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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Role_in_quantum_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Role in quantum mechanics</span> </div> </a> <ul id="toc-Role_in_quantum_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Measurements" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Measurements"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Measurements</span> </div> </a> <ul id="toc-Measurements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eigenstates_and_pure_states" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Eigenstates_and_pure_states"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Eigenstates and pure states</span> </div> </a> <ul id="toc-Eigenstates_and_pure_states-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Representations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Representations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Representations</span> </div> </a> <ul id="toc-Representations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wave_function_representations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Wave_function_representations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Wave function representations</span> </div> </a> <button aria-controls="toc-Wave_function_representations-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 Wave function representations subsection</span> </button> <ul id="toc-Wave_function_representations-sublist" class="vector-toc-list"> <li id="toc-Pure_states_of_wave_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pure_states_of_wave_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Pure states of wave functions</span> </div> </a> <ul id="toc-Pure_states_of_wave_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mixed_states_of_wave_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mixed_states_of_wave_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Mixed states of wave functions</span> </div> </a> <ul id="toc-Mixed_states_of_wave_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Schrödinger_picture_vs._Heisenberg_picture" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Schrödinger_picture_vs._Heisenberg_picture"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Schrödinger picture vs. Heisenberg picture</span> </div> </a> <ul id="toc-Schrödinger_picture_vs._Heisenberg_picture-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Formalism_in_quantum_physics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formalism_in_quantum_physics"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Formalism in quantum physics</span> </div> </a> <button aria-controls="toc-Formalism_in_quantum_physics-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 Formalism in quantum physics subsection</span> </button> <ul id="toc-Formalism_in_quantum_physics-sublist" class="vector-toc-list"> <li id="toc-Pure_states_as_rays_in_a_complex_Hilbert_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pure_states_as_rays_in_a_complex_Hilbert_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Pure states as rays in a complex Hilbert space</span> </div> </a> <ul id="toc-Pure_states_as_rays_in_a_complex_Hilbert_space-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">7.2</span> <span>Spin</span> </div> </a> <ul id="toc-Spin-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Many-body_states_and_particle_statistics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Many-body_states_and_particle_statistics"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Many-body states and particle statistics</span> </div> </a> <ul id="toc-Many-body_states_and_particle_statistics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Basis_states_of_one-particle_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Basis_states_of_one-particle_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4</span> <span>Basis states of one-particle systems</span> </div> </a> <ul id="toc-Basis_states_of_one-particle_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pure_states_vs._bound_states" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pure_states_vs._bound_states"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.5</span> <span>Pure states vs. bound states</span> </div> </a> <ul id="toc-Pure_states_vs._bound_states-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Superposition_of_pure_states" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Superposition_of_pure_states"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.6</span> <span>Superposition of pure states</span> </div> </a> <ul id="toc-Superposition_of_pure_states-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mixed_states" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mixed_states"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.7</span> <span>Mixed states</span> </div> </a> <ul id="toc-Mixed_states-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mathematical_generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mathematical_generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Mathematical generalizations</span> </div> </a> <ul id="toc-Mathematical_generalizations-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">9</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">10</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">11</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">12</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">Quantum state</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 41 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-41" 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">41 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%AD%D8%A7%D9%84%D8%A9_%D9%83%D9%85%D9%88%D9%85%D9%8A%D8%A9" title="حالة كمومية – Arabic" lang="ar" hreflang="ar" data-title="حالة كمومية" 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%85%E0%A6%AC%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BE" title="কোয়ান্টাম অবস্থা – Bangla" lang="bn" hreflang="bn" data-title="কোয়ান্টাম অবস্থা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%BE_%D1%81%D1%8A%D1%81%D1%82%D0%BE%D1%8F%D0%BD%D0%B8%D0%B5" title="Квантово състояние – Bulgarian" lang="bg" hreflang="bg" data-title="Квантово състояние" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Estat_qu%C3%A0ntic" title="Estat quàntic – Catalan" lang="ca" hreflang="ca" data-title="Estat quàntic" 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_%D1%82%C4%83%D1%80%C4%83%D0%BC" title="Квантла тăрăм – Chuvash" lang="cv" hreflang="cv" data-title="Квантла тăрăм" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kvantov%C3%BD_stav" title="Kvantový stav – Czech" lang="cs" hreflang="cs" data-title="Kvantový stav" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Zustand_(Quantenmechanik)" title="Zustand (Quantenmechanik) – German" lang="de" hreflang="de" data-title="Zustand (Quantenmechanik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kvantolek" title="Kvantolek – Estonian" lang="et" hreflang="et" data-title="Kvantolek" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%B2%CE%B1%CE%BD%CF%84%CE%B9%CE%BA%CE%AE_%CE%BA%CE%B1%CF%84%CE%AC%CF%83%CF%84%CE%B1%CF%83%CE%B7" title="Κβαντική κατάσταση – Greek" lang="el" hreflang="el" data-title="Κβαντική κατάσταση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Estado_cu%C3%A1ntico" title="Estado cuántico – Spanish" lang="es" hreflang="es" data-title="Estado cuántico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kvantuma_stato" title="Kvantuma stato – Esperanto" lang="eo" hreflang="eo" data-title="Kvantuma stato" 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/Egoera_kuantiko" title="Egoera kuantiko – Basque" lang="eu" hreflang="eu" data-title="Egoera kuantiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AD%D8%A7%D9%84%D8%AA_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85%DB%8C" title="حالت کوانتومی – Persian" lang="fa" hreflang="fa" data-title="حالت کوانتومی" 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/%C3%89tat_quantique" title="État quantique – French" lang="fr" hreflang="fr" data-title="État quantique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%94%D5%BE%D5%A1%D5%B6%D5%BF%D5%A1%D5%B5%D5%AB%D5%B6_%D5%BE%D5%AB%D5%B3%D5%A1%D5%AF" title="Քվանտային վիճակ – Armenian" lang="hy" hreflang="hy" data-title="Քվանտային վիճակ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Keadaan_kuantum" title="Keadaan kuantum – Indonesian" lang="id" hreflang="id" data-title="Keadaan kuantum" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Stato_quantico" title="Stato quantico – Italian" lang="it" hreflang="it" data-title="Stato quantico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A6%D7%91_%D7%A7%D7%95%D7%95%D7%A0%D7%98%D7%99" title="מצב קוונטי – Hebrew" lang="he" hreflang="he" data-title="מצב קוונטי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Kvantum%C3%A1llapot" title="Kvantumállapot – Hungarian" lang="hu" hreflang="hu" data-title="Kvantumállapot" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kwantumtoestand" title="Kwantumtoestand – Dutch" lang="nl" hreflang="nl" data-title="Kwantumtoestand" 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%E7%8A%B6%E6%85%8B" title="量子状態 – Japanese" lang="ja" hreflang="ja" data-title="量子状態" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kvantetilstand" title="Kvantetilstand – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kvantetilstand" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A9%81%E0%A8%86%E0%A8%82%E0%A8%9F%E0%A8%AE_%E0%A8%85%E0%A8%B5%E0%A8%B8%E0%A8%A5%E0%A8%BE" title="ਕੁਆਂਟਮ ਅਵਸਥਾ – Punjabi" lang="pa" hreflang="pa" data-title="ਕੁਆਂਟਮ ਅਵਸਥਾ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Stan_kwantowy" title="Stan kwantowy – Polish" lang="pl" hreflang="pl" data-title="Stan kwantowy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Estado_qu%C3%A2ntico" title="Estado quântico – Portuguese" lang="pt" hreflang="pt" data-title="Estado quântico" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Stare_cuantic%C4%83" title="Stare cuantică – Romanian" lang="ro" hreflang="ro" data-title="Stare cuantică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%BE%D0%B5_%D1%81%D0%BE%D1%81%D1%82%D0%BE%D1%8F%D0%BD%D0%B8%D0%B5" title="Квантовое состояние – Russian" lang="ru" hreflang="ru" data-title="Квантовое состояние" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Quantum_state" title="Quantum state – Simple English" lang="en-simple" hreflang="en-simple" data-title="Quantum state" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kvantno_stanje" title="Kvantno stanje – Slovenian" lang="sl" hreflang="sl" data-title="Kvantno stanje" data-language-autonym="Slovenščina" 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Eigenstate&redirect=no" class="mw-redirect" title="Eigenstate">Eigenstate</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical entity to describe the probability of each possible measurement on a system</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output 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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 class="mw-selflink selflink">State</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li></ul></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Experiments</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell's inequality</a></li> <li><a href="/wiki/CHSH_inequality" title="CHSH inequality">CHSH inequality</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Leggett_inequality" title="Leggett inequality">Leggett inequality</a></li> <li><a href="/wiki/Leggett%E2%80%93Garg_inequality" title="Leggett–Garg inequality">Leggett–Garg inequality</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li></ul> </div> <ul><li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a> <ul><li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice</a></li></ul></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed-choice</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Formulations</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Overview</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase-space</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Sum-over-histories (path integral)</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective-collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Advanced topics</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Yakir_Aharonov" title="Yakir Aharonov">Aharonov</a></li> <li><a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">Bell</a></li> <li><a href="/wiki/Hans_Bethe" title="Hans Bethe">Bethe</a></li> <li><a href="/wiki/Patrick_Blackett" title="Patrick Blackett">Blackett</a></li> <li><a href="/wiki/Felix_Bloch" title="Felix Bloch">Bloch</a></li> <li><a href="/wiki/David_Bohm" title="David Bohm">Bohm</a></li> <li><a href="/wiki/Niels_Bohr" title="Niels Bohr">Bohr</a></li> <li><a href="/wiki/Max_Born" title="Max Born">Born</a></li> <li><a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Bose</a></li> <li><a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">de Broglie</a></li> <li><a href="/wiki/Arthur_Compton" title="Arthur Compton">Compton</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Clinton_Davisson" title="Clinton Davisson">Davisson</a></li> <li><a href="/wiki/Peter_Debye" title="Peter Debye">Debye</a></li> <li><a href="/wiki/Paul_Ehrenfest" title="Paul