Millikan</a></li> <li><a href="/wiki/Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Onnes</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/Isidor_Isaac_Rabi" title="Isidor Isaac Rabi">Rabi</a></li> <li><a href="/wiki/C._V._Raman" title="C. V. Raman">Raman</a></li> <li><a href="/wiki/Johannes_Rydberg" title="Johannes Rydberg">Rydberg</a></li> <li><a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödinger</a></li> <li><a href="/wiki/Michelle_Simmons" title="Michelle Simmons">Simmons</a></li> <li><a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Sommerfeld</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Wilhelm_Wien" title="Wilhelm Wien">Wien</a></li> <li><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner</a></li> <li><a href="/wiki/Pieter_Zeeman" title="Pieter Zeeman">Zeeman</a></li> <li><a href="/wiki/Anton_Zeilinger" title="Anton Zeilinger">Zeilinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar" style="border-top:1px solid #aaa;padding-top:0.1em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_mechanics" title="Template:Quantum mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics" title="Template talk:Quantum mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics" title="Special:EditPage/Template:Quantum mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>The <b>mathematical formulations of quantum mechanics</b> are those <a href="/wiki/Formalism_(mathematics)" class="mw-redirect" title="Formalism (mathematics)">mathematical formalisms</a> that permit a <a href="/wiki/Rigour#Mathematics" title="Rigour">rigorous</a> description of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>. This mathematical formalism uses mainly a part of <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>, especially <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a>, which are a kind of <a href="/wiki/Linear_space" class="mw-redirect" title="Linear space">linear space</a>. Such are distinguished from mathematical formalisms for physics theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a> (<a href="/wiki/L2_space" class="mw-redirect" title="L2 space"><i>L</i><sup>2</sup> space</a> mainly), and <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">operators</a> on these spaces. In brief, values of physical <a href="/wiki/Observable" title="Observable">observables</a> such as <a href="/wiki/Energy" title="Energy">energy</a> and <a href="/wiki/Momentum" title="Momentum">momentum</a> were no longer considered as values of <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> on <a href="/wiki/Phase_space" title="Phase space">phase space</a>, but as <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a>; more precisely as <a href="/wiki/Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">spectral values</a> of linear <a href="/wiki/Operator_(physics)" title="Operator (physics)">operators</a> in Hilbert space.<sup id="cite_ref-FOOTNOTEByronFuller1992277_1-0" class="reference"><a href="#cite_note-FOOTNOTEByronFuller1992277-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>These formulations of quantum mechanics continue to be used today. At the heart of the description are ideas of <i><a href="/wiki/Quantum_state" title="Quantum state">quantum state</a></i> and <i>quantum observables</i>, which are radically different from those used in previous <a href="/wiki/Mathematical_model" title="Mathematical model">models</a> of physical reality. While the mathematics permits calculation of many quantities that can be measured experimentally, there is a definite theoretical limit to values that can be simultaneously measured. This limitation was first elucidated by <a href="/wiki/Heisenberg_uncertainty_relations" class="mw-redirect" title="Heisenberg uncertainty relations">Heisenberg</a> through a <a href="/wiki/Thought_experiment" title="Thought experiment">thought experiment</a>, and is represented mathematically in the new formalism by the <a href="/wiki/Non-commutative" class="mw-redirect" title="Non-commutative">non-commutativity</a> of operators representing quantum observables. </p><p>Prior to the development of quantum mechanics as a separate <a href="/wiki/Theory" title="Theory">theory</a>, the mathematics used in physics consisted mainly of formal <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, beginning with <a href="/wiki/Calculus" title="Calculus">calculus</a>, and increasing in complexity up to <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a> and <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a>. <a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a> was used in <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>. Geometric intuition played a strong role in the first two and, accordingly, <a href="/wiki/Relativity_physics" class="mw-redirect" title="Relativity physics">theories of relativity</a> were formulated entirely in terms of differential geometric concepts. The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the development of quantum mechanics (around 1925) physicists continued to think of quantum theory within the confines of what is now called <a href="/wiki/Classical_physics" title="Classical physics">classical physics</a>, and in particular within the same mathematical structures. The most sophisticated example of this is the <a href="/wiki/Sommerfeld%E2%80%93Wilson%E2%80%93Ishiwara_quantization" class="mw-redirect" title="Sommerfeld–Wilson–Ishiwara quantization">Sommerfeld–Wilson–Ishiwara quantization</a> rule, which was formulated entirely on the classical <a href="/wiki/Phase_space" title="Phase space">phase space</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History_of_the_formalism">History of the formalism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=1" title="Edit section: History of the formalism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="The_"old_quantum_theory"_and_the_need_for_new_mathematics"><span id="The_.22old_quantum_theory.22_and_the_need_for_new_mathematics"></span>The "old quantum theory" and the need for new mathematics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=2" title="Edit section: The "old quantum theory" and the need for new mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></div> <p>In the 1890s, <a href="/wiki/Max_Planck" title="Max Planck">Planck</a> was able to derive the <a href="/wiki/Blackbody_spectrum" class="mw-redirect" title="Blackbody spectrum">blackbody spectrum</a>, which was later used to avoid the classical <a href="/wiki/Ultraviolet_catastrophe" title="Ultraviolet catastrophe">ultraviolet catastrophe</a> by making the unorthodox assumption that, in the interaction of <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a> with <a href="/wiki/Matter" title="Matter">matter</a>, energy could only be exchanged in discrete units which he called <a href="/wiki/Quantum" title="Quantum">quanta</a>. Planck postulated a direct proportionality between the frequency of radiation and the quantum of energy at that frequency. The proportionality constant, <span class="texhtml"><i>h</i></span>, is now called the <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a> in his honor. </p><p>In 1905, <a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a> explained certain features of the <a href="/wiki/Photoelectric_effect" title="Photoelectric effect">photoelectric effect</a> by assuming that Planck's energy quanta were actual particles, which were later dubbed <a href="/wiki/Photons" class="mw-redirect" title="Photons">photons</a>. </p> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Bohr_atom_model_(mul).svg" class="mw-file-description" title="light at the right frequency"><img alt="light at the right frequency" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Bohr_atom_model_%28mul%29.svg/300px-Bohr_atom_model_%28mul%29.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Bohr_atom_model_%28mul%29.svg/450px-Bohr_atom_model_%28mul%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Bohr_atom_model_%28mul%29.svg/600px-Bohr_atom_model_%28mul%29.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption> light at the right frequency</figcaption></figure> <p>All of these developments were <a href="/wiki/Phenomenology_(particle_physics)" class="mw-redirect" title="Phenomenology (particle physics)">phenomenological</a> and challenged the theoretical physics of the time. <a href="/wiki/Old_quantum_theory" title="Old quantum theory">Bohr and Sommerfeld</a> went on to modify <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> in an attempt to deduce the <a href="/wiki/Bohr_model" title="Bohr model">Bohr model</a> from first principles. They proposed that, of all closed classical orbits traced by a mechanical system in its phase space, only the ones that enclosed an area which was a multiple of the Planck constant were actually allowed. The most sophisticated version of this formalism was the so-called <a href="/wiki/Sommerfeld%E2%80%93Wilson%E2%80%93Ishiwara_quantization" class="mw-redirect" title="Sommerfeld–Wilson–Ishiwara quantization">Sommerfeld–Wilson–Ishiwara quantization</a>. Although the Bohr model of the hydrogen atom could be explained in this way, the spectrum of the helium atom (classically an unsolvable <a href="/wiki/3-body_problem" class="mw-redirect" title="3-body problem">3-body problem</a>) could not be predicted. The mathematical status of quantum theory remained uncertain for some time. </p><p>In 1923, <a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">de Broglie</a> proposed that <a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">wave–particle duality</a> applied not only to photons but to electrons and every other physical system. </p><p>The situation changed rapidly in the years 1925–1930, when working mathematical foundations were found through the groundbreaking work of <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a>, <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a>, <a href="/wiki/Max_Born" title="Max Born">Max Born</a>, <a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Pascual Jordan</a>, and the foundational work of <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a>, <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> and <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a>, and it became possible to unify several different approaches in terms of a fresh set of ideas. The physical interpretation of the theory was also clarified in these years after <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a> discovered the uncertainty relations and <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a> introduced the idea of <a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)">complementarity</a>. </p> <div class="mw-heading mw-heading3"><h3 id="The_"new_quantum_theory""><span id="The_.22new_quantum_theory.22"></span>The "new quantum theory"</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=3" title="Edit section: The "new quantum theory""><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Werner Heisenberg's <a href="/wiki/Matrix_mechanics" title="Matrix mechanics">matrix mechanics</a> was the first successful attempt at replicating the observed quantization of <a href="/wiki/Atomic_spectra" class="mw-redirect" title="Atomic spectra">atomic spectra</a>. Later in the same year, Schrödinger created his <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">wave mechanics</a>. Schrödinger's formalism was considered easier to understand, visualize and calculate as it led to <a href="/wiki/Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a>, which physicists were already familiar with solving. Within a year, it was shown that the two theories were equivalent. </p><p>Schrödinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought that the <a href="/wiki/Absolute_value#Complex_numbers" title="Absolute value">absolute square</a> of the wave function of an <a href="/wiki/Electron" title="Electron">electron</a> should be interpreted as the <a href="/wiki/Charge_density" title="Charge density">charge density</a> of an object smeared out over an extended, possibly infinite, volume of space. It was <a href="/wiki/Max_Born" title="Max Born">Max Born</a> who introduced the interpretation of the <a href="/wiki/Absolute_value#Complex_numbers" title="Absolute value">absolute square</a> of the wave function as the probability distribution of the position of a <a href="/wiki/Point_particle" title="Point particle"><i>pointlike</i> object</a>. Born's idea was soon taken over by Niels Bohr in Copenhagen who then became the "father" of the <a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen interpretation</a> of quantum mechanics. Schrödinger's <a href="/wiki/Wave_function" title="Wave function">wave function</a> can be seen to be closely related to the classical <a href="/wiki/Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a>. The correspondence to classical mechanics was even more explicit, although somewhat more formal, in Heisenberg's matrix mechanics. In his PhD thesis project, <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a><sup id="cite_ref-FOOTNOTEDirac1925_2-0" class="reference"><a href="#cite_note-FOOTNOTEDirac1925-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> discovered that the equation for the operators in the <a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg representation</a>, as it is now called, closely translates to classical equations for the dynamics of certain quantities in the Hamiltonian formalism of classical mechanics, when one expresses them through <a href="/wiki/Poisson_bracket" title="Poisson bracket">Poisson brackets</a>, a procedure now known as <a href="/wiki/Canonical_quantization" title="Canonical quantization">canonical quantization</a>. </p><p>Already before Schrödinger, the young postdoctoral fellow Werner Heisenberg invented his <a href="/wiki/Matrix_mechanics" title="Matrix mechanics">matrix mechanics</a>, which was the first correct quantum mechanics – the essential breakthrough. Heisenberg's matrix mechanics formulation was based on algebras of infinite matrices, a very radical formulation in light of the mathematics of classical physics, although he started from the index-terminology of the experimentalists of that time, not even aware that his "index-schemes" were matrices, as Born soon pointed out to him. In fact, in these early years, <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> was not generally popular with physicists in its present form. </p><p>Although Schrödinger himself after a year proved the equivalence of his wave-mechanics and Heisenberg's matrix mechanics, the reconciliation of the two approaches and their modern abstraction as motions in Hilbert space is generally attributed to Paul Dirac, who wrote a lucid account in his 1930 classic <i><a href="/wiki/The_Principles_of_Quantum_Mechanics" title="The Principles of Quantum Mechanics">The Principles of Quantum Mechanics</a></i>. He is the third, and possibly most important, pillar of that field (he soon was the only one to have discovered a relativistic generalization of the theory). In his above-mentioned account, he introduced the <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">bra–ket notation</a>, together with an abstract formulation in terms of the <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> used in functional analysis; he showed that Schrödinger's and Heisenberg's approaches were two different representations of the same theory, and found a third, most general one, which represented the dynamics of the system. His work was particularly fruitful in many types of generalizations of the field. </p><p>The first complete mathematical formulation of this approach, known as the <a href="/wiki/Dirac%E2%80%93von_Neumann_axioms" title="Dirac–von Neumann axioms">Dirac–von Neumann axioms</a>, is generally credited to <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a>'s 1932 book <i><a href="/wiki/Mathematical_Foundations_of_Quantum_Mechanics" title="Mathematical Foundations of Quantum Mechanics">Mathematical Foundations of Quantum Mechanics</a></i>, although <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> had already referred to Hilbert spaces (which he called <i>unitary spaces</i>) in his 1927 classic paper and book. It was developed in parallel with a new approach to the mathematical <a href="/wiki/Spectral_theory" title="Spectral theory">spectral theory</a> based on linear operators rather than the <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic forms</a> that were <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>'s approach a generation earlier. Though theories of quantum mechanics continue to evolve to this day, there is a basic framework for the mathematical formulation of quantum mechanics which underlies most approaches and can be traced back to the mathematical work of John von Neumann. In other words, discussions about <a href="/wiki/Interpretation_of_quantum_mechanics" class="mw-redirect" title="Interpretation of quantum mechanics"><i>interpretation</i> of the theory</a>, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations. </p> <div class="mw-heading mw-heading3"><h3 id="Later_developments">Later developments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=4" title="Edit section: Later developments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The application of the new quantum theory to electromagnetism resulted in <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, which was developed starting around 1930. Quantum field theory has driven the development of more sophisticated formulations of quantum mechanics, of which the ones presented here are simple special