Ehrenfest">Ehrenfest</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Hugh_Everett_III" title="Hugh Everett III">Everett</a></li> <li><a href="/wiki/Vladimir_Fock" title="Vladimir Fock">Fock</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a></li> <li><a href="/wiki/Roy_J._Glauber" title="Roy J. Glauber">Glauber</a></li> <li><a href="/wiki/Martin_Gutzwiller" title="Martin Gutzwiller">Gutzwiller</a></li> <li><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a></li> <li><a href="/wiki/Hans_Kramers" title="Hans Kramers">Kramers</a></li> <li><a href="/wiki/Willis_Lamb" title="Willis Lamb">Lamb</a></li> <li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li> <li><a href="/wiki/Max_von_Laue" title="Max von Laue">Laue</a></li> <li><a href="/wiki/Henry_Moseley" title="Henry Moseley">Moseley</a></li> <li><a href="/wiki/Robert_Andrews_Millikan" title="Robert Andrews Millikan">Millikan</a></li> <li><a href="/wiki/Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Onnes</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/Isidor_Isaac_Rabi" title="Isidor Isaac Rabi">Rabi</a></li> <li><a href="/wiki/C._V._Raman" title="C. V. Raman">Raman</a></li> <li><a href="/wiki/Johannes_Rydberg" title="Johannes Rydberg">Rydberg</a></li> <li><a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödinger</a></li> <li><a href="/wiki/Michelle_Simmons" title="Michelle Simmons">Simmons</a></li> <li><a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Sommerfeld</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Wilhelm_Wien" title="Wilhelm Wien">Wien</a></li> <li><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner</a></li> <li><a href="/wiki/Pieter_Zeeman" title="Pieter Zeeman">Zeeman</a></li> <li><a href="/wiki/Anton_Zeilinger" title="Anton Zeilinger">Zeilinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar" style="border-top:1px solid #aaa;padding-top:0.1em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_mechanics" title="Template:Quantum mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics" title="Template talk:Quantum mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics" title="Special:EditPage/Template:Quantum mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Quantum_physics" class="mw-redirect" title="Quantum physics">quantum physics</a>, a <b>quantum state</b> is a mathematical entity that embodies the knowledge of a quantum system. <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a> specifies the construction, evolution, and <a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">measurement</a> of a quantum state. The result is a prediction for the system represented by the state. Knowledge of the quantum state, and the rules for the system's evolution in time, exhausts all that can be known about a quantum system. </p><p>Quantum states may be defined differently for different kinds of systems or problems. Two broad categories are </p> <ul><li><a href="/wiki/Wave_functions" class="mw-redirect" title="Wave functions">wave functions</a> describing quantum systems using position or momentum variables and</li> <li>the more abstract <a href="#Formalism_in_quantum_physics">vector quantum states</a>.</li></ul> <p>Historical, educational, and application-focused problems typically feature wave functions; modern professional physics uses the abstract vector states. In both categories, quantum states divide into <a href="#Pure_state">pure</a> versus <a href="#Mixed_state">mixed states</a>, or into <a href="/wiki/Coherent_state" title="Coherent state">coherent states</a> and incoherent states. Categories with special properties include <a href="/wiki/Stationary_state" title="Stationary state">stationary states</a> for time independence and <a href="/wiki/Quantum_vacuum_state" title="Quantum vacuum state">quantum vacuum states</a> in <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="From_the_states_of_classical_mechanics">From the states of classical mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_state&action=edit&section=1" title="Edit section: From the states of classical mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As a tool for physics, quantum states grew out of states in <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a>. A classical dynamical state consists of a set of dynamical variables with well-defined <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real</a> values at each instant of time.<sup id="cite_ref-messiah_1-0" class="reference"><a href="#cite_note-messiah-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 3">: 3 </span></sup> For example, the state of a cannon ball would consist of its position and velocity. The state values evolve under equations of motion and thus remain strictly determined. If we know the position of a cannon and the exit velocity of its projectiles, then we can use equations containing the force of gravity to predict the trajectory of a cannon ball precisely. </p><p>Similarly, quantum states consist of sets of dynamical variables that evolve under equations of motion. However, the values derived from quantum states are <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a>, quantized, limited by <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty relations</a>,<sup id="cite_ref-messiah_1-1" class="reference"><a href="#cite_note-messiah-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 159">: 159 </span></sup> and only provide a <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> for the outcomes for a system. These constraints alter the nature of quantum dynamic variables. For example, the quantum state of an electron in a <a href="/wiki/Double-slit_experiment" title="Double-slit experiment">double-slit experiment</a> would consist of complex values over the detection region and, when squared, only predict the probability distribution of electron counts across the detector. </p> <div class="mw-heading mw-heading2"><h2 id="Role_in_quantum_mechanics">Role in quantum mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_state&action=edit&section=2" title="Edit section: Role in quantum mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The process of describing a quantum system with quantum mechanics begins with identifying a set of variables defining the quantum state of the system.<sup id="cite_ref-messiah_1-2" class="reference"><a href="#cite_note-messiah-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 204">: 204 </span></sup> The set will contain <a href="/wiki/Observable#Compatible_and_incompatible_observables_in_quantum_mechanics" title="Observable">compatible and incompatible variables</a>. Simultaneous measurement of a <a href="/wiki/Complete_set_of_commuting_observables" title="Complete set of commuting observables">complete set of compatible variables</a> prepares the system in a unique state. The state then evolves deterministically according to the <a href="/wiki/Equations_of_motion" title="Equations of motion">equations of motion</a>. Subsequent measurement of the state produces a sample from a probability distribution predicted by the quantum mechanical <a href="/wiki/Operator_(physics)#Operators_in_quantum_mechanics" title="Operator (physics)">operator</a> corresponding to the measurement. </p><p>The fundamentally statistical or probabilisitic nature of quantum measurements changes the role of quantum states in quantum mechanics compared to classical states in classical mechanics. In classical mechanics, the initial state of one or more bodies is measured; the state evolves according to the equations of motion; measurements of the final state are compared to predictions. In quantum mechanics, ensembles of identically prepared quantum states evolve according to the equations of motion and many repeated measurements are compared to predicted probability distributions.<sup id="cite_ref-messiah_1-3" class="reference"><a href="#cite_note-messiah-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 204">: 204 </span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Measurements">Measurements</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_state&action=edit&section=3" title="Edit section: Measurements"><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/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement in quantum mechanics</a></div> <p>Measurements, macroscopic operations on quantum states, filter the state.<sup id="cite_ref-messiah_1-4" class="reference"><a href="#cite_note-messiah-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 196">: 196 </span></sup> Whatever the input quantum state might be, repeated identical measurements give consistent values. For this reason, measurements 'prepare' quantum states for experiments, placing the system in a partially defined state. Subsequent measurements may either further prepare the system – these are compatible measurements – or it may alter the state, redefining it – these are called incompatible or complementary measurements. For example, we may measure the momentum of a state along the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> axis any number of times and get the same result, but if we measure the position after once measuring the momentum, subsequent measurements of momentum are changed. The quantum state appears unavoidably altered by incompatible measurements. This is known as the <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Eigenstates_and_pure_states">Eigenstates and pure states</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_state&action=edit&section=4" title="Edit section: Eigenstates and pure states"><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">See also: <a href="/wiki/Eigenvalues_and_eigenvectors#Schrödinger_equation" title="Eigenvalues and eigenvectors">Eigenvalues and eigenvectors § Schrödinger equation</a></div> <p><span class="anchor" id="Eigenstate"></span> The quantum state after a measurement is in an <b>eigenstate</b> corresponding to that measurement and the value measured.<sup id="cite_ref-messiah_1-5" class="reference"><a href="#cite_note-messiah-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 202">: 202 </span></sup> Other aspects of the state may be unknown. Repeating the measurement will not alter the state. In some cases, compatible measurements can further refine the state, causing it to be an eigenstate corresponding to all these measurements.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> A full set of compatible measurements produces a <b>pure state</b>. Any state that is not pure is called a <b>mixed state</b> as discussed in more depth <a href="#Mixed_states_of_wave_functions">below</a>.<sup id="cite_ref-messiah_1-6" class="reference"><a href="#cite_note-messiah-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 204">: 204 </span></sup><sup id="cite_ref-peres_3-0" class="reference"><a href="#cite_note-peres-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 73">: 73 </span></sup> </p><p>The eigenstate solutions to the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> can be formed into pure states. Experiments rarely produce pure states. Therefore statistical mixtures of solutions must be compared to experiments.<sup id="cite_ref-messiah_1-7" class="reference"><a href="#cite_note-messiah-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 204">: 204 </span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Representations">Representations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_state&action=edit&section=5" title="Edit section: Representations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The same physical quantum state can be expressed mathematically in different ways called <b>representations</b>.<sup id="cite_ref-messiah_1-8" class="reference"><a href="#cite_note-messiah-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The position wave function is one representation often seen first in introductions to quantum mechanics. The equivalent momentum wave function is another wave function based representation. Representations are analogous to coordinate systems<sup id="cite_ref-messiah_1-9" class="reference"><a href="#cite_note-messiah-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 244">: 244 </span></sup> or similar mathematical devices like <a href="/wiki/Parametric_equation" title="Parametric equation">parametric equations</a>. Selecting a representation will make some aspects of a problem easier at the cost of making other things difficult. </p><p>In formal quantum mechanics (see <i><a href="#Formalism_in_quantum_physics">§ Formalism in quantum physics</a></i> below) the theory develops in terms of abstract '<a href="/wiki/Vector_space" title="Vector space">vector space</a>', avoiding any particular representation. This allows many elegant concepts of quantum mechanics to be expressed and to be applied even in cases where no classical analog exists.<sup id="cite_ref-messiah_1-10" class="reference"><a href="#cite_note-messiah-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 244">: 244 </span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Wave_function_representations">Wave function representations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_state&action=edit&section=6" title="Edit section: Wave function representations"><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/Wave_function" title="Wave function">Wave function</a></div> <p><a href="/wiki/Wave_function" title="Wave function">Wave functions</a> represent quantum states, particularly when they are functions of position or of <a href="/wiki/Momentum" title="Momentum">momentum</a>. Historically, definitions of quantum states used wavefunctions before the more formal methods were developed.