cases. </p> <ul><li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase-space formulation</a> of quantum mechanics & <a href="/wiki/Geometric_quantization" title="Geometric quantization">geometric quantization</a></li> <li><a href="/wiki/Quantum_field_theory_in_curved_spacetime" title="Quantum field theory in curved spacetime">quantum field theory in curved spacetime</a></li> <li><a href="/wiki/Wightman_axioms" title="Wightman axioms">axiomatic</a>, <a href="/wiki/Local_quantum_physics" class="mw-redirect" title="Local quantum physics">algebraic</a> and <a href="/wiki/Constructive_quantum_field_theory" title="Constructive quantum field theory">constructive quantum field theory</a></li> <li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a> <a href="/wiki/Formalism_(mathematics)" class="mw-redirect" title="Formalism (mathematics)">formalism</a></li> <li><a href="/wiki/POVM" title="POVM">Generalized statistical model of quantum mechanics</a></li></ul> <p>A related topic is the relationship to classical mechanics. Any new physical theory is supposed to reduce to successful old theories in some approximation. For quantum mechanics, this translates into the need to study the so-called <a href="/wiki/Classical_limit_of_quantum_mechanics" class="mw-redirect" title="Classical limit of quantum mechanics">classical limit of quantum mechanics</a>. Also, as Bohr emphasized, human cognitive abilities and language are inextricably linked to the classical realm, and so classical descriptions are intuitively more accessible than quantum ones. In particular, <a href="/wiki/Quantization_(physics)" title="Quantization (physics)">quantization</a>, namely the construction of a quantum theory whose classical limit is a given and known classical theory, becomes an important area of quantum physics in itself. </p><p>Finally, some of the originators of quantum theory (notably Einstein and Schrödinger) were unhappy with what they thought were the philosophical implications of quantum mechanics. In particular, Einstein took the position that quantum mechanics must be incomplete, which motivated research into so-called <a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">hidden-variable theories</a>. The issue of hidden variables has become in part an experimental issue with the help of <a href="/wiki/Quantum_optics" title="Quantum optics">quantum optics</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Postulates_of_quantum_mechanics">Postulates of quantum mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=5" title="Edit section: Postulates of quantum mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A physical system is generally described by three basic ingredients: <a href="/wiki/Quantum_state" title="Quantum state">states</a>; <a href="/wiki/Observable" title="Observable">observables</a>; and <a href="/wiki/Dynamics_(mechanics)" class="mw-redirect" title="Dynamics (mechanics)">dynamics</a> (or law of <a href="/wiki/Time_evolution" title="Time evolution">time evolution</a>) or, more generally, a <a href="/wiki/Gauge_invariance" class="mw-redirect" title="Gauge invariance">group of physical symmetries</a>. A classical description can be given in a fairly direct way by a phase space <a href="/wiki/Model_(abstract)" class="mw-redirect" title="Model (abstract)">model</a> of mechanics: states are points in a phase space formulated by <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">symplectic manifold</a>, observables are real-valued functions on it, time evolution is given by a one-parameter <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> of symplectic transformations of the phase space, and physical symmetries are realized by symplectic transformations. A quantum description normally consists of a <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> of states, observables are <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">self-adjoint operators</a> on the space of states, time evolution is given by a <a href="/wiki/Stone%27s_theorem_on_one-parameter_unitary_groups" title="Stone's theorem on one-parameter unitary groups">one-parameter group</a> of unitary transformations on the Hilbert space of states, and physical symmetries are realized by <a href="/wiki/Unitary_transformation" title="Unitary transformation">unitary transformations</a>. (It is possible, to map this Hilbert-space picture to a <a href="/wiki/Phase_space_formulation" class="mw-redirect" title="Phase space formulation">phase space formulation</a>, invertibly. See below.) </p><p>The following summary of the mathematical framework of quantum mechanics can be partly traced back to the <a href="/wiki/Dirac%E2%80%93von_Neumann_axioms" title="Dirac–von Neumann axioms">Dirac–von Neumann axioms</a>.<sup id="cite_ref-FOOTNOTECohen-TannoudjiDiuLaloë2020_3-0" class="reference"><a href="#cite_note-FOOTNOTECohen-TannoudjiDiuLaloë2020-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Description_of_the_state_of_a_system">Description of the state of a system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=6" title="Edit section: Description of the state of a system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Each isolated physical system is associated with a (topologically) <a href="/wiki/Separable_space" title="Separable space">separable</a> <a href="/wiki/Complex_number" title="Complex number">complex</a> <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> <span class="texhtml"><i>H</i></span> with <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a> <span class="texhtml">⟨<i>φ</i>|<i>ψ</i>⟩</span>. </p> <style data-mw-deduplicate="TemplateStyles:r1224211176">.mw-parser-output .quotebox{background-color:#F9F9F9;border:1px solid #aaa;box-sizing:border-box;padding:10px;font-size:88%;max-width:100%}.mw-parser-output .quotebox.floatleft{margin:.5em 1.4em .8em 0}.mw-parser-output .quotebox.floatright{margin:.5em 0 .8em 1.4em}.mw-parser-output .quotebox.centered{overflow:hidden;position:relative;margin:.5em auto .8em auto}.mw-parser-output .quotebox.floatleft span,.mw-parser-output .quotebox.floatright span{font-style:inherit}.mw-parser-output .quotebox>blockquote{margin:0;padding:0;border-left:0;font-family:inherit;font-size:inherit}.mw-parser-output .quotebox-title{text-align:center;font-size:110%;font-weight:bold}.mw-parser-output .quotebox-quote>:first-child{margin-top:0}.mw-parser-output .quotebox-quote:last-child>:last-child{margin-bottom:0}.mw-parser-output .quotebox-quote.quoted:before{font-family:"Times New Roman",serif;font-weight:bold;font-size:large;color:gray;content:" “ ";vertical-align:-45%;line-height:0}.mw-parser-output .quotebox-quote.quoted:after{font-family:"Times New Roman",serif;font-weight:bold;font-size:large;color:gray;content:" ” ";line-height:0}.mw-parser-output .quotebox .left-aligned{text-align:left}.mw-parser-output .quotebox .right-aligned{text-align:right}.mw-parser-output .quotebox .center-aligned{text-align:center}.mw-parser-output .quotebox .quote-title,.mw-parser-output .quotebox .quotebox-quote{display:block}.mw-parser-output .quotebox cite{display:block;font-style:normal}@media screen and (max-width:640px){.mw-parser-output .quotebox{width:100%!important;margin:0 0 .8em!important;float:none!important}}</style><div class="quotebox pullquote centered" style="width:50%; ;"> <div class="quotebox-title" style="">Postulate I</div> <blockquote class="quotebox-quote center-aligned" style=""> <p>The state of an isolated physical system is represented, at a fixed time <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>,</span> by a state vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"></span> belonging to a Hilbert space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}"></span> called the <i>state space</i>. </p> </blockquote> </div> <p><a href="/wiki/Hilbert_space#Separable_spaces" title="Hilbert space">Separability</a> is a mathematically convenient hypothesis, with the physical interpretation that the state is uniquely determined by countably many observations. Quantum states can be identified with <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a> in <span class="texhtml"><i>H</i></span>, where two vectors (of length 1) represent the same state if they differ only by a <a href="/wiki/Phase_factor" title="Phase factor">phase factor</a>.<sup id="cite_ref-FOOTNOTEBäuerlede_Kerf1990330_4-0" class="reference"><a href="#cite_note-FOOTNOTEBäuerlede_Kerf1990330-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> As such, quantum states form a <a href="/wiki/Wigner%27s_theorem#Rays_and_ray_space" title="Wigner's theorem"><b>ray</b></a> in <a href="/wiki/Projective_Hilbert_space" title="Projective Hilbert space">projective Hilbert space</a>, not a <i>vector</i>. Many textbooks fail to make this distinction, which could be partly a result of the fact that the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> itself involves Hilbert-space "vectors", with the result that the imprecise use of "state vector" rather than <i>ray</i> is very difficult to avoid.<sup id="cite_ref-FOOTNOTESolemBiedenharn1993_5-0" class="reference"><a href="#cite_note-FOOTNOTESolemBiedenharn1993-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Accompanying Postulate I is the composite system postulate:<sup id="cite_ref-FOOTNOTEJauchWignerYanase1997_6-0" class="reference"><a href="#cite_note-FOOTNOTEJauchWignerYanase1997-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1224211176"><div class="quotebox pullquote centered" style="width:50%; ;"> <div class="quotebox-title" style="">Composite system postulate</div> <blockquote class="quotebox-quote center-aligned" style=""> <p>The Hilbert space of a composite system is the Hilbert space <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a> of the state spaces associated with the component systems. For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles. </p> </blockquote> </div> <p>In the presence of <a href="/wiki/Quantum_entanglement" title="Quantum entanglement">quantum entanglement</a>, the quantum state of the composite system cannot be factored as a tensor product of states of its local constituents; Instead, it is expressed as a sum, or <a href="/wiki/Quantum_superposition" title="Quantum superposition">superposition</a>, of tensor products of states of component subsystems. A subsystem in an entangled composite system generally cannot be described by a state vector (or a ray), but instead is described by a <a href="/wiki/Density_matrix" title="Density matrix">density operator</a>; Such quantum state is known as a <a href="/wiki/Mixed_state_(physics)" class="mw-redirect" title="Mixed state (physics)">mixed state</a>. The <a href="/wiki/Density_operator" class="mw-redirect" title="Density operator">density operator</a> of a mixed state is a <a href="/wiki/Trace_class" title="Trace class">trace class</a>, nonnegative (<a href="/wiki/Positive_semi-definite_matrix" class="mw-redirect" title="Positive semi-definite matrix">positive semi-definite</a>) <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">self-adjoint</a> operator <span class="texhtml"><i>ρ</i></span> normalized to be of <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> 1. In turn, any <a href="/wiki/Density_operator" class="mw-redirect" title="Density operator">density operator</a> of a mixed state can be represented as a subsystem of a larger composite system in a pure state (see <a href="/wiki/Schr%C3%B6dinger%E2%80%93HJW_theorem" title="Schrödinger–HJW theorem">purification theorem</a>). </p><p>In the absence of quantum entanglement, the quantum state of the composite system is called a <a href="/wiki/Separable_state" title="Separable state">separable state</a>. The density matrix of a bipartite system in a separable state can be expressed as <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 \rho =\sum _{k}p_{k}\rho _{1}^{k}\otimes \rho _{2}^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo>⊗</mo> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =\sum _{k}p_{k}\rho _{1}^{k}\otimes \rho _{2}^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c00e6733edf87e0a9f15defb47eaa19c70227dfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.722ex; height:5.509ex;" alt="{\displaystyle \rho =\sum _{k}p_{k}\rho _{1}^{k}\otimes \rho _{2}^{k}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\sum _{k}p_{k}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thickmathspace" /> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \;\sum _{k}p_{k}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94aa2c10f42d40d5f1928f435fb9770be4b4eabf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.906ex; height:5.509ex;" alt="{\displaystyle \;\sum _{k}p_{k}=1}"></span>. If there is only a single non-zero <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01084a31964201514f3e6bd0136989e11ea6e58a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.348ex; height:2.009ex;" alt="{\displaystyle p_{k}}"></span>, then the state can be expressed just as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \rho =\rho _{1}\otimes \rho _{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \rho =\rho _{1}\otimes \rho _{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/608e503b9fde60b0a1e50faf42de437f4a138758" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.3ex; height:2.509ex;" alt="{\textstyle \rho =\rho _{1}\otimes \rho _{2},}"></span> and is called simply separable or product state. </p> <div class="mw-heading mw-heading3"><h3 id="Measurement_on_a_system">Measurement on a system <span class="anchor" id="Measurement"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=7" title="Edit section: Measurement on a system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Description_of_physical_quantities">Description of physical quantities</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=8" title="Edit section: Description of physical quantities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Physical observables are represented by <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian</a> matrices on <span class="texhtml"><i>H</i></span>. Since these operators are Hermitian, their <a href="/wiki/Eigenvalues" class="mw-redirect" title="Eigenvalues">eigenvalues</a> are always real, and represent the possible outcomes/results from measuring the corresponding observable. If the spectrum of the observable is <a href="/wiki/Discrete_spectrum" class="mw-redirect" title="Discrete spectrum">discrete</a>, then the possible results are <i>quantized</i>. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1224211176"><div class="quotebox pullquote centered" style="width:50%; ;"> <div class="quotebox-title" style="">Postulate II.a</div> <blockquote class="quotebox-quote center-aligned" style=""> <p>Every measurable physical quantity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}"></span> is described by a Hermitian operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> acting in the state space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}"></span>. This operator is an observable, meaning that its <a href="/wiki/Eigenvectors" class="mw-redirect" title="Eigenvectors">eigenvectors</a> form a basis for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}"></span>. The result of measuring a physical quantity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}"></span> must be one of the eigenvalues of the corresponding observable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>. </p> </blockquote> </div> <div class="mw-heading mw-heading4"><h4 id="Results_of_measurement">Results of measurement</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=9" title="Edit section: Results of measurement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By spectral theory, we can associate a <a href="/wiki/Probability_measure" title="Probability measure">probability measure</a> to the values of <span class="texhtml"><i>A</i></span> in any state <span class="texhtml"><i>ψ</i></span>. We can also show that the possible values of the observable <span class="texhtml"><i>A</i></span> in any state must belong to the <a href="/wiki/Spectrum_of_an_operator" class="mw-redirect" title="Spectrum of an operator">spectrum</a> of <span class="texhtml"><i>A</i></span>. The <a href="/wiki/Expected_value" title="Expected value">expectation value</a> (in the sense of probability theory) of the observable <span class="texhtml"><i>A</i></span> for the system in state represented by the unit vector <span class="texhtml"><i>ψ</i></span> ∈ <i>H</i> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \psi |A|\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi |A|\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/252318a6d8c7175811b621b61aee7c17a60f919a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.872ex; height:2.843ex;" alt="{\displaystyle \langle \psi |A|\psi \rangle }"></span>. If we represent the state <span class="texhtml"><i>ψ</i></span> in the basis formed by the eigenvectors of <span class="texhtml"><i>A</i></span>, then the square of the modulus of the component attached to a given eigenvector is the probability of observing its corresponding eigenvalue. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1224211176"><div class="quotebox pullquote centered" style="width:50%; ;"> <div class="quotebox-title" style="">Postulate II.b</div> <blockquote class="quotebox-quote center-aligned" style=""> <p>When the physical quantity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}"></span> is measured on a system in a normalized state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"></span>, the probability of obtaining an eigenvalue (denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span> for discrete spectra and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> for continuous spectra) of the corresponding observable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is given by the <i>amplitude squared</i> of the appropriate wave function (projection onto corresponding eigenvector). </p><p><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 {\begin{aligned}\mathbb {P} (a_{n})&=|\langle a_{n}|\psi \rangle |^{2}&{\text{(Discrete, nondegenerate spectrum)}}\\\mathbb {P} (a_{n})&=\sum _{i}^{g_{n}}|\langle a_{n}^{i}|\psi \rangle |^{2}&{\text{(Discrete, degenerate spectrum)}}\\d\mathbb {P} (\alpha )&=|\langle \alpha |\psi \rangle |^{2}d\alpha &{\text{(Continuous, nondegenerate spectrum)}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(Discrete, nondegenerate spectrum)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(Discrete, degenerate spectrum)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>α</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(Continuous, nondegenerate spectrum)</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbb {P} (a_{n})&=|\langle a_{n}|\psi \rangle |^{2}&{\text{(Discrete, nondegenerate spectrum)}}\\\mathbb {P} (a_{n})&=\sum _{i}^{g_{n}}|\langle a_{n}^{i}|\psi \rangle |^{2}&{\text{(Discrete, degenerate spectrum)}}\\d\mathbb {P} (\alpha )&=|\langle \alpha |\psi \rangle |^{2}d\alpha &{\text{(Continuous, nondegenerate spectrum)}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33be915f8293bed64902bac80763a604f1933b52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; margin-top: -0.295ex; width:66.116ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}\mathbb {P} (a_{n})&=|\langle a_{n}|\psi \rangle |^{2}&{\text{(Discrete, nondegenerate spectrum)}}\\\mathbb {P} (a_{n})&=\sum _{i}^{g_{n}}|\langle a_{n}^{i}|\psi \rangle |^{2}&{\text{(Discrete, degenerate spectrum)}}\\d\mathbb {P} (\alpha )&=|\langle \alpha |\psi \rangle |^{2}d\alpha &{\text{(Continuous, nondegenerate spectrum)}}\end{aligned}}}"></span> </p> </blockquote> </div> <p>For a mixed state <span class="texhtml"><i>ρ</i></span>, the expected value of <span class="texhtml"><i>A</i></span> in the state <span class="texhtml"><i>ρ</i></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (A\rho )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mi>ρ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (A\rho )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d28cfa063e18471d75fcb864c7bb141c7fc1b42f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.571ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} (A\rho )}"></span>, and the probability of obtaining an eigenvalue <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span> in a discrete, nondegenerate spectrum of the corresponding observable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} (a_{n})=\operatorname {tr} (|a_{n}\rangle \langle a_{n}|\rho )=\langle a_{n}|\rho |a_{n}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (a_{n})=\operatorname {tr} (|a_{n}\rangle \langle a_{n}|\rho )=\langle a_{n}|\rho |a_{n}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/278d40a63b7baf5ddb6541fdef996d30c4723720" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.903ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (a_{n})=\operatorname {tr} (|a_{n}\rangle \langle a_{n}|\rho )=\langle a_{n}|\rho |a_{n}\rangle }"></span>. </p><p>If the eigenvalue <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span> has <a href="/wiki/Degenerate_energy_levels" title="Degenerate energy levels">degenerate</a>, orthonormal eigenvectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{|a_{n1}\rangle ,|a_{n2}\rangle ,\dots ,|a_{nm}\rangle \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{|a_{n1}\rangle ,|a_{n2}\rangle ,\dots ,|a_{nm}\rangle \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09af23960c514128535baa221d0d2cb83f48b812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.623ex; height:2.843ex;" alt="{\displaystyle \{|a_{n1}\rangle ,|a_{n2}\rangle ,\dots ,|a_{nm}\rangle \}}"></span>, then the <a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">projection operator</a> onto the eigensubspace can be defined as the identity operator in the eigensubspace: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}=|a_{n1}\rangle \langle a_{n1}|+|a_{n2}\rangle \langle a_{n2}|+\dots +|a_{nm}\rangle \langle a_{nm}|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n}=|a_{n1}\rangle \langle a_{n1}|+|a_{n2}\rangle \langle a_{n2}|+\dots +|a_{nm}\rangle \langle a_{nm}|,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b90f1269526c13f58761555a9d837ad9a76e4e85" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.873ex; height:2.843ex;" alt="{\displaystyle P_{n}=|a_{n1}\rangle \langle a_{n1}|+|a_{n2}\rangle \langle a_{n2}|+\dots +|a_{nm}\rangle \langle a_{nm}|,}"></span> and 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 \mathbb {P} (a_{n})=\operatorname {tr} (P_{n}\rho )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>ρ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (a_{n})=\operatorname {tr} (P_{n}\rho )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c3c2033116d52e813427113d89bc0dd11fe204f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.315ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (a_{n})=\operatorname {tr} (P_{n}\rho )}"></span>. </p><p>Postulates II.a and II.b are collectively known as the <a href="/wiki/Born_rule" title="Born rule">Born rule</a> of quantum mechanics. </p> <div class="mw-heading mw-heading4"><h4 id="Effect_of_measurement_on_the_state">Effect of measurement on the state</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=10" title="Edit section: Effect of measurement on the state"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> When a measurement is performed, only one result is obtained (according to some <a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">interpretations of quantum mechanics</a>). This is modeled mathematically as the processing of additional information from the measurement, confining the probabilities of an immediate second measurement of the same observable. In the case of a discrete, non-degenerate spectrum, two sequential measurements of the same observable will always give the same value assuming the second immediately follows the first. Therefore, the state vector must change as a result of measurement, and <i>collapse</i> onto the eigensubspace associated with the eigenvalue measured. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1224211176"></p><div class="quotebox pullquote centered" style="width:50%; ;"> <div class="quotebox-title" style="">Postulate II.c</div> <blockquote class="quotebox-quote center-aligned" style=""> <p>If the measurement of the physical quantity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280ae03440942ab348c2ca9b8db6b56ffa9618f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.343ex;" alt="{\displaystyle {\mathcal {A}}}"></span> on the system in the state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"></span> gives the result <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span>, then the state of the system immediately after the measurement is the normalized projection of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"></span> onto the eigensubspace associated with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span><br /><br /> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle \quad {\overset {a_{n}}{\Longrightarrow }}\quad {\frac {P_{n}|\psi \rangle }{\sqrt {\langle \psi |P_{n}|\psi \rangle }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo stretchy="false">⟹</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mover> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <msqrt> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle \quad {\overset {a_{n}}{\Longrightarrow }}\quad {\frac {P_{n}|\psi \rangle }{\sqrt {\langle \psi |P_{n}|\psi \rangle }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c64edb7cb949008084af6d89fc28d014366376" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.515ex; height:7.009ex;" alt="{\displaystyle |\psi \rangle \quad {\overset {a_{n}}{\Longrightarrow }}\quad {\frac {P_{n}|\psi \rangle }{\sqrt {\langle \psi |P_{n}|\psi \rangle }}}}"></span> </p> </blockquote> </div> <p>For a mixed state <span class="texhtml"><i>ρ</i></span>, after obtaining an eigenvalue <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span> in a discrete, nondegenerate spectrum of the corresponding observable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, the updated state is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \rho '={\frac {P_{n}\rho P_{n}^{\dagger }}{\operatorname {tr} (P_{n}\rho P_{n}^{\dagger })}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>ρ</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>ρ</mi> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> </mrow> <mrow> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>ρ</mi> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \rho '={\frac {P_{n}\rho P_{n}^{\dagger }}{\operatorname {tr} (P_{n}\rho P_{n}^{\dagger })}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae9468a1335a8622e9d82a14f3f38f61617b508" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.3ex; height:5.843ex;" alt="{\textstyle \rho '={\frac {P_{n}\rho P_{n}^{\dagger }}{\operatorname {tr} (P_{n}\rho P_{n}^{\dagger })}}}"></span>. If the eigenvalue <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span> has degenerate, orthonormal eigenvectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{|a_{n1}\rangle ,|a_{n2}\rangle ,\dots ,|a_{nm}\rangle \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{|a_{n1}\rangle ,|a_{n2}\rangle ,\dots ,|a_{nm}\rangle \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09af23960c514128535baa221d0d2cb83f48b812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.623ex; height:2.843ex;" alt="{\displaystyle \{|a_{n1}\rangle ,|a_{n2}\rangle ,\dots ,|a_{nm}\rangle \}}"></span>, then the <a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">projection operator</a> onto the eigensubspace is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}=|a_{n1}\rangle \langle a_{n1}|+|a_{n2}\rangle \langle a_{n2}|+\dots +|a_{nm}\rangle \langle a_{nm}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n}=|a_{n1}\rangle \langle a_{n1}|+|a_{n2}\rangle \langle a_{n2}|+\dots +|a_{nm}\rangle \langle a_{nm}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dedc74167dcda12b1d758e8bfdbae71a2b839c0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.226ex; height:2.843ex;" alt="{\displaystyle P_{n}=|a_{n1}\rangle \langle a_{n1}|+|a_{n2}\rangle \langle a_{n2}|+\dots +|a_{nm}\rangle \langle a_{nm}|}"></span>. </p><p>Postulates II.c is sometimes called the "state update rule" or "collapse rule"; Together with the Born rule (Postulates II.a and II.b), they form a complete representation of <a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">measurements</a>, and are sometimes collectively called the measurement postulate(s). </p><p>Note that the <a href="/wiki/Projection-valued_measure" title="Projection-valued measure">projection-valued measures</a> (PVM) described in the measurement postulate(s) can be generalized to <a href="/wiki/POVM" title="POVM">positive operator-valued measures</a> (POVM), which is the most general kind of measurement in quantum mechanics. A POVM can be understood as the effect on a component subsystem when a PVM is performed on a larger, composite system (see <a href="/wiki/Naimark%27s_dilation_theorem" title="Naimark's dilation theorem">Naimark's dilation theorem</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Time_evolution_of_a_system">Time evolution of a system</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=11" title="Edit section: Time evolution of a system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Though it is possible to derive the Schrödinger equation, which describes how a state vector evolves in time, most texts assert the equation as a postulate. Common derivations include using the <a href="/wiki/De_Broglie_Hypothesis" class="mw-redirect" title="De Broglie Hypothesis">de Broglie hypothesis</a> or <a href="/wiki/Relation_between_Schr%C3%B6dinger%27s_equation_and_the_path_integral_formulation_of_quantum_mechanics" title="Relation between Schrödinger's equation and the path integral formulation of quantum mechanics">path integrals</a>. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1224211176"><div class="quotebox pullquote centered" style="width:50%; ;"> <div class="quotebox-title" style="">Postulate III</div> <blockquote class="quotebox-quote center-aligned" style=""> <p>The time evolution of the state vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi (t)\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi (t)\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe09d6c91bfdbae7a3aaa7f0ae7ff6b96f521eca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.714ex; height:2.843ex;" alt="{\displaystyle |\psi (t)\rangle }"></span> is governed by the Schrödinger equation, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d1b6c8837aed2794e7b52afa88ad371f1d275fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.713ex; height:2.843ex;" alt="{\displaystyle H(t)}"></span> is the observable associated with the total energy of the system (called the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>)<br /><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {d}{dt}}|\psi (t)\rangle =H(t)|\psi (t)\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {d}{dt}}|\psi (t)\rangle =H(t)|\psi (t)\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2ed1452137707fc2285fc3ba091b7354ccfdfe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.239ex; height:5.509ex;" alt="{\displaystyle i\hbar {\frac {d}{dt}}|\psi (t)\rangle =H(t)|\psi (t)\rangle }"></span> </p> </blockquote> </div> <p>Equivalently, the time evolution postulate can be stated as: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1224211176"><div class="quotebox pullquote centered" style="width:50%; ;"> <div class="quotebox-title" style="">Postulate III</div> <blockquote class="quotebox-quote center-aligned" style=""> <p>The time evolution of a <a href="/wiki/Closed_system#In_quantum_physics" title="Closed system">closed system</a> is described by a <a href="/wiki/Unitary_transformation_(quantum_mechanics)" title="Unitary transformation (quantum mechanics)">unitary transformation</a> on the initial state.<br /><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi (t)\rangle =U(t;t_{0})|\psi (t_{0})\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>;</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi (t)\rangle =U(t;t_{0})|\psi (t_{0})\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06d099717b3d59bdf1dd63356381981e62a4bb88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.939ex; height:2.843ex;" alt="{\displaystyle |\psi (t)\rangle =U(t;t_{0})|\psi (t_{0})\rangle }"></span> </p> </blockquote> </div> <p>For a closed system in a mixed state <span class="texhtml"><i>ρ</i></span>, the time evolution is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (t)=U(t;t_{0})\rho (t_{0})U^{\dagger }(t;t_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>;</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>;</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (t)=U(t;t_{0})\rho (t_{0})U^{\dagger }(t;t_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c14975f37bef25c3cea88f070e2a26661790eee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.594ex; height:3.176ex;" alt="{\displaystyle \rho (t)=U(t;t_{0})\rho (t_{0})U^{\dagger }(t;t_{0})}"></span>. </p><p>The evolution of an <a href="/wiki/Open_quantum_system" title="Open quantum system">open quantum system</a> can be described by <a href="/wiki/Quantum_operation" title="Quantum operation">quantum operations</a> (in an <a href="/wiki/Quantum_operation#Statement_of_the_theorem" title="Quantum operation">operator sum</a> formalism) and <a href="/wiki/Quantum_instrument" title="Quantum instrument">quantum instruments</a>, and generally does not have to be unitary. </p> <div class="mw-heading mw-heading3"><h3 id="Other_implications_of_the_postulates">Other implications of the postulates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=12" title="Edit section: Other implications of the postulates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Physical symmetries act on the Hilbert space of quantum states <a href="/wiki/Unitary_operator" title="Unitary operator">unitarily</a> or <a href="/wiki/Antiunitary" class="mw-redirect" title="Antiunitary">antiunitarily</a> due to <a href="/wiki/Wigner%27s_theorem" title="Wigner's theorem">Wigner's theorem</a> (<a href="/wiki/Supersymmetry" title="Supersymmetry">supersymmetry</a> is another matter entirely).</li> <li>Density operators are those that are in the closure of the <a href="/wiki/Convex_hull" title="Convex hull">convex hull</a> of the one-dimensional orthogonal projectors. Conversely, one-dimensional orthogonal projectors are <a href="/wiki/Extreme_point" title="Extreme point">extreme points</a> of the set of density operators. Physicists also call one-dimensional orthogonal projectors <i>pure states</i> and other density operators <i>mixed states</i>.</li> <li>One can in this formalism state Heisenberg's <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a> and <a href="/wiki/Mathematical_proof" title="Mathematical proof">prove</a> it as a theorem, although the exact historical sequence of events, concerning who derived what and under which framework, is the subject of historical investigations outside the scope of this article.