<sup id="cite_ref-Whittaker_4-0" class="reference"><a href="#cite_note-Whittaker-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 268">: 268 </span></sup> The wave function is a complex-valued function of any complete set of commuting or compatible <a href="/wiki/Degrees_of_freedom" title="Degrees of freedom">degrees of freedom</a>. For example, one set could be the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbeca34b28f569a407ef74a955d041df9f360268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}"></span> spatial coordinates of an electron. Preparing a system by measuring the complete set of compatible observables produces a <b>pure quantum state</b>. More common, incomplete preparation produces a <b>mixed quantum state</b>. Wave function solutions of <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger's equations of motion</a> for operators corresponding to measurements can readily be expressed as pure states; they must be combined with statistical weights matching experimental preparation to compute the expected probability distribution.<sup id="cite_ref-messiah_1-11" class="reference"><a href="#cite_note-messiah-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 205">: 205 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Pure_states_of_wave_functions">Pure states of wave functions <span class="anchor" id="Pure_state"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_state&action=edit&section=7" title="Edit section: Pure states of wave functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Hydrogen_Density_Plots.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Hydrogen_Density_Plots.png/450px-Hydrogen_Density_Plots.png" decoding="async" width="450" height="409" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Hydrogen_Density_Plots.png/675px-Hydrogen_Density_Plots.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Hydrogen_Density_Plots.png/900px-Hydrogen_Density_Plots.png 2x" data-file-width="2200" data-file-height="2000" /></a><figcaption><a href="/wiki/Probability_density" class="mw-redirect" title="Probability density">Probability densities</a> for the electron of a hydrogen atom in different quantum states.</figcaption></figure> <p>Numerical or analytic solutions in quantum mechanics can be expressed as <b>pure states</b>. These solution states, called <a href="/wiki/Introduction_to_eigenstates" class="mw-redirect" title="Introduction to eigenstates">eigenstates</a>, are labeled with quantized values, typically <a href="/wiki/Quantum_numbers" class="mw-redirect" title="Quantum numbers">quantum numbers</a>. For example, when dealing with the <a href="/wiki/Spectrum#Energy_spectrum" title="Spectrum">energy spectrum</a> of the <a href="/wiki/Electron" title="Electron">electron</a> in a <a href="/wiki/Hydrogen_atom" title="Hydrogen atom">hydrogen atom</a>, the relevant pure states are identified by the <a href="/wiki/Principal_quantum_number" title="Principal quantum number">principal quantum number</a> <span class="texhtml"><i>n</i></span>, the <a href="/wiki/Angular_momentum_quantum_number" class="mw-redirect" title="Angular momentum quantum number">angular momentum quantum number</a> <span class="texhtml"><i>ℓ</i></span>, the <a href="/wiki/Magnetic_quantum_number" title="Magnetic quantum number">magnetic quantum number</a> <span class="texhtml"><i>m</i></span>, and the <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> <i>z</i>-component <span class="texhtml"><i>s</i><sub><i>z</i></sub></span>. For another example, if the spin of an electron is measured in any direction, e.g. with a <a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach experiment</a>, there are two possible results: up or down. A pure state here is represented by a two-dimensional <a href="/wiki/Complex_number" title="Complex number">complex</a> vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e26787affffd22458b880995530dfd4fa175e694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.843ex;" alt="{\displaystyle (\alpha ,\beta )}"></span>, with a length of one; that is, with <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 |\alpha |^{2}+|\beta |^{2}=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>β</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\alpha |^{2}+|\beta |^{2}=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17f71821d3f8bd5f9b0ace13a933534cb47cd33a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.264ex; height:3.343ex;" alt="{\displaystyle |\alpha |^{2}+|\beta |^{2}=1,}"></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 |\alpha |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </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/fa0a69f9288fbf3cd30a6daacb659e3175a0fb9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.781ex; height:2.843ex;" alt="{\displaystyle |\alpha |}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\beta |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\beta |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f49d8960bd5eb7e064f64859b7b1e71b4d4ce35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.626ex; height:2.843ex;" alt="{\displaystyle |\beta |}"></span> are the <a href="/wiki/Complex_number#Modulus_and_argument" title="Complex number">absolute values</a> 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 \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> 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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>. </p><p>The <a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">postulates of quantum mechanics</a> state that pure states, at a given time <span class="texhtml"><i> t</i></span>, correspond to <a href="/wiki/Vector_space" title="Vector space">vectors</a> in a <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#In_quantum_mechanics" title="Hilbert space">Hilbert space</a>, while each measurable physical quantity (such as the energy or momentum of a <a href="/wiki/Particle" title="Particle">particle</a>) is associated with a mathematical <a href="/wiki/Operator_(physics)" title="Operator (physics)">operator</a> called the <b><a href="/wiki/Observable" title="Observable">observable</a></b>. The operator serves as a <a href="/wiki/Linear_mapping" class="mw-redirect" title="Linear mapping">linear function</a> that acts on the states of the system. The <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of the operator correspond to the possible values of the observable. For example, it is possible to observe a particle with a momentum of 1 kg⋅m/s if and only if one of the eigenvalues of the momentum operator is 1 kg⋅m/s. The corresponding <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvector</a> (which physicists call an <b>eigenstate</b>) with eigenvalue 1 kg⋅m/s would be a quantum state with a definite, well-defined value of momentum of 1 kg⋅m/s, with no <a href="/wiki/Heisenberg_uncertainty_principle" class="mw-redirect" title="Heisenberg uncertainty principle">quantum uncertainty</a>. If its momentum were measured, the result is guaranteed to be 1 kg⋅m/s. </p><p>On the other hand, a pure state described as a <a href="/wiki/Superposition" class="mw-redirect" title="Superposition">superposition</a> of multiple different eigenstates <i>does</i> in general have quantum uncertainty for the given observable. Using <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">bra–ket notation</a>, this <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of eigenstates can be represented as:<sup id="cite_ref-sakurai_5-0" class="reference"><a href="#cite_note-sakurai-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 22, 171, 172">: 22, 171, 172 </span></sup> <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 (t)\rangle =\sum _{n}C_{n}(t)|\Phi _{n}\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 stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi (t)\rangle =\sum _{n}C_{n}(t)|\Phi _{n}\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5b33f6c757da88ca3f8e5834d094d396af9bad4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.473ex; height:5.509ex;" alt="{\displaystyle |\Psi (t)\rangle =\sum _{n}C_{n}(t)|\Phi _{n}\rangle .}"></span> The coefficient that corresponds to a particular state in the linear combination is a complex number, thus allowing interference effects between states. The coefficients are time dependent. How a quantum state changes in time is governed by the <a href="/wiki/Hamiltonian_(quantum_mechanics)#Schrödinger_equation" title="Hamiltonian (quantum mechanics)">time evolution operator</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Mixed_states_of_wave_functions">Mixed states of wave functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_state&action=edit&section=8" title="Edit section: Mixed states of wave functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A mixed quantum state corresponds to a probabilistic mixture of pure states; however, different distributions of pure states can generate equivalent (i.e., physically indistinguishable) mixed states. A <a href="/wiki/Mixture_distribution" title="Mixture distribution">mixture</a> of quantum states is again a quantum state. </p><p>A mixed state for electron spins, in the density-matrix formulation, has the structure of a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}"></span> matrix that is <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian</a> and positive semi-definite, and has <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> 1.<sup id="cite_ref-rieffel_6-0" class="reference"><a href="#cite_note-rieffel-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> A more complicated case is given (in <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">bra–ket notation</a>) by the <a href="/wiki/Singlet_state" title="Singlet state">singlet state</a>, which exemplifies <a href="/wiki/Quantum_entanglement" title="Quantum entanglement">quantum entanglement</a>: <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 ={\frac {1}{\sqrt {2}}}{\bigl (}\left|\uparrow \downarrow \right\rangle -\left|\downarrow \uparrow \right\rangle {\bigr )},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mi>ψ</mi> <mo>⟩</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow> <mo>|</mo> <mrow> <mo stretchy="false">↑↓</mo> </mrow> <mo>⟩</mo> </mrow> <mo>−</mo> <mrow> <mo>|</mo> <mrow> <mo stretchy="false">↓↑</mo> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\psi \right\rangle ={\frac {1}{\sqrt {2}}}{\bigl (}\left|\uparrow \downarrow \right\rangle -\left|\downarrow \uparrow \right\rangle {\bigr )},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d06827dd5b1bd12008b13f66d4d86eed6cdbc850" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:24.242ex; height:6.176ex;" alt="{\displaystyle \left|\psi \right\rangle ={\frac {1}{\sqrt {2}}}{\bigl (}\left|\uparrow \downarrow \right\rangle -\left|\downarrow \uparrow \right\rangle {\bigr )},}"></span> which involves <a href="/wiki/Quantum_superposition" title="Quantum superposition">superposition</a> of joint spin states for two particles with spin 1/2. The singlet state satisfies the property that if the particles' spins are measured along the same direction then either the spin of the first particle is observed up and the spin of the second particle is observed down, or the first one is observed down and the second one is observed up, both possibilities occurring with equal probability. </p><p>A pure quantum state can be represented by a <a href="/wiki/Ray_(quantum_theory)" class="mw-redirect" title="Ray (quantum theory)">ray</a> in a <a href="/wiki/Projective_Hilbert_space" title="Projective Hilbert space">projective Hilbert space</a> over the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, while mixed states are represented by <a href="/wiki/Density_matrix" title="Density matrix">density matrices</a>, which are <a href="/wiki/Definiteness_of_a_matrix" class="mw-redirect" title="Definiteness of a matrix">positive semidefinite operators</a> that act on Hilbert spaces.<sup id="cite_ref-holevo_7-0" class="reference"><a href="#cite_note-holevo-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-peres_3-1" class="reference"><a href="#cite_note-peres-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Schr%C3%B6dinger%E2%80%93HJW_theorem" title="Schrödinger–HJW theorem">Schrödinger–HJW theorem</a> classifies the multitude of ways to write a given mixed state as a <a href="/wiki/Convex_combination" title="Convex combination">convex combination</a> of pure states.