</li> <li>Recent research has shown<sup id="cite_ref-FOOTNOTECarcassiMacconeAidala2021_7-0" class="reference"><a href="#cite_note-FOOTNOTECarcassiMacconeAidala2021-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> that the composite system postulate (tensor product postulate) can be derived from the state postulate (Postulate I) and the measurement postulates (Postulates II); Moreover, it has also been shown<sup id="cite_ref-FOOTNOTEMasanesGalleyMüller2019_8-0" class="reference"><a href="#cite_note-FOOTNOTEMasanesGalleyMüller2019-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> that the measurement postulates (Postulates II) can be derived from "unitary quantum mechanics", which includes only the state postulate (Postulate I), the composite system postulate (tensor product postulate) and the unitary evolution postulate (Postulate III).</li></ul> <p>Furthermore, to the postulates of quantum mechanics one should also add basic statements on the properties of <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> and Pauli's <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">exclusion principle</a>, see below. </p> <div class="mw-heading mw-heading3"><h3 id="Spin">Spin</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=13" title="Edit section: Spin"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In addition to their other properties, all particles possess a quantity called <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a>, an <i>intrinsic angular momentum</i>. Despite the name, particles do not literally spin around an axis, and quantum mechanical spin has no correspondence in classical physics. In the position representation, a spinless wavefunction has position <span class="texhtml"><b>r</b></span> and time <span class="texhtml"><i>t</i></span> as continuous variables, <span class="texhtml"><i>ψ</i> = <i>ψ</i>(<b>r</b>, <i>t</i>)</span>. For spin wavefunctions the spin is an additional discrete variable: <span class="texhtml"><i>ψ</i> = <i>ψ</i>(<b>r</b>, <i>t</i>, <i>σ</i>)</span>, where <span class="texhtml"><i>σ</i></span> takes the values; <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =-S\hbar ,-(S-1)\hbar ,\dots ,0,\dots ,+(S-1)\hbar ,+S\hbar \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mo>−</mo> <mi>S</mi> <mi class="MJX-variant">ℏ</mi> <mo>,</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi class="MJX-variant">ℏ</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi class="MJX-variant">ℏ</mi> <mo>,</mo> <mo>+</mo> <mi>S</mi> <mi class="MJX-variant">ℏ</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =-S\hbar ,-(S-1)\hbar ,\dots ,0,\dots ,+(S-1)\hbar ,+S\hbar \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d964af556126ace732f6cd03a29584089b32ae1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.128ex; height:2.843ex;" alt="{\displaystyle \sigma =-S\hbar ,-(S-1)\hbar ,\dots ,0,\dots ,+(S-1)\hbar ,+S\hbar \,.}"></span> </p><p>That is, the state of a single particle with spin <span class="texhtml"><i>S</i></span> is represented by a <span class="texhtml">(2<i>S</i> + 1)</span>-component <a href="/wiki/Spinor" title="Spinor">spinor</a> of complex-valued wave functions. </p><p>Two classes of particles with <i>very different</i> behaviour are <a href="/wiki/Boson" title="Boson">bosons</a> which have integer spin (<span class="texhtml"><i>S</i> = 0, 1, 2, ...</span>), and <a href="/wiki/Fermion" title="Fermion">fermions</a> possessing half-integer spin (<span class="texhtml"><i>S</i> = <style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">3</span>⁄<span class="den">2</span></span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">5</span>⁄<span class="den">2</span></span>, ...</span>). </p> <div class="mw-heading mw-heading3"><h3 id="Symmetrization_postulate">Symmetrization postulate</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=14" title="Edit section: Symmetrization postulate"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Identical_particles" class="mw-redirect" title="Identical particles">Identical particles</a></div> <p>In quantum mechanics, two particles can be distinguished from one another using two methods. By performing a measurement of intrinsic properties of each particle, particles of different types can be distinguished. Otherwise, if the particles are identical, their trajectories can be tracked which distinguishes the particles based on the locality of each particle. While the second method is permitted in classical mechanics, (i.e. all classical particles are treated with distinguishability), the same cannot be said for quantum mechanical particles since the process is infeasible due to the fundamental uncertainty principles that govern small scales. Hence the requirement of indistinguishability of quantum particles is presented by the symmetrization postulate. The postulate is applicable to a system of bosons or fermions, for example, in predicting the spectra of <a href="/wiki/Helium_atom" title="Helium atom">helium atom</a>. The postulate, explained in the following sections, can be stated as follows: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1224211176"><div class="quotebox pullquote centered" style="width:50%; ;"> <div class="quotebox-title" style="">Symmetrization postulate<sup id="cite_ref-FOOTNOTESakuraiNapolitano2021443_9-0" class="reference"><a href="#cite_note-FOOTNOTESakuraiNapolitano2021443-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></div> <blockquote class="quotebox-quote center-aligned" style=""> <p>The wavefunction of a system of <i>N</i> identical particles (in 3D) is either totally symmetric (Bosons) or totally antisymmetric (Fermions) under interchange of any pair of particles. </p> </blockquote> </div> <p>Exceptions can occur when the particles are constrained to two spatial dimensions where existence of particles known as <a href="/wiki/Anyon" title="Anyon">anyons</a> are possible which are said to have a continuum of statistical properties spanning the range between fermions and bosons.<sup id="cite_ref-FOOTNOTESakuraiNapolitano2021443_9-1" class="reference"><a href="#cite_note-FOOTNOTESakuraiNapolitano2021443-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The connection between behaviour of identical particles and their spin is given by <a href="/wiki/Spin%E2%80%93statistics_theorem" title="Spin–statistics theorem">spin statistics theorem</a>. </p><p>It can be shown that two particles localized in different regions of space can still be represented using a symmetrized/antisymmetrized wavefunction and that independent treatment of these wavefunctions gives the same result.<sup id="cite_ref-FOOTNOTESakuraiNapolitano2021434-437_10-0" class="reference"><a href="#cite_note-FOOTNOTESakuraiNapolitano2021434-437-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Hence the symmetrization postulate is applicable in the general case of a system of identical particles. </p> <div class="mw-heading mw-heading4"><h4 id="Exchange_Degeneracy">Exchange Degeneracy</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=15" title="Edit section: Exchange Degeneracy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a system of identical particles, let <i>P</i> be known as exchange operator that acts on the wavefunction 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 P{\bigg (}\cdots |\psi \rangle |\phi \rangle \cdots {\bigg )}\equiv \cdots |\phi \rangle |\psi \rangle \cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mo>⋯</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>⋯</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mo>≡</mo> <mo>⋯</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P{\bigg (}\cdots |\psi \rangle |\phi \rangle \cdots {\bigg )}\equiv \cdots |\phi \rangle |\psi \rangle \cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5319b1a03a89e0b26e1604d1e652a1a66b5828" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.484ex; height:6.176ex;" alt="{\displaystyle P{\bigg (}\cdots |\psi \rangle |\phi \rangle \cdots {\bigg )}\equiv \cdots |\phi \rangle |\psi \rangle \cdots }"></span></dd></dl> <p>If a physical system of identical particles is given, wavefunction of all particles can be well known from observation but these cannot be labelled to each particle. Thus, the above exchanged wavefunction represents the same physical state as the original state which implies that the wavefunction is not unique. This is known as exchange degeneracy.<sup id="cite_ref-FOOTNOTECohen-TannoudjiDiuLaloë20201375–1377_11-0" class="reference"><a href="#cite_note-FOOTNOTECohen-TannoudjiDiuLaloë20201375–1377-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>More generally, consider a linear combination of such states, <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 \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e77f6b1e903837c5765c9683da41dd93199621c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |\Psi \rangle }"></span>. For the best representation of the physical system, we expect this to be an eigenvector of <i>P</i> since exchange operator is not excepted to give completely different vectors in projective Hilbert space. Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/179a59cf0e092f5b41c771bbf84ba484200b44c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.137ex; height:2.676ex;" alt="{\displaystyle P^{2}=1}"></span>, the possible eigenvalues of <i>P</i> are +1 and −1. 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 |\Psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e77f6b1e903837c5765c9683da41dd93199621c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |\Psi \rangle }"></span> states for identical particle system are represented as symmetric for +1 eigenvalue or antisymmetric for -1 eigenvalue as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P|\cdots n_{i},n_{j}\cdots ;S\rangle =+|\cdots n_{i},n_{j}\cdots ;S\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋯</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>⋯</mo> <mo>;</mo> <mi>S</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋯</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>⋯</mo> <mo>;</mo> <mi>S</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P|\cdots n_{i},n_{j}\cdots ;S\rangle =+|\cdots n_{i},n_{j}\cdots ;S\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03c442a87a1bbbcfb048c39ebfeb727db4fc71c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.877ex; height:3.009ex;" alt="{\displaystyle P|\cdots n_{i},n_{j}\cdots ;S\rangle =+|\cdots n_{i},n_{j}\cdots ;S\rangle }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P|\cdots n_{i},n_{j}\cdots ;A\rangle =-|\cdots n_{i},n_{j}\cdots ;A\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋯</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>⋯</mo> <mo>;</mo> <mi>A</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋯</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>⋯</mo> <mo>;</mo> <mi>A</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P|\cdots n_{i},n_{j}\cdots ;A\rangle =-|\cdots n_{i},n_{j}\cdots ;A\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e981d7b64c8e9683b9a4bb46477945e22f647e51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.364ex; height:3.009ex;" alt="{\displaystyle P|\cdots n_{i},n_{j}\cdots ;A\rangle =-|\cdots n_{i},n_{j}\cdots ;A\rangle }"></span></dd></dl> <p>The explicit symmetric/antisymmetric form of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e77f6b1e903837c5765c9683da41dd93199621c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.36ex; height:2.843ex;" alt="{\displaystyle |\Psi \rangle }"></span> is <a href="/wiki/Identical_particles#Wavefunction_representation" class="mw-redirect" title="Identical particles">constructed</a> using a symmetrizer or <a href="/wiki/Antisymmetrizer" title="Antisymmetrizer">antisymmetrizer</a> operator. Particles that form symmetric states are called <a href="/wiki/Boson" title="Boson">bosons</a> and those that form antisymmetric states are called as fermions. The relation of spin with this classification is given from <a href="/wiki/Spin%E2%80%93statistics_theorem" title="Spin–statistics theorem">spin statistics theorem</a> which shows that integer spin particles are bosons and half integer spin particles are fermions. </p> <div class="mw-heading mw-heading4"><h4 id="Pauli_exclusion_principle">Pauli exclusion principle</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=16" title="Edit section: Pauli exclusion principle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The property of spin relates to another basic property concerning systems of <span class="texhtml"><i>N</i></span> identical particles: the <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a>, which is a consequence of the following permutation behaviour of an <span class="texhtml"><i>N</i></span>-particle wave function; again in the position representation one must postulate that for the transposition of any two of the <span class="texhtml"><i>N</i></span> particles one always should have </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: #50C878; color: inherit;text-align: center; display: table"><b>Pauli principle</b> <p><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 (\dots ,\,\mathbf {r} _{i},\sigma _{i},\,\dots ,\,\mathbf {r} _{j},\sigma _{j},\,\dots )=(-1)^{2S}\cdot \psi (\dots ,\,\mathbf {r} _{j},\sigma _{j},\,\dots ,\mathbf {r} _{i},\sigma _{i},\,\dots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>S</mi> </mrow> </msup> <mo>⋅</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\dots ,\,\mathbf {r} _{i},\sigma _{i},\,\dots ,\,\mathbf {r} _{j},\sigma _{j},\,\dots )=(-1)^{2S}\cdot \psi (\dots ,\,\mathbf {r} _{j},\sigma _{j},\,\dots ,\mathbf {r} _{i},\sigma _{i},\,\dots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/998ad382ec46921e1281154a441747adf1997b9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:67.877ex; height:3.343ex;" alt="{\displaystyle \psi (\dots ,\,\mathbf {r} _{i},\sigma _{i},\,\dots ,\,\mathbf {r} _{j},\sigma _{j},\,\dots )=(-1)^{2S}\cdot \psi (\dots ,\,\mathbf {r} _{j},\sigma _{j},\,\dots ,\mathbf {r} _{i},\sigma _{i},\,\dots )}"></span> </p> </div> <p>i.e., on <a href="/wiki/Transposition_(mathematics)" class="mw-redirect" title="Transposition (mathematics)">transposition</a> of the arguments of any two particles the wavefunction should <i>reproduce</i>, apart from a prefactor <span class="texhtml">(−1)<sup>2<i>S</i></sup></span> which is <span class="texhtml">+1</span> for bosons, but (<span class="texhtml">−1</span>) for <a href="/wiki/Fermions" class="mw-redirect" title="Fermions">fermions</a>. Electrons are fermions with <span class="texhtml"><i>S</i> = 1/2</span>; quanta of light are bosons with <span class="texhtml"><i>S</i> = 1</span>. </p><p>Due to the form of anti-symmetrized wavefunction: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{n_{1}\cdots n_{N}}^{(A)}(x_{1},\ldots ,x_{N})={\frac {1}{\sqrt {N!}}}\left|{\begin{matrix}\psi _{n_{1}}(x_{1})&\psi _{n_{1}}(x_{2})&\cdots &\psi _{n_{1}}(x_{N})\\\psi _{n_{2}}(x_{1})&\psi _{n_{2}}(x_{2})&\cdots &\psi _{n_{2}}(x_{N})\\\vdots &\vdots &\ddots &\vdots \\\psi _{n_{N}}(x_{1})&\psi _{n_{N}}(x_{2})&\cdots &\psi _{n_{N}}(x_{N})\\\end{matrix}}\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>N</mi> <mo>!</mo> </msqrt> </mfrac> </mrow> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{n_{1}\cdots n_{N}}^{(A)}(x_{1},\ldots ,x_{N})={\frac {1}{\sqrt {N!}}}\left|{\begin{matrix}\psi _{n_{1}}(x_{1})&\psi _{n_{1}}(x_{2})&\cdots &\psi _{n_{1}}(x_{N})\\\psi _{n_{2}}(x_{1})&\psi _{n_{2}}(x_{2})&\cdots &\psi _{n_{2}}(x_{N})\\\vdots &\vdots &\ddots &\vdots \\\psi _{n_{N}}(x_{1})&\psi _{n_{N}}(x_{2})&\cdots &\psi _{n_{N}}(x_{N})\\\end{matrix}}\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3aed21bd5caf4c26f6223a4217d75aa86330f43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:66.93ex; height:14.843ex;" alt="{\displaystyle \Psi _{n_{1}\cdots n_{N}}^{(A)}(x_{1},\ldots ,x_{N})={\frac {1}{\sqrt {N!}}}\left|{\begin{matrix}\psi _{n_{1}}(x_{1})&\psi _{n_{1}}(x_{2})&\cdots &\psi _{n_{1}}(x_{N})\\\psi _{n_{2}}(x_{1})&\psi _{n_{2}}(x_{2})&\cdots &\psi _{n_{2}}(x_{N})\\\vdots &\vdots &\ddots &\vdots \\\psi _{n_{N}}(x_{1})&\psi _{n_{N}}(x_{2})&\cdots &\psi _{n_{N}}(x_{N})\\\end{matrix}}\right|}"></span></dd></dl> <p>if the wavefunction of each particle is completely determined by a set of quantum number, then two fermions cannot share the same set of quantum numbers since the resulting function cannot be anti-symmetrized (i.e. above formula gives zero). The same cannot be said of Bosons since their wavefunction is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x_{1}x_{2}\cdots x_{N};S\rangle ={\frac {\prod _{j}n_{j}!}{N!}}\sum _{p}\left|x_{p(1)}\right\rangle \left|x_{p(2)}\right\rangle \cdots \left|x_{p(N)}\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>;</mo> <mi>S</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>!</mo> </mrow> <mrow> <mi>N</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munder> <mrow> <mo>|</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⟩</mo> </mrow> <mrow> <mo>|</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⟩</mo> </mrow> <mo>⋯</mo> <mrow> <mo>|</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x_{1}x_{2}\cdots x_{N};S\rangle ={\frac {\prod _{j}n_{j}!}{N!}}\sum _{p}\left|x_{p(1)}\right\rangle \left|x_{p(2)}\right\rangle \cdots \left|x_{p(N)}\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58476bba63e99495e69aeea4efe81d4b7f84e470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.102ex; height:7.343ex;" alt="{\displaystyle |x_{1}x_{2}\cdots x_{N};S\rangle ={\frac {\prod _{j}n_{j}!}{N!