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Before a particular <a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">measurement</a> is performed on a quantum system, the theory gives only a <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> for the outcome, and the form that this distribution takes is completely determined by the quantum state and the <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operators</a> describing the measurement. Probability distributions for different measurements exhibit tradeoffs exemplified by the <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a>: a state that implies a narrow spread of possible outcomes for one experiment necessarily implies a wide spread of possible outcomes for another. </p><p>Statistical mixtures of states are a different type of linear combination. A statistical mixture of states is a <a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">statistical ensemble</a> of independent systems. Statistical mixtures represent the degree of knowledge whilst the uncertainty within quantum mechanics is fundamental. Mathematically, a statistical mixture is not a combination using complex coefficients, but rather a combination using real-valued, positive probabilities of different 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 \Phi _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c3c96a91205fb1ae9d97b9e93b763b424bbac5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.897ex; height:2.509ex;" alt="{\displaystyle \Phi _{n}}"></span>. A number <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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5949c8b1de44005a1af3a11188361f2a830842d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.711ex; height:2.509ex;" alt="{\displaystyle P_{n}}"></span> represents the probability of a randomly selected system being 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 \Phi _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c3c96a91205fb1ae9d97b9e93b763b424bbac5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.897ex; height:2.509ex;" alt="{\displaystyle \Phi _{n}}"></span>. Unlike the linear combination case each system is in a definite eigenstate.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>The expectation value <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 A\rangle }_{\sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\langle A\rangle }_{\sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/269b0760a9a2e1d28023b1bd1f9a37304bc59bc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.725ex; height:3.009ex;" alt="{\displaystyle {\langle A\rangle }_{\sigma }}"></span> of an observable <span class="texhtml"><i>A</i></span> is a statistical mean of measured values of the observable. It is this mean, and the distribution of probabilities, that is predicted by physical theories. </p><p>There is no state that is simultaneously an eigenstate for <i>all</i> observables. For example, we cannot prepare a state such that both the position measurement <span class="texhtml"><i>Q</i>(<i>t</i>)</span> and the momentum measurement <span class="texhtml"><i>P</i>(<i>t</i>)</span> (at the same time <span class="texhtml mvar" style="font-style:italic;">t</span>) are known exactly; at least one of them will have a range of possible values.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> This is the content of the <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Heisenberg uncertainty relation</a>. </p><p>Moreover, in contrast to classical mechanics, it is unavoidable that <i>performing a measurement on the system generally changes its state</i>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 4">: 4 </span></sup> More precisely: After measuring an observable <i>A</i>, the system will be in an eigenstate of <i>A</i>; thus the state has changed, unless the system was already in that eigenstate. This expresses a kind of logical consistency: If we measure <i>A</i> twice in the same run of the experiment, the measurements being directly consecutive in time,<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> then they will produce the same results. This has some strange consequences, however, as follows. </p><p>Consider two <a href="/wiki/Incompatible_observables" class="mw-redirect" title="Incompatible observables">incompatible observables</a>, <span class="texhtml"><i>A</i></span> and <span class="texhtml"><i>B</i></span>, where <span class="texhtml"><i>A</i></span> corresponds to a measurement earlier in time than <span class="texhtml"><i>B</i></span>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> Suppose that the system is in an eigenstate of <span class="texhtml"><i>B</i></span> at the experiment's beginning. If we measure only <span class="texhtml"><i>B</i></span>, all runs of the experiment will yield the same result. If we measure first <span class="texhtml"><i>A</i></span> and then <span class="texhtml"><i>B</i></span> in the same run of the experiment, the system will transfer to an eigenstate of <span class="texhtml"><i>A</i></span> after the first measurement, and we will generally notice that the results of <span class="texhtml"><i>B</i></span> are statistical. Thus: <i>Quantum mechanical measurements influence one another</i>, and the order in which they are performed is important. </p><p>Another feature of quantum states becomes relevant if we consider a physical system that consists of multiple subsystems; for example, an experiment with two particles rather than one. Quantum physics allows for certain states, called <i>entangled states</i>, that show certain statistical correlations between measurements on the two particles which cannot be explained by classical theory. For details, see <i><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Quantum entanglement</a></i>. These entangled states lead to experimentally testable properties (<a href="/wiki/Bell%27s_theorem" title="Bell's theorem">Bell's theorem</a>) that allow us to distinguish between quantum theory and alternative classical (non-quantum) models. </p> <div class="mw-heading mw-heading3"><h3 id="Schrödinger_picture_vs._Heisenberg_picture"><span id="Schr.C3.B6dinger_picture_vs._Heisenberg_picture"></span>Schrödinger picture vs. Heisenberg picture</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_state&action=edit&section=9" title="Edit section: Schrödinger picture vs. Heisenberg picture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One can take the observables to be dependent on time, while the state <span class="texhtml"><i>σ</i></span> was fixed once at the beginning of the experiment. This approach is called the <a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg picture</a>. (This approach was taken in the later part of the discussion above, with time-varying observables <span class="texhtml"><i>P</i>(<i>t</i>)</span>, <span class="texhtml"><i>Q</i>(<i>t</i>)</span>.) One can, equivalently, treat the observables as fixed, while the state of the system depends on time; that is known as the <a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger picture</a>. (This approach was taken in the earlier part of the discussion above, with a time-varying 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="{\textstyle |\Psi (t)\rangle =\sum _{n}C_{n}(t)|\Phi _{n}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle |\Psi (t)\rangle =\sum _{n}C_{n}(t)|\Phi _{n}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e06f22478cb71eb6a1101744cec14629a4422fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.144ex; height:3.009ex;" alt="{\textstyle |\Psi (t)\rangle =\sum _{n}C_{n}(t)|\Phi _{n}\rangle }"></span>.) Conceptually (and mathematically), the two approaches are equivalent; choosing one of them is a matter of convention. </p><p>Both viewpoints are used in quantum theory. While non-relativistic <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> is usually formulated in terms of the Schrödinger picture, the Heisenberg picture is often preferred in a relativistic context, that is, for <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>. Compare with <a href="/wiki/Dirac_picture" class="mw-redirect" title="Dirac picture">Dirac picture</a>.