}}\sum _{p}\left|x_{p(1)}\right\rangle \left|x_{p(2)}\right\rangle \cdots \left|x_{p(N)}\right\rangle }"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2c97b31dca4eeefe57123a12e69e6ea73f3dcd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.304ex; height:2.343ex;" alt="{\displaystyle n_{j}}"></span> is the number of particles with same wavefunction. </p> <div class="mw-heading mw-heading4"><h4 id="Exceptions_for_symmetrization_postulate">Exceptions for symmetrization postulate</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=17" title="Edit section: Exceptions for symmetrization postulate"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In nonrelativistic quantum mechanics all particles are either bosons or <a href="/wiki/Fermions" class="mw-redirect" title="Fermions">fermions</a>; in relativistic quantum theories also <a href="/wiki/Supersymmetry" title="Supersymmetry">"supersymmetric"</a> theories exist, where a particle is a linear combination of a bosonic and a fermionic part. Only in dimension <span class="texhtml"><i>d</i> = 2</span> can one construct entities where <span class="texhtml">(−1)<sup>2<i>S</i></sup></span> is replaced by an arbitrary complex number with magnitude 1, called <a href="/wiki/Anyon" title="Anyon">anyons</a>. In relativistic quantum mechanics, spin statistic theorem can prove that under certain set of assumptions that the integer spins particles are classified as bosons and half spin particles are classified as <a href="/wiki/Fermion" title="Fermion">fermions</a>. Anyons which form neither symmetric nor antisymmetric states are said to have fractional spin. </p><p>Although <i>spin</i> and the <i>Pauli principle</i> can only be derived from relativistic generalizations of quantum mechanics, the properties mentioned in the last two paragraphs belong to the basic postulates already in the non-relativistic limit. Especially, many important properties in natural science, e.g. the <a href="/wiki/Periodic_system" class="mw-redirect" title="Periodic system">periodic system</a> of chemistry, are consequences of the two properties. </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_structure_of_quantum_mechanics">Mathematical structure of quantum mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=18" title="Edit section: Mathematical structure of quantum mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Pictures_of_dynamics">Pictures of dynamics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=19" title="Edit section: Pictures of dynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Dynamical_pictures" title="Dynamical pictures">Dynamical pictures</a></div> <div><ul><li>In the so-called <a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger picture</a> of quantum mechanics, the dynamics is given as follows: <p>The <a href="/wiki/Time_evolution" title="Time evolution">time evolution</a> of the state is given by a differentiable function from the real numbers <span class="texhtml"><b>R</b></span>, representing instants of time, to the Hilbert space of system states. This map is characterized by a differential equation as follows: If <span class="texhtml"><span class="nowrap">|<i>ψ</i>(<i>t</i>)⟩</span></span> denotes the state of the system at any one time <span class="texhtml"><i>t</i></span>, the following Schrödinger equation holds: </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: #50C878; color: inherit;text-align: center; display: table"><b>Schrödinger equation </b> <i>(general)</i> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {d}{dt}}\left|\psi (t)\right\rangle =H\left|\psi (t)\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>|</mo> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mi>H</mi> <mrow> <mo>|</mo> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {d}{dt}}\left|\psi (t)\right\rangle =H\left|\psi (t)\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b692255e8d80417f422629cb61362d7dbc9b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.364ex; height:5.509ex;" alt="{\displaystyle i\hbar {\frac {d}{dt}}\left|\psi (t)\right\rangle =H\left|\psi (t)\right\rangle }"></span> </p> </div> <p>where <span class="texhtml"><i>H</i></span> is a densely defined self-adjoint operator, called the system <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>, <span class="texhtml"><i>i</i></span> is the <a href="/wiki/Complex_number" title="Complex number">imaginary unit</a> and <span class="texhtml"><i>ħ</i></span> is the <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a>. As an observable, <span class="texhtml"><i>H</i></span> corresponds to the total <a href="/wiki/Energy" title="Energy">energy</a> of the system. </p><p>Alternatively, by <a href="/wiki/Stone%27s_theorem_on_one-parameter_unitary_groups" title="Stone's theorem on one-parameter unitary groups">Stone's theorem</a> one can state that there is a strongly continuous one-parameter unitary map <span class="texhtml"><i>U</i>(<i>t</i>)</span>: <span class="texhtml"><i>H</i> → <i>H</i></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\psi (t+s)\right\rangle =U(t)\left|\psi (s)\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow> <mo>|</mo> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\psi (t+s)\right\rangle =U(t)\left|\psi (s)\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e56df7a8a0e743e7f5ffde2916f0d80cfbffafbe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.526ex; height:2.843ex;" alt="{\displaystyle \left|\psi (t+s)\right\rangle =U(t)\left|\psi (s)\right\rangle }"></span> for all times <span class="texhtml"><i>s</i>, <i>t</i></span>. The existence of a self-adjoint Hamiltonian <span class="texhtml"><i>H</i></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(t)=e^{-(i/\hbar )tH}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> <mo stretchy="false">)</mo> <mi>t</mi> <mi>H</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(t)=e^{-(i/\hbar )tH}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d7034b4e9543aebae32482a5fe6a607ce47c4a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.77ex; height:3.343ex;" alt="{\displaystyle U(t)=e^{-(i/\hbar )tH}}"></span> is a consequence of Stone's theorem on one-parameter unitary groups. It is assumed that <span class="texhtml"><i>H</i></span> does not depend on time and that the perturbation starts at <span class="texhtml"><i>t</i><sub>0</sub> = 0</span>; otherwise one must use the <a href="/wiki/Dyson_series" title="Dyson series">Dyson series</a>, formally written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(t)={\mathcal {T}}\left[\exp \left(-{\frac {i}{\hbar }}\int _{t_{0}}^{t}dt'\,H(t')\right)\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mrow> <mo>[</mo> <mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>d</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mi>H</mi> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(t)={\mathcal {T}}\left[\exp \left(-{\frac {i}{\hbar }}\int _{t_{0}}^{t}dt'\,H(t')\right)\right],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/442eb5c646ec193e2cc7e5056896a8648396079b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.855ex; height:6.509ex;" alt="{\displaystyle U(t)={\mathcal {T}}\left[\exp \left(-{\frac {i}{\hbar }}\int _{t_{0}}^{t}dt'\,H(t')\right)\right],}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}"></span> is Dyson's <a href="/wiki/Time-ordering" class="mw-redirect" title="Time-ordering">time-ordering</a> symbol. </p><p>(This symbol permutes a product of noncommuting operators of the form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{1}(t_{1})\cdot B_{2}(t_{2})\cdot \dots \cdot B_{n}(t_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo>⋯</mo> <mo>⋅</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{1}(t_{1})\cdot B_{2}(t_{2})\cdot \dots \cdot B_{n}(t_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/838f98fb88bd073c9595c749b0b82099fc7f8437" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.653ex; height:2.843ex;" alt="{\displaystyle B_{1}(t_{1})\cdot B_{2}(t_{2})\cdot \dots \cdot B_{n}(t_{n})}"></span> into the uniquely determined re-ordered expression <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{i_{1}}(t_{i_{1}})\cdot B_{i_{2}}(t_{i_{2}})\cdot \dots \cdot B_{i_{n}}(t_{i_{n}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo>⋯</mo> <mo>⋅</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{i_{1}}(t_{i_{1}})\cdot B_{i_{2}}(t_{i_{2}})\cdot \dots \cdot B_{i_{n}}(t_{i_{n}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f7b4818a96164af9f340022bc5bd2709321bee9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.053ex; height:3.009ex;" alt="{\displaystyle B_{i_{1}}(t_{i_{1}})\cdot B_{i_{2}}(t_{i_{2}})\cdot \dots \cdot B_{i_{n}}(t_{i_{n}})}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i_{1}}\geq t_{i_{2}}\geq \dots \geq t_{i_{n}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>≥</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo>≥</mo> <mo>⋯</mo> <mo>≥</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i_{1}}\geq t_{i_{2}}\geq \dots \geq t_{i_{n}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3650a56b252693b70247bdbf4248034bdae5a4fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.598ex; height:2.676ex;" alt="{\displaystyle t_{i_{1}}\geq t_{i_{2}}\geq \dots \geq t_{i_{n}}\,.}"></span> </p> The result is a causal chain, the primary <i>cause</i> in the past on the utmost r.h.s., and finally the present <i>effect</i> on the utmost l.h.s. .)</li><li>The <a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg picture</a> of quantum mechanics focuses on observables and instead of considering states as varying in time, it regards the states as fixed and the observables as changing. To go from the Schrödinger to the Heisenberg picture one needs to define time-independent states and time-dependent operators thus: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\psi \right\rangle =\left|\psi (0)\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mi>ψ</mi> <mo>⟩</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\psi \right\rangle =\left|\psi (0)\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d23b83fe688e0cdb40e8ffba01a2f0f0488b804c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.2ex; height:2.843ex;" alt="{\displaystyle \left|\psi \right\rangle =\left|\psi (0)\right\rangle }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(t)=U(-t)AU(t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>A</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(t)=U(-t)AU(t).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5afc0a80548c87e308a4ae27fef667a822813a14" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.552ex; height:2.843ex;" alt="{\displaystyle A(t)=U(-t)AU(t).}"></span> It is then easily checked that the expected values of all observables are the same in both pictures <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \psi \mid A(t)\mid \psi \rangle =\langle \psi (t)\mid A\mid \psi (t)\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mo>∣</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>A</mi> <mo>∣</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi \mid A(t)\mid \psi \rangle =\langle \psi (t)\mid A\mid \psi (t)\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b8c0e9169ab1c57391f6480d49ace5bcb36a32d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.951ex; height:2.843ex;" alt="{\displaystyle \langle \psi \mid A(t)\mid \psi \rangle =\langle \psi (t)\mid A\mid \psi (t)\rangle }"></span> and that the time-dependent Heisenberg operators satisfy <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"><b>Heisenberg picture</b> <i>(general)</i> <p><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 {d}{dt}}A(t)={\frac {i}{\hbar }}[H,A(t)]+{\frac {\partial A(t)}{\partial t}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>H</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>A</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}A(t)={\frac {i}{\hbar }}[H,A(t)]+{\frac {\partial A(t)}{\partial t}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d714d59756ef17027e03f666547917ba07d8057" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.341ex; height:5.843ex;" alt="{\displaystyle {\frac {d}{dt}}A(t)={\frac {i}{\hbar }}[H,A(t)]+{\frac {\partial A(t)}{\partial t}},}"></span> </p> </div> which is true for time-dependent <span class="texhtml"><i>A</i> = <i>A</i>(<i>t</i>)</span>. Notice the commutator expression is purely formal when one of the operators is <a href="/wiki/Unbounded_operator" title="Unbounded operator">unbounded</a>. One would specify a representation for the expression to make sense of it.</li><li>The so-called <a href="/wiki/Dirac_picture" class="mw-redirect" title="Dirac picture">Dirac picture</a> or <a href="/wiki/Interaction_picture" title="Interaction picture">interaction picture</a> has time-dependent <i>states</i> and observables, evolving with respect to different Hamiltonians. This picture is most useful when the evolution of the observables can be solved exactly, confining any complications to the evolution of the states. For this reason, the Hamiltonian for the observables is called "free Hamiltonian" and the Hamiltonian for the states is called "interaction Hamiltonian". In symbols: <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"><b>Dirac picture</b> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {d}{dt}}\left|\psi (t)\right\rangle ={H}_{\rm {int}}(t)\left|\psi (t)\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>|</mo> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow> <mo>|</mo> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {d}{dt}}\left|\psi (t)\right\rangle ={H}_{\rm {int}}(t)\left|\psi (t)\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/540045c7e87fab9ef7dd5996d8388f88cd36cb6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.124ex; height:5.509ex;" alt="{\displaystyle i\hbar {\frac {d}{dt}}\left|\psi (t)\right\rangle ={H}_{\rm {int}}(t)\left|\psi (t)\right\rangle }"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {d}{dt}}A(t)=[A(t),H_{0}].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {d}{dt}}A(t)=[A(t),H_{0}].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fe8dccc41b7f0b775dcb47dde6daca9ae2f080a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.843ex; height:5.509ex;" alt="{\displaystyle i\hbar {\frac {d}{dt}}A(t)=[A(t),H_{0}].}"></span> </p> </div> <p>The interaction picture does not always exist, though. In interacting quantum field theories, <a href="/wiki/Haag%27s_theorem" title="Haag's theorem">Haag's theorem</a> states that the interaction picture does not exist. This is because the Hamiltonian cannot be split into a free and an interacting part within a <a href="/wiki/Superselection_sector" class="mw-redirect" title="Superselection sector">superselection sector</a>. Moreover, even if in the Schrödinger picture the Hamiltonian does not depend on time, e.g. <span class="texhtml"><i>H</i> = <i>H</i><sub>0</sub> + <i>V</i></span>, in the interaction picture it does, at least, if <span class="texhtml"><i>V</i></span> does not commute with <span class="texhtml"><i>H</i><sub>0</sub></span>, since <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\rm {int}}(t)\equiv e^{{(i/\hbar })tH_{0}}\,V\,e^{{(-i/\hbar })tH_{0}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> <mo stretchy="false">)</mo> <mi>t</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msup> <mspace width="thinmathspace" /> <mi>V</mi> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> <mo stretchy="false">)</mo> <mi>t</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{\rm {int}}(t)\equiv e^{{(i/\hbar })tH_{0}}\,V\,e^{{(-i/\hbar })tH_{0}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/959f447d3d7581eaa714c73e2b9e19e6888477ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.807ex; height:3.343ex;" alt="{\displaystyle H_{\rm {int}}(t)\equiv e^{{(i/\hbar })tH_{0}}\,V\,e^{{(-i/\hbar })tH_{0}}.}"></span> </p><p>So the above-mentioned Dyson-series has to be used anyhow. </p><p>The Heisenberg picture is the closest to classical Hamiltonian mechanics (for example, the commutators appearing in the above equations directly translate into the classical <a href="/wiki/Poisson_bracket" title="Poisson bracket">Poisson brackets</a>); but this is already rather "high-browed", and the Schrödinger picture is considered easiest to visualize and understand by most people, to judge from pedagogical accounts of quantum mechanics. The Dirac picture is the one used in <a href="/wiki/Perturbation_theory" title="Perturbation theory">perturbation theory</a>, and is specially associated to <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> and <a href="/wiki/Many-body_theory" class="mw-redirect" title="Many-body theory">many-body physics</a>. </p> Similar equations can be written for any one-parameter unitary group of symmetries of the physical system. Time would be replaced by a suitable coordinate parameterizing the unitary group (for instance, a rotation angle, or a translation distance) and the Hamiltonian would be replaced by the conserved quantity associated with the symmetry (for instance, angular or linear momentum).</li></ul></div> <p><b>Summary</b>: </p> <table class="wikitable" style="padding:0.3em; clear:right; margin:1em auto; text-align:center;"> <tbody><tr> <td rowspan="2" style="background-color:#E0FFEE;">Evolution of: </td> <td colspan="3" style="background-color:#E0FFEE;"><a href="/wiki/Dynamical_pictures" title="Dynamical pictures">Picture</a> (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Pictures_in_quantum_mechanics" title="Template:Pictures in quantum mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Pictures_in_quantum_mechanics" title="Template talk:Pictures in quantum mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Pictures_in_quantum_mechanics" title="Special:EditPage/Template:Pictures in quantum mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div>) </td></tr> <tr> <td style="background-color:#E0F0FF;"><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a> (S) </td> <td style="background-color:#E0F0FF;"><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a> (H) </td> <td style="background-color:#E0F0FF;"><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a> (I) </td></tr> <tr> <td style="background-color:#D0FFDD;"><a href="/wiki/Bra-ket_notation" class="mw-redirect" title="Bra-ket notation">Ket state</a> </td> <td><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 _{\rm {S}}(t)\rangle =e^{-iH_{\rm {S}}~t/\hbar }|\psi _{\rm {S}}(0)\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi _{\rm {S}}(t)\rangle =e^{-iH_{\rm {S}}~t/\hbar }|\psi _{\rm {S}}(0)\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f7e8476598d735d3fa5be6979777f685fb8ea2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.495ex; height:3.343ex;" alt="{\displaystyle |\psi _{\rm {S}}(t)\rangle =e^{-iH_{\rm {S}}~t/\hbar }|\psi _{\rm {S}}(0)\rangle }"></span> </td> <td>constant </td> <td><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 _{\rm {I}}(t)\rangle =e^{iH_{0,\mathrm {S} }~t/\hbar }|\psi _{\rm {S}}(t)\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi _{\rm {I}}(t)\rangle =e^{iH_{0,\mathrm {S} }~t/\hbar }|\psi _{\rm {S}}(t)\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6239a430d0eac4e07abc9f7926d65b12b4755c8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.612ex; height:3.343ex;" alt="{\displaystyle |\psi _{\rm {I}}(t)\rangle =e^{iH_{0,\mathrm {S} }~t/\hbar }|\psi _{\rm {S}}(t)\rangle }"></span> </td></tr> <tr> <td style="background-color:#D0FFDD;"><a href="/wiki/Observable" title="Observable">Observable</a> </td> <td>constant </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\rm {H}}(t)=e^{iH_{\rm {S}}~t/\hbar }A_{\rm {S}}e^{-iH_{\rm {S}}~t/\hbar }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\rm {H}}(t)=e^{iH_{\rm {S}}~t/\hbar }A_{\rm {S}}e^{-iH_{\rm {S}}~t/\hbar }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fed2a30b52c2506bf999ba176813e78a5cd6f28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.273ex; height:3.343ex;" alt="{\displaystyle A_{\rm {H}}(t)=e^{iH_{\rm {S}}~t/\hbar }A_{\rm {S}}e^{-iH_{\rm {S}}~t/\hbar }}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }A_{\rm {S}}e^{-iH_{0,\mathrm {S} }~t/\hbar }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }A_{\rm {S}}e^{-iH_{0,\mathrm {S} }~t/\hbar }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e17359cb3e0e75ad9db7c54218cb5ddedcb7605e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.712ex; height:3.343ex;" alt="{\displaystyle A_{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }A_{\rm {S}}e^{-iH_{0,\mathrm {S} }~t/\hbar }}"></span> </td></tr> <tr> <td style="background-color:#D0FFDD;"><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a> </td> <td><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 \rho _{\rm {S}}(t)=e^{-iH_{\rm {S}}~t/\hbar }\rho _{\rm {S}}(0)e^{iH_{\rm {S}}~t/\hbar }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\rm {S}}(t)=e^{-iH_{\rm {S}}~t/\hbar }\rho _{\rm {S}}(0)e^{iH_{\rm {S}}~t/\hbar }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/238d1dec8256c54d7316d3656ca666eb32784d6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.844ex; height:3.343ex;" alt="{\displaystyle \rho _{\rm {S}}(t)=e^{-iH_{\rm {S}}~t/\hbar }\rho _{\rm {S}}(0)e^{iH_{\rm {S}}~t/\hbar }}"></span> </td> <td>constant </td> <td><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 \rho _{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }\rho _{\rm {S}}(t)e^{-iH_{0,\mathrm {S} }~t/\hbar }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }\rho _{\rm {S}}(t)e^{-iH_{0,\mathrm {S} }~t/\hbar }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f578fe89295524333c363ccf0b9781275894c7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.278ex; height:3.343ex;" alt="{\displaystyle \rho _{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }\rho _{\rm {S}}(t)e^{-iH_{0,\mathrm {S} }~t/\hbar }}"></span> </td></tr> </tbody></table> <div class="mw-heading mw-heading3"><h3 id="Representations">Representations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=20" title="Edit section: Representations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The original form of the Schrödinger equation depends on choosing a particular representation of Heisenberg's <a href="/wiki/Canonical_commutation_relations" class="mw-redirect" title="Canonical commutation relations">canonical commutation relations</a>. The <a href="/wiki/Stone%E2%80%93von_Neumann_theorem" title="Stone–von Neumann theorem">Stone–von Neumann theorem</a> dictates that all irreducible representations of the finite-dimensional Heisenberg commutation relations are unitarily equivalent. A systematic understanding of its consequences has led to the <a href="/wiki/Phase_space_formulation" class="mw-redirect" title="Phase space formulation">phase space formulation</a> of quantum mechanics, which works in full <a href="/wiki/Phase_space" title="Phase space">phase space</a> instead of <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>, so then with a more intuitive link to the <a href="/wiki/Classical_limit" title="Classical limit">classical limit</a> thereof. This picture also simplifies considerations of <a href="/wiki/Quantization_(physics)" title="Quantization (physics)">quantization</a>, the deformation extension from classical to quantum mechanics. </p><p>The <a href="/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillator</a> is an exactly solvable system where the different representations are easily compared. There, apart from the Heisenberg, or Schrödinger (position or momentum), or phase-space representations, one also encounters the Fock (number) representation and the <a href="/wiki/Oscillator_representation" title="Oscillator representation">Segal–Bargmann (Fock-space or coherent state) representation</a> (named after <a href="/wiki/Irving_Segal" title="Irving Segal">Irving Segal</a> and <a href="/wiki/Valentine_Bargmann" title="Valentine Bargmann">Valentine Bargmann</a>). All four are unitarily equivalent. </p> <div class="mw-heading mw-heading3"><h3 id="Time_as_an_operator">Time as an operator</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=21" title="Edit section: Time as an operator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The framework presented so far singles out time as <i>the</i> parameter that everything depends on. It is possible to formulate mechanics in such a way that time becomes itself an observable associated with a self-adjoint operator. At the classical level, it is possible to arbitrarily parameterize the trajectories of particles in terms of an unphysical parameter <span class="texhtml"><i>s</i></span>, and in that case the time <i>t</i> becomes an additional generalized coordinate of the physical system. At the quantum level, translations in <span class="texhtml"><i>s</i></span> would be generated by a "Hamiltonian" <span class="texhtml"><i>H</i> − <i>E</i></span>, where <i>E</i> is the energy operator and <span class="texhtml"><i>H</i></span> is the "ordinary" Hamiltonian. However, since <i>s</i> is an unphysical parameter, <i>physical</i> states must be left invariant by "<i>s</i>-evolution", and so the physical state space is the kernel of <span class="texhtml"><i>H</i> − <i>E</i></span> (this requires the use of a <a href="/wiki/Rigged_Hilbert_space" title="Rigged Hilbert space">rigged Hilbert space</a> and a renormalization of the norm). </p><p>This is related to the <a href="/wiki/Dirac_bracket" title="Dirac bracket">quantization of constrained systems</a> and <a href="/wiki/Quantization_of_gauge_theories" class="mw-redirect" title="Quantization of gauge theories">quantization of gauge theories</a>. It is also possible to formulate a quantum theory of "events" where time becomes an observable.<sup id="cite_ref-FOOTNOTEEdwards1979_12-0" class="reference"><a href="#cite_note-FOOTNOTEEdwards1979-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Problem_of_measurement">Problem of measurement</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=22" title="Edit section: Problem of measurement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement in quantum mechanics</a></div> <p>The picture given in the preceding paragraphs is sufficient for description of a completely isolated system. However, it fails to account for one of the main differences between quantum mechanics and classical mechanics, that is, the effects of <a href="/wiki/Measurement" title="Measurement">measurement</a>.<sup id="cite_ref-FOOTNOTEGreensteinZajonc2006215_13-0" class="reference"><a href="#cite_note-FOOTNOTEGreensteinZajonc2006215-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The von Neumann description of quantum measurement of an observable <span class="texhtml"><i>A</i></span>, when the system is prepared in a pure state <span class="texhtml"><i>ψ</i></span> is the following (note, however, that von Neumann's description dates back to the 1930s and is based on experiments as performed during that time – more specifically the <a href="/wiki/Compton_scattering" title="Compton scattering">Compton–Simon experiment</a>; it is not applicable to most present-day measurements within the quantum domain): </p> <ul><li>Let <span class="texhtml"><i>A</i></span> have spectral resolution <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\int \lambda \,d\operatorname {E} _{A}(\lambda ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo>∫</mo> <mi>λ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\int \lambda \,d\operatorname {E} _{A}(\lambda ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72d10f58c0ca5130e744051e21e5b8b9ddb6d587" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.627ex; height:5.676ex;" alt="{\displaystyle A=\int \lambda \,d\operatorname {E} _{A}(\lambda ),}"></span> where <span class="texhtml">E<sub><i>A</i></sub></span> is the <a href="/wiki/Resolution_of_the_identity" class="mw-redirect" title="Resolution of the identity">resolution of the identity</a> (also called <a href="/wiki/Projection-valued_measure" title="Projection-valued measure">projection-valued measure</a>) associated with <span class="texhtml"><i>A</i></span>. Then the probability of the measurement outcome lying in an interval <span class="texhtml"><i>B</i></span> of <span class="texhtml"><b>R</b></span> is <span class="texhtml">|E<sub><i>A</i></sub>(<i>B</i>) <i>ψ</i>|<sup>2</sup></span>. In other words, the probability is obtained by integrating the characteristic function of <span class="texhtml"><i>B</i></span> against the countably additive measure <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \psi \mid \operatorname {E} _{A}\psi \rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mo>∣</mo> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⁡</mo> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi \mid \operatorname {E} _{A}\psi \rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc91e39cb8fc76670267ca3c7315d346b46adad" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.854ex; height:2.843ex;" alt="{\displaystyle \langle \psi \mid \operatorname {E} _{A}\psi \rangle .}"></span></li> <li>If the measured value is contained in <span class="texhtml"><i>B</i></span>, then immediately after the measurement, the system will be in the (generally non-normalized) state <span class="texhtml">E<sub><i>A</i></sub>(<i>B</i>)<i>ψ</i></span>. If the measured value does not lie in <span class="texhtml"><i>B</i></span>, replace <span class="texhtml"><i>B</i></span> by its complement for the above state.</li></ul> <p>For example, suppose the state space is the <span class="texhtml"><i>n</i></span>-dimensional complex Hilbert space <span class="texhtml"><b>C</b><sup><i>n</i></sup></span> and <span class="texhtml"><i>A</i></span> is a Hermitian matrix with eigenvalues <span class="texhtml"><i>λ</i><sub><i>i</i></sub></span>, with corresponding eigenvectors <span class="texhtml"><i>ψ<sub>i</sub></i></span>. The projection-valued measure associated with <span class="texhtml"><i>A</i></span>, <span class="texhtml">E<sub><i>A</i></sub></span>, is then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} _{A}(B)=|\psi _{i}\rangle \langle \psi _{i}|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} _{A}(B)=|\psi _{i}\rangle \langle \psi _{i}|,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a17e2730e3d187b7a2f3e7f4501a4c67bfb3a9c6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.095ex; height:2.843ex;" alt="{\displaystyle \operatorname {E} _{A}(B)=|\psi _{i}\rangle \langle \psi _{i}|,}"></span> where <span class="texhtml"><i>B</i></span> is a Borel set containing only the single eigenvalue <span class="texhtml"><i>λ<sub>i</sub></i></span>. If the system is prepared in state <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"></span> Then the probability of a measurement returning the value <span class="texhtml"><i>λ<sub>i</sub></i></span> can be calculated by integrating the spectral measure <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \psi \mid \operatorname {E} _{A}\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mo>∣</mo> <msub> <mi mathvariant="normal">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⁡</mo> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi \mid \operatorname {E} _{A}\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7df757d215e5bfcfd06e6f5733890f61d2d7bca9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.208ex; height:2.843ex;" alt="{\displaystyle \langle \psi \mid \operatorname {E} _{A}\psi \rangle }"></span> over <span class="texhtml"><i>B<sub>i</sub></i></span>. This gives trivially <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \psi |\psi _{i}\rangle \langle \psi _{i}\mid \psi \rangle =|\langle \psi \mid \psi _{i}\rangle |^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∣</mo> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mo>∣</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi |\psi _{i}\rangle \langle \psi _{i}\mid \psi \rangle =|\langle \psi \mid \psi _{i}\rangle |^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d448cca751f067300353031820cdb20b114915a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.52ex; height:3.343ex;" alt="{\displaystyle \langle \psi |\psi _{i}\rangle \langle \psi _{i}\mid \psi \rangle =|\langle \psi \mid \psi _{i}\rangle |^{2}.}"></span> </p><p>The characteristic property of the von Neumann measurement scheme is that repeating the same measurement will give the same results. This is also called the <i>projection postulate</i>. </p><p>A more general formulation replaces the projection-valued measure with a positive-operator valued measure (POVM). To illustrate, take again the finite-dimensional case. Here we would replace the rank-1 projections <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi _{i}\rangle \langle \psi _{i}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi _{i}\rangle \langle \psi _{i}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5014b6330cd16bdcd79e204c8b602f26837c464a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.729ex; height:2.843ex;" alt="{\displaystyle |\psi _{i}\rangle \langle \psi _{i}|}"></span> by a finite set of positive operators <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{i}F_{i}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{i}F_{i}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ea0fd3eaffd19d5905a5ef5b75f82f67a9bb202" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.163ex; height:2.843ex;" alt="{\displaystyle F_{i}F_{i}^{*}}"></span> whose sum is still the identity operator as before (the resolution of identity). Just as a set of possible outcomes <span class="texhtml">{<i>λ</i><sub>1</sub> ... <i>λ<sub>n</sub></i>} </span> is associated to a projection-valued measure, the same can be said for a POVM. Suppose the measurement outcome is <span class="texhtml"><i>λ<sub>i</sub></i></span>. Instead of collapsing to the (unnormalized) state <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi _{i}\rangle \langle \psi _{i}|\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi _{i}\rangle \langle \psi _{i}|\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5fb9ab146b1b34d30a2f23df3c6efeff377b84e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.146ex; height:2.843ex;" alt="{\displaystyle |\psi _{i}\rangle \langle \psi _{i}|\psi \rangle }"></span> after the measurement, the system now will be in the state <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{i}|\psi \rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{i}|\psi \rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66faab68701ab346299c84389b228fa481841c6b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.006ex; height:2.843ex;" alt="{\displaystyle F_{i}|\psi \rangle .}"></span> </p><p>Since the <span class="texhtml"><i>F<sub>i</sub> F<sub>i</sub>*</i></span> operators need not be mutually orthogonal projections, the projection postulate of von Neumann no longer holds. </p><p>The same formulation applies to general <a href="/wiki/Mixed_state_(physics)" class="mw-redirect" title="Mixed state (physics)">mixed states</a>. </p><p>In von Neumann's approach, the state transformation due to measurement is distinct from that due to time evolution in several ways. For example, time evolution is deterministic and unitary whereas measurement is non-deterministic and non-unitary. However, since both types of state transformation take one quantum state to another, this difference was viewed by many as unsatisfactory. The POVM formalism views measurement as one among many other <a href="/wiki/Quantum_operation" title="Quantum operation">quantum operations</a>, which are described by <a href="/wiki/Completely_positive_map" title="Completely positive map">completely positive maps</a> which do not increase the trace. </p><p>In any case it seems that the above-mentioned problems can only be resolved if the time evolution included not only the quantum system, but also, and essentially, the classical measurement apparatus (see above). </p> <div class="mw-heading mw-heading2"><h2 id="List_of_mathematical_tools">List of mathematical tools</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=23" title="Edit section: List of mathematical tools"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Part of the folklore of the subject concerns the <a href="/wiki/Mathematical_physics" title="Mathematical physics">mathematical physics</a> textbook <a href="/wiki/Methods_of_Mathematical_Physics" class="mw-redirect" title="Methods of Mathematical Physics">Methods of Mathematical Physics</a> put together by <a href="/wiki/Richard_Courant" title="Richard Courant">Richard Courant</a> from <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a>'s <a href="/wiki/G%C3%B6ttingen_University" class="mw-redirect" title="Göttingen University">Göttingen University</a> courses. The story is told (by mathematicians) that physicists had dismissed the material as not interesting in the current research areas, until the advent of Schrödinger's equation. At that point it was realised that the mathematics of the new quantum mechanics was already laid out in it. It is also said that Heisenberg had consulted Hilbert about his matrix mechanics, and Hilbert observed that his own experience with infinite-dimensional matrices had derived from differential equations, advice which Heisenberg ignored, missing the opportunity to unify the theory as Weyl and Dirac did a few years later. Whatever the basis of the anecdotes, the mathematics of the theory was conventional at the time, whereas the physics was radically new. </p><p>The main tools include: </p> <ul><li><a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>: <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a>, <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvectors</a>, <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>: <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert spaces</a>, <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operators</a>, <a href="/wiki/Spectral_theory" title="Spectral theory">spectral theory</a></li> <li><a href="/wiki/Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a>: <a href="/wiki/Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a>, <a href="/wiki/Separation_of_variables" title="Separation of variables">separation of variables</a>, <a href="/wiki/Ordinary_differential_equations" class="mw-redirect" title="Ordinary differential equations">ordinary differential equations</a>, <a href="/wiki/Sturm%E2%80%93Liouville_theory" title="Sturm–Liouville theory">Sturm–Liouville theory</a>, <a href="/wiki/Eigenfunction" title="Eigenfunction">eigenfunctions</a></li> <li><a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a>: <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transforms</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=24" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/List_of_mathematical_topics_in_quantum_theory" title="List of mathematical topics in quantum theory">List of mathematical topics in quantum theory</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=25" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTEByronFuller1992277-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEByronFuller1992277_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFByronFuller1992">Byron & Fuller 1992</a>, p. 277.</span> </li> <li id="cite_note-FOOTNOTEDirac1925-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDirac1925_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDirac1925">Dirac 1925</a>.</span> </li> <li id="cite_note-FOOTNOTECohen-TannoudjiDiuLaloë2020-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECohen-TannoudjiDiuLaloë2020_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCohen-TannoudjiDiuLaloë2020">Cohen-Tannoudji, Diu & Laloë 2020</a>.</span> </li> <li id="cite_note-FOOTNOTEBäuerlede_Kerf1990330-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBäuerlede_Kerf1990330_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBäuerlede_Kerf1990">Bäuerle & de Kerf 1990</a>, p. 330.</span> </li> <li id="cite_note-FOOTNOTESolemBiedenharn1993-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESolemBiedenharn1993_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSolemBiedenharn1993">Solem & Biedenharn 1993</a>.</span> </li> <li id="cite_note-FOOTNOTEJauchWignerYanase1997-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEJauchWignerYanase1997_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFJauchWignerYanase1997">Jauch, Wigner & Yanase 1997</a>.</span> </li> <li id="cite_note-FOOTNOTECarcassiMacconeAidala2021-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECarcassiMacconeAidala2021_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCarcassiMacconeAidala2021">Carcassi, Maccone & Aidala 2021</a>.</span> </li> <li id="cite_note-FOOTNOTEMasanesGalleyMüller2019-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMasanesGalleyMüller2019_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMasanesGalleyMüller2019">Masanes, Galley & Müller 2019</a>.</span> </li> <li id="cite_note-FOOTNOTESakuraiNapolitano2021443-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTESakuraiNapolitano2021443_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTESakuraiNapolitano2021443_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFSakuraiNapolitano2021">Sakurai & Napolitano 2021</a>, p. 443.</span> </li> <li id="cite_note-FOOTNOTESakuraiNapolitano2021434-437-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESakuraiNapolitano2021434-437_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSakuraiNapolitano2021">Sakurai & Napolitano 2021</a>, p. 434-437.</span> </li> <li id="cite_note-FOOTNOTECohen-TannoudjiDiuLaloë20201375–1377-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECohen-TannoudjiDiuLaloë20201375–1377_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCohen-TannoudjiDiuLaloë2020">Cohen-Tannoudji, Diu & Laloë 2020</a>, p. 1375–1377.</span> </li> <li id="cite_note-FOOTNOTEEdwards1979-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEEdwards1979_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFEdwards1979">Edwards 1979</a>.</span> </li> <li id="cite_note-FOOTNOTEGreensteinZajonc2006215-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGreensteinZajonc2006215_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGreensteinZajonc2006">Greenstein & Zajonc 2006</a>, p. 215.</span> </li> </ol></div></div> <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=Mathematical_formulation_of_quantum_mechanics&action=edit&section=26" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBäuerlede_Kerf1990" class="citation book cs1">Bäuerle, Gerard G. 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"Understanding geometrical phases in quantum mechanics: An elementary example". <i>Foundations of Physics</i>. <b>23</b> (2): <span class="nowrap">185–</span>195. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993FoPh...23..185S">1993FoPh...23..185S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01883623">10.1007/BF01883623</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121930907">121930907</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Foundations+of+Physics&rft.atitle=Understanding+geometrical+phases+in+quantum+mechanics%3A+An+elementary+example&rft.volume=23&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E185-%3C%2Fspan%3E195&rft.date=1993&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121930907%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01883623&rft_id=info%3Abibcode%2F1993FoPh...23..185S&rft.aulast=Solem&rft.aufirst=J.+C.&rft.au=Biedenharn%2C+L.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStreaterWightman2000" class="citation book cs1">Streater, Raymond Frederick; Wightman, Arthur Strong (2000). <i>PCT, Spin and Statistics, and All that</i>. Princeton, NJ: Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-07062-9" title="Special:BookSources/978-0-691-07062-9"><bdi>978-0-691-07062-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=PCT%2C+Spin+and+Statistics%2C+and+All+that&rft.place=Princeton%2C+NJ&rft.pub=Princeton+University+Press&rft.date=2000&rft.isbn=978-0-691-07062-9&rft.aulast=Streater&rft.aufirst=Raymond+Frederick&rft.au=Wightman%2C+Arthur+Strong&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSakuraiNapolitano2021" class="citation book cs1">Sakurai, Jun John; Napolitano, Jim (2021). <i>Modern quantum mechanics</i> (3rd ed.). Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-108-47322-4" title="Special:BookSources/978-1-108-47322-4"><bdi>978-1-108-47322-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+quantum+mechanics&rft.place=Cambridge&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=2021&rft.isbn=978-1-108-47322-4&rft.aulast=Sakurai&rft.aufirst=Jun+John&rft.au=Napolitano%2C+Jim&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_formulation_of_quantum_mechanics&action=edit&section=27" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAuyang1995" class="citation book cs1">Auyang, Sunny Y. (1995). <i>How is Quantum Field Theory Possible?</i>. New York, NY: Oxford University Press on Demand. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-509344-5" title="Special:BookSources/978-0-19-509344-5"><bdi>978-0-19-509344-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=How+is+Quantum+Field+Theory+Possible%3F&rft.place=New+York%2C+NY&rft.pub=Oxford+University+Press+on+Demand&rft.date=1995&rft.isbn=978-0-19-509344-5&rft.aulast=Auyang&rft.aufirst=Sunny+Y.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEmch1972" class="citation book cs1">Emch, Gérard G. (1972). <i>Algebraic Methods in Statistical Mechanics and Quantum Field Theory</i>. New York: John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-23900-3" title="Special:BookSources/0-471-23900-3"><bdi>0-471-23900-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Methods+in+Statistical+Mechanics+and+Quantum+Field+Theory&rft.place=New+York&rft.pub=John+Wiley+%26+Sons&rft.date=1972&rft.isbn=0-471-23900-3&rft.aulast=Emch&rft.aufirst=G%C3%A9rard+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGiachettaMangiarottiSardanashvily2005" class="citation book cs1">Giachetta, Giovanni; Mangiarotti, Luigi; Sardanashvily, Gennadi (2005). <i>Geometric and Algebraic Topological Methods in Quantum Mechanics</i>. WORLD SCIENTIFIC. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math-ph/0410040">math-ph/0410040</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F5731">10.1142/5731</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-256-129-9" title="Special:BookSources/978-981-256-129-9"><bdi>978-981-256-129-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometric+and+Algebraic+Topological+Methods+in+Quantum+Mechanics&rft.pub=WORLD+SCIENTIFIC&rft.date=2005&rft_id=info%3Aarxiv%2Fmath-ph%2F0410040&rft_id=info%3Adoi%2F10.1142%2F5731&rft.isbn=978-981-256-129-9&rft.aulast=Giachetta&rft.aufirst=Giovanni&rft.au=Mangiarotti%2C+Luigi&rft.au=Sardanashvily%2C+Gennadi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGleason1957" class="citation journal cs1">Gleason, Andrew M. (1957). <a rel="nofollow" class="external text" href="http://www.jstor.org/stable/24900629">"Measures on the Closed Subspaces of a Hilbert Space"</a>. <i>Journal of Mathematics and Mechanics</i>. <b>6</b> (6). Indiana University Mathematics Department: <span class="nowrap">885–</span>893. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/24900629">24900629</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematics+and+Mechanics&rft.atitle=Measures+on+the+Closed+Subspaces+of+a+Hilbert+Space&rft.volume=6&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E885-%3C%2Fspan%3E893&rft.date=1957&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F24900629%23id-name%3DJSTOR&rft.aulast=Gleason&rft.aufirst=Andrew+M.&rft_id=http%3A%2F%2Fwww.jstor.org%2Fstable%2F24900629&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall2013" class="citation book cs1">Hall, Brian C. (2013). <i>Quantum Theory for Mathematicians</i>. Graduate Texts in Mathematics. Vol. 267. New York, NY: Springer New York. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013qtm..book.....H">2013qtm..book.....H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4614-7116-5">10.1007/978-1-4614-7116-5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4614-7115-8" title="Special:BookSources/978-1-4614-7115-8"><bdi>978-1-4614-7115-8</bdi></a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0072-5285">0072-5285</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:117837329">117837329</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Theory+for+Mathematicians&rft.place=New+York%2C+NY&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer+New+York&rft.date=2013&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A117837329%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2013qtm..book.....H&rft.issn=0072-5285&rft_id=info%3Adoi%2F10.1007%2F978-1-4614-7116-5&rft.isbn=978-1-4614-7115-8&rft.aulast=Hall&rft.aufirst=Brian+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJauch1968" class="citation book cs1">Jauch, Josef Maria (1968). <i>Foundations of Quantum Mechanics</i>. Reading, Mass.: Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-03298-8" title="Special:BookSources/0-201-03298-8"><bdi>0-201-03298-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Quantum+Mechanics&rft.place=Reading%2C+Mass.&rft.pub=Addison-Wesley&rft.date=1968&rft.isbn=0-201-03298-8&rft.aulast=Jauch&rft.aufirst=Josef+Maria&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJost1965" class="citation book cs1">Jost, R. (1965). <i>The General Theory of Quantized Fields</i>. Lectures in applied mathematics. American Mathematical Society.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+General+Theory+of+Quantized+Fields&rft.series=Lectures+in+applied+mathematics&rft.pub=American+Mathematical+Society&rft.date=1965&rft.aulast=Jost&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKuhn1987" class="citation book cs1">Kuhn, Thomas S. (1987). <i>Black-Body Theory and the Quantum Discontinuity, 1894-1912</i>. Chicago: University of Chicago Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-226-45800-7" title="Special:BookSources/978-0-226-45800-7"><bdi>978-0-226-45800-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Black-Body+Theory+and+the+Quantum+Discontinuity%2C+1894-1912&rft.place=Chicago&rft.pub=University+of+Chicago+Press&rft.date=1987&rft.isbn=978-0-226-45800-7&rft.aulast=Kuhn&rft.aufirst=Thomas+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandsman2017" class="citation book cs1">Landsman, Klaas (2017). <i>Foundations of Quantum Theory</i>. Fundamental Theories of Physics. Vol. 188. Cham: Springer International Publishing. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-51777-3">10.1007/978-3-319-51777-3</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-51776-6" title="Special:BookSources/978-3-319-51776-6"><bdi>978-3-319-51776-6</bdi></a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0168-1222">0168-1222</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Quantum+Theory&rft.place=Cham&rft.series=Fundamental+Theories+of+Physics&rft.pub=Springer+International+Publishing&rft.date=2017&rft.issn=0168-1222&rft_id=info%3Adoi%2F10.1007%2F978-3-319-51777-3&rft.isbn=978-3-319-51776-6&rft.aulast=Landsman&rft.aufirst=Klaas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMackey2004" class="citation book cs1">Mackey, George W. (2004). <i>Mathematical Foundations of Quantum Mechanics</i>. Mineola, N.Y: Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-43517-6" title="Special:BookSources/978-0-486-43517-6"><bdi>978-0-486-43517-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Foundations+of+Quantum+Mechanics&rft.place=Mineola%2C+N.Y&rft.pub=Courier+Corporation&rft.date=2004&rft.isbn=978-0-486-43517-6&rft.aulast=Mackey&rft.aufirst=George+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcMahon2013" class="citation book cs1">McMahon, David (2013). <a rel="nofollow" class="external text" href="https://theswissbay.ch/pdf/Gentoomen%20Library/Misc/Demystified%20Series/McGraw-Hill%20-%20Quantum%20Mechanics%20Demystified%20%282006%29.pdf"><i>Quantum Mechanics Demystified, 2nd Edition</i></a> <span class="cs1-format">(PDF)</span>. New York, NY: McGraw-Hill Prof Med/Tech. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-176563-3" title="Special:BookSources/978-0-07-176563-3"><bdi>978-0-07-176563-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics+Demystified%2C+2nd+Edition&rft.place=New+York%2C+NY&rft.pub=McGraw-Hill+Prof+Med%2FTech&rft.date=2013&rft.isbn=978-0-07-176563-3&rft.aulast=McMahon&rft.aufirst=David&rft_id=https%3A%2F%2Ftheswissbay.ch%2Fpdf%2FGentoomen%2520Library%2FMisc%2FDemystified%2520Series%2FMcGraw-Hill%2520-%2520Quantum%2520Mechanics%2520Demystified%2520%25282006%2529.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoretti2017" class="citation book cs1">Moretti, Valter (2017). <i>Spectral Theory and Quantum Mechanics</i>. Unitext. Vol. 110. Cham: Springer International Publishing. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-70706-8">10.1007/978-3-319-70706-8</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-70705-1" title="Special:BookSources/978-3-319-70705-1"><bdi>978-3-319-70705-1</bdi></a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2038-5714">2038-5714</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:125121522">125121522</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spectral+Theory+and+Quantum+Mechanics&rft.place=Cham&rft.series=Unitext&rft.pub=Springer+International+Publishing&rft.date=2017&rft_id=info%3Adoi%2F10.1007%2F978-3-319-70706-8&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A125121522%23id-name%3DS2CID&rft.issn=2038-5714&rft.isbn=978-3-319-70705-1&rft.aulast=Moretti&rft.aufirst=Valter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoretti2019" class="citation book cs1">Moretti, Valter (2019). <i>Fundamental Mathematical Structures of Quantum Theory</i>. Cham: Springer International Publishing. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-030-18346-2">10.1007/978-3-030-18346-2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-18345-5" title="Special:BookSources/978-3-030-18345-5"><bdi>978-3-030-18345-5</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:197485828">197485828</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamental+Mathematical+Structures+of+Quantum+Theory&rft.place=Cham&rft.pub=Springer+International+Publishing&rft.date=2019&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A197485828%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-3-030-18346-2&rft.isbn=978-3-030-18345-5&rft.aulast=Moretti&rft.aufirst=Valter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPrugovecki2006" class="citation book cs1">Prugovecki, Eduard (2006). <i>Quantum Mechanics in Hilbert Space</i>. Mineola, NY: Courier Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-45327-9" title="Special:BookSources/978-0-486-45327-9"><bdi>978-0-486-45327-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics+in+Hilbert+Space&rft.place=Mineola%2C+NY&rft.pub=Courier+Dover+Publications&rft.date=2006&rft.isbn=978-0-486-45327-9&rft.aulast=Prugovecki&rft.aufirst=Eduard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReedSimon1972" class="citation book cs1">Reed, Michael; Simon, Barry (1972). <i>Methods of Modern Mathematical Physics</i>. New York: Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-585001-8" title="Special:BookSources/978-0-12-585001-8"><bdi>978-0-12-585001-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Methods+of+Modern+Mathematical+Physics&rft.place=New+York&rft.pub=Academic+Press&rft.date=1972&rft.isbn=978-0-12-585001-8&rft.aulast=Reed&rft.aufirst=Michael&rft.au=Simon%2C+Barry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShankar2013" class="citation book cs1">Shankar, R. (2013). <a rel="nofollow" class="external text" href="https://www.fisica.net/mecanica-quantica/Shankar%20-%20Principles%20of%20quantum%20mechanics.pdf"><i>Principles of Quantum Mechanics</i></a> <span class="cs1-format">(PDF)</span>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4615-7675-4" title="Special:BookSources/978-1-4615-7675-4"><bdi>978-1-4615-7675-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Quantum+Mechanics&rft.pub=Springer&rft.date=2013&rft.isbn=978-1-4615-7675-4&rft.aulast=Shankar&rft.aufirst=R.&rft_id=https%3A%2F%2Fwww.fisica.net%2Fmecanica-quantica%2FShankar%2520-%2520Principles%2520of%2520quantum%2520mechanics.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTeschl2009" class="citation book cs1">Teschl, Gerald (2009). <a rel="nofollow" class="external text" href="https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf"><i>Mathematical Methods in Quantum Mechanics</i></a> <span class="cs1-format">(PDF)</span>. Providence, R.I: American Mathematical Soc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4660-5" title="Special:BookSources/978-0-8218-4660-5"><bdi>978-0-8218-4660-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Methods+in+Quantum+Mechanics&rft.place=Providence%2C+R.I&rft.pub=American+Mathematical+Soc.&rft.date=2009&rft.isbn=978-0-8218-4660-5&rft.aulast=Teschl&rft.aufirst=Gerald&rft_id=https%3A%2F%2Fwww.mat.univie.ac.at%2F~gerald%2Fftp%2Fbook-schroe%2Fschroe.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Neumann2018" class="citation book cs1">von Neumann, John (2018). <i>Mathematical Foundations of Quantum Mechanics</i>. Princeton Oxford: Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-17856-1" title="Special:BookSources/978-0-691-17856-1"><bdi>978-0-691-17856-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Foundations+of+Quantum+Mechanics&rft.place=Princeton+Oxford&rft.pub=Princeton+University+Press&rft.date=2018&rft.isbn=978-0-691-17856-1&rft.aulast=von+Neumann&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeaver2001" class="citation book cs1">Weaver, Nik (2001). <i>Mathematical Quantization</i>. Chapman and Hall/CRC. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1201%2F9781420036237">10.1201/9781420036237</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-429-07514-8" title="Special:BookSources/978-0-429-07514-8"><bdi>978-0-429-07514-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Quantization&rft.pub=Chapman+and+Hall%2FCRC&rft.date=2001&rft_id=info%3Adoi%2F10.1201%2F9781420036237&rft.isbn=978-0-429-07514-8&rft.aulast=Weaver&rft.aufirst=Nik&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeyl1950" class="citation book cs1">Weyl, Hermann (1950). <i>The Theory of Groups and Quantum Mechanics</i>. Mineola, NY: Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-60269-1" title="Special:BookSources/978-0-486-60269-1"><bdi>978-0-486-60269-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Groups+and+Quantum+Mechanics&rft.place=Mineola%2C+NY&rft.pub=Courier+Corporation&rft.date=1950&rft.isbn=978-0-486-60269-1&rft.aulast=Weyl&rft.aufirst=Hermann&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+formulation+of+quantum+mechanics" class="Z3988"></span></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output 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.navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Quantum_mechanics332" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_mechanics_topics" title="Template:Quantum mechanics topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics_topics" title="Template talk:Quantum mechanics topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics_topics" title="Special:EditPage/Template:Quantum mechanics topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_mechanics332" style="font-size:114%;margin:0 4em"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a> <ul><li><a href="/wiki/Timeline_of_quantum_mechanics" title="Timeline of quantum mechanics">Timeline</a></li></ul></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamentals</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Born_rule" title="Born rule">Born rule</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li> <li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)"> Complementarity</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a> <ul><li><a href="/wiki/Ground_state" title="Ground state">Ground state</a></li> <li><a href="/wiki/Excited_state" title="Excited state">Excited state</a></li> <li><a href="/wiki/Degenerate_energy_levels" title="Degenerate energy levels">Degenerate levels</a></li> <li><a href="/wiki/Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li></ul></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Formulations</a></li> <li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Equations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Weyl_equation" title="Weyl equation">Weyl</a></li> <li><a href="/wiki/Majorana_equation" title="Majorana equation">Majorana</a></li> <li><a href="/wiki/Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell test</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li> <li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed choice</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_nanoscience" class="mw-redirect" title="Quantum nanoscience">Science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_biology" title="Quantum biology">Quantum biology</a></li> <li><a href="/wiki/Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li> <li><a href="/wiki/Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li> <li><a href="/wiki/Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li> <li><a href="/wiki/Quantum_mind" title="Quantum mind">Quantum mind</a></li> <li><a href="/wiki/Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li> <li><a href="/wiki/Quantum_spacetime" title="Quantum spacetime">Quantum spacetime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li> <li><a href="/wiki/Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">Quantum bus</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a> <ul><li><a href="/wiki/Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li> <li><a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">Quantum electronics</a></li> <li><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li> <li><a href="/wiki/Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li> <li><a href="/wiki/Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li> <li><a href="/wiki/Quantum_machine" title="Quantum machine">Quantum machine</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li> <li><a href="/wiki/Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li> <li><a href="/wiki/Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">Quantum network</a></li> <li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li> <li><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Extensions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li> <li><a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a> <ul><li><a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></li></ul></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat_in_popular_culture" title="Schrödinger's cat in popular culture">in popular culture</a></li></ul></li> <li><a href="/wiki/Wigner%27s_friend" title="Wigner's friend">Wigner's friend</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Quantum_mysticism" title="Quantum mysticism">Quantum mysticism</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Quantum_mechanics" title="Category:Quantum mechanics">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Functional_analysis_(topics_–_glossary)364" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Functional_analysis" title="Template:Functional analysis"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Functional_analysis" title="Template talk:Functional analysis"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Functional_analysis" title="Special:EditPage/Template:Functional analysis"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Functional_analysis_(topics_–_glossary)364" style="font-size:114%;margin:0 4em"><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a> (<a href="/wiki/List_of_functional_analysis_topics" title="List of functional analysis topics">topics</a> – <a href="/wiki/Glossary_of_functional_analysis" title="Glossary of functional analysis">glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Spaces</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><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_space" title="Banach space">Banach</a></li> <li><a href="/wiki/Besov_space" title="Besov space">Besov</a></li> <li><a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet</a></li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert</a></li> <li><a href="/wiki/H%C3%B6lder_space" class="mw-redirect" title="Hölder space">Hölder</a></li> <li><a href="/wiki/Nuclear_space" title="Nuclear space">Nuclear</a></li> <li><a href="/wiki/Orlicz_space" title="Orlicz space">Orlicz</a></li> <li><a href="/wiki/Schwartz_space" title="Schwartz space">Schwartz</a></li> <li><a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev</a></li> <li><a href="/wiki/Topological_vector_space" title="Topological vector space">Topological vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</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/Barrelled_space" title="Barrelled space">Barrelled</a></li> <li><a href="/wiki/Complete_topological_vector_space" title="Complete topological vector space">Complete</a></li> <li><a href="/wiki/Dual_space" title="Dual space">Dual</a> (<a href="/wiki/Dual_space#Algebraic_dual_space" title="Dual space">Algebraic</a> / <a href="/wiki/Dual_space#Continuous_dual_space" title="Dual space">Topological</a>)</li> <li><a href="/wiki/Locally_convex_topological_vector_space" title="Locally convex topological vector space">Locally convex</a></li> <li><a href="/wiki/Reflexive_space" title="Reflexive space">Reflexive</a></li> <li><a href="/wiki/Separable_space" title="Separable space">Separable</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Theorems_in_functional_analysis" title="Category:Theorems in functional analysis">Theorems</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/Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach</a></li> <li><a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation</a></li> <li><a href="/wiki/Closed_graph_theorem_(functional_analysis)" title="Closed graph theorem (functional analysis)">Closed graph</a></li> <li><a href="/wiki/Uniform_boundedness_principle" title="Uniform boundedness principle">Uniform boundedness principle</a></li> <li><a href="/wiki/Kakutani_fixed-point_theorem#Infinite-dimensional_generalizations" title="Kakutani fixed-point theorem">Kakutani fixed-point</a></li> <li><a href="/wiki/Krein%E2%80%93Milman_theorem" title="Krein–Milman theorem">Krein–Milman</a></li> <li><a href="/wiki/Min-max_theorem" title="Min-max theorem">Min–max</a></li> <li><a href="/wiki/Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark</a></li> <li><a href="/wiki/Banach%E2%80%93Alaoglu_theorem" title="Banach–Alaoglu theorem">Banach–Alaoglu</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operators</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/Adjoint_operator" class="mw-redirect" title="Adjoint operator">Adjoint</a></li> <li><a href="/wiki/Bounded_operator" title="Bounded operator">Bounded</a></li> <li><a href="/wiki/Compact_operator" title="Compact operator">Compact</a></li> <li><a href="/wiki/Hilbert%E2%80%93Schmidt_operator" title="Hilbert–Schmidt operator">Hilbert–Schmidt</a></li> <li><a href="/wiki/Normal_operator" title="Normal operator">Normal</a></li> <li><a href="/wiki/Nuclear_operator" title="Nuclear operator">Nuclear</a></li> <li><a href="/wiki/Trace_class" title="Trace class">Trace class</a></li> <li><a href="/wiki/Transpose_of_a_linear_map" title="Transpose of a linear map">Transpose</a></li> <li><a href="/wiki/Unbounded_operator" title="Unbounded operator">Unbounded</a></li> <li><a href="/wiki/Unitary_operator" title="Unitary operator">Unitary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebras</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/Banach_algebra" title="Banach algebra">Banach algebra</a></li> <li><a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a></li> <li><a href="/wiki/Spectrum_of_a_C*-algebra" title="Spectrum of a C*-algebra">Spectrum of a C*-algebra</a></li> <li><a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></li> <li><a href="/wiki/Group_algebra_of_a_locally_compact_group" title="Group algebra of a locally compact group">Group algebra of a locally compact group</a></li> <li><a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">Von Neumann algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Open problems</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/Invariant_subspace_problem" title="Invariant subspace problem">Invariant subspace problem</a></li> <li><a href="/wiki/Mahler%27s_conjecture" class="mw-redirect" title="Mahler's conjecture">Mahler's conjecture</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</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/Hardy_space" title="Hardy space">Hardy space</a></li> <li><a href="/wiki/Spectral_theory_of_ordinary_differential_equations" title="Spectral theory of ordinary differential equations">Spectral theory of ordinary differential equations</a></li> <li><a href="/wiki/Heat_kernel" title="Heat kernel">Heat kernel</a></li> <li><a href="/wiki/Index_theorem" class="mw-redirect" title="Index theorem">Index theorem</a></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Calculus of variations</a></li> <li><a href="/wiki/Functional_calculus" title="Functional calculus">Functional calculus</a></li> <li><a href="/wiki/Integral_linear_operator" title="Integral linear operator">Integral linear operator</a></li> <li><a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomial</a></li> <li><a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">Topological quantum field theory</a></li> <li><a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a></li> <li><a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a></li> <li><a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">Distribution</a> (or <a href="/wiki/Generalized_function" title="Generalized function">Generalized functions</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Advanced topics</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/Approximation_property" title="Approximation property">Approximation property</a></li> <li><a href="/wiki/Balanced_set" title="Balanced set">Balanced set</a></li> <li><a href="/wiki/Choquet_theory" title="Choquet theory">Choquet theory</a></li> <li><a href="/wiki/Weak_topology" title="Weak topology">Weak topology</a></li> <li><a href="/wiki/Banach%E2%80%93Mazur_distance" class="mw-redirect" title="Banach–Mazur distance">Banach–Mazur distance</a></li> <li><a href="/wiki/Tomita%E2%80%93Takesaki_theory" title="Tomita–Takesaki theory">Tomita–Takesaki theory</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" 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Mathematical formulation of quantum mechanics - Wikipedia