<sup id="cite_ref-Gottfried_(2013)_17-0" class="reference"><a href="#cite_note-Gottfried_(2013)-17"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 65">: <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8gFX-9YcvIYC&pg=PA65">65</a> </span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Formalism_in_quantum_physics">Formalism in quantum physics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_state&action=edit&section=10" title="Edit section: Formalism in quantum physics"><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">See also: <a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Mathematical formulation of quantum mechanics</a></div> <div class="mw-heading mw-heading3"><h3 id="Pure_states_as_rays_in_a_complex_Hilbert_space">Pure states as rays in a complex Hilbert space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_state&action=edit&section=11" title="Edit section: Pure states as rays in a complex Hilbert space"><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">See also: <a href="/wiki/Wigner%27s_theorem#Rays_and_ray_space" title="Wigner's theorem">Wigner's theorem § Rays and ray space</a></div> <p>Quantum physics is most commonly formulated in terms of <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, as follows. Any given system is identified with some finite- or infinite-dimensional <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>. The pure states correspond to vectors of <a href="/wiki/Normed_vector_space" title="Normed vector space">norm</a> 1. Thus the set of all pure states corresponds to the <a href="/wiki/Unit_sphere" title="Unit sphere">unit sphere</a> in the Hilbert space, because the unit sphere is defined as the set of all vectors with norm 1. </p><p>Multiplying a pure state by a <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a> is physically inconsequential (as long as the state is considered by itself). If a vector in a complex 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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> can be obtained from another vector by multiplying by some non-zero complex number, the two vectors in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> are said to correspond to the same <a href="/wiki/Ray_(quantum_theory)" class="mw-redirect" title="Ray (quantum theory)">ray</a> in the <a href="/wiki/Projective_Hilbert_space" title="Projective Hilbert space">projective Hilbert space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} (H)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} (H)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df9ab9357902a5f4ee6c4be4e4eae7896c676de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.7ex; height:2.843ex;" alt="{\displaystyle \mathbf {P} (H)}"></span> 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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>. Note that although the word <i>ray</i> is used, properly speaking, a point in the projective Hilbert space corresponds to a <i>line</i> passing through the origin of the Hilbert space, rather than a <a href="/wiki/Line_(geometry)#Ray" title="Line (geometry)">half-line</a>, or <i>ray</i> in the <a href="/wiki/Line_(geometry)#Ray" title="Line (geometry)">geometrical sense</a>. </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=Quantum_state&action=edit&section=12" title="Edit section: Spin"><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/Mathematical_formulation_of_quantum_mechanics#Spin" title="Mathematical formulation of quantum mechanics">Mathematical formulation of quantum mechanics § Spin</a></div> <p>The <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> has the same dimension (<a href="/wiki/Mass" title="Mass">M</a>·<a href="/wiki/Length" title="Length">L</a><sup>2</sup>·<a href="/wiki/Time" title="Time">T</a><sup>−1</sup>) as the <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a> and, at quantum scale, behaves as a <i>discrete</i> degree of freedom of a quantum system. Most particles possess a kind of intrinsic angular momentum that does not appear at all in classical mechanics and arises from Dirac's relativistic generalization of the theory. Mathematically it is described with <a href="/wiki/Spinor" title="Spinor">spinors</a>. In non-relativistic quantum mechanics the <a href="/wiki/Representation_theory_of_SU(2)" title="Representation theory of SU(2)">group representations</a> of the <a href="/wiki/Lie_group" title="Lie group">Lie group</a> SU(2) are used to describe this additional freedom. For a given particle, the choice of representation (and hence the range of possible values of the spin observable) is specified by a non-negative number <span class="texhtml"><i>S</i></span> that, in units of the <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a> <span class="texhtml"><i>ħ</i></span>, is either an <a href="/wiki/Integer" title="Integer">integer</a> (0, 1, 2, ...) or a <a href="/wiki/Half-integer" title="Half-integer">half-integer</a> (1/2, 3/2, 5/2, ...). For a <a href="/wiki/Rest_mass" class="mw-redirect" title="Rest mass">massive</a> particle with spin <span class="texhtml"><i>S</i></span>, its <a href="/wiki/Spin_quantum_number" title="Spin quantum number">spin quantum number</a> <span class="texhtml"><i>m</i></span> always assumes one of the <span class="texhtml">2<i>S</i> + 1</span> possible values in the set <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 \{-S,-S+1,\ldots ,S-1,S\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mi>S</mi> <mo>,</mo> <mo>−</mo> <mi>S</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>S</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>S</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{-S,-S+1,\ldots ,S-1,S\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61bc546a1df79863fe4d31986fb04dabc16274e7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.19ex; height:2.843ex;" alt="{\displaystyle \{-S,-S+1,\ldots ,S-1,S\}}"></span> </p><p>As a consequence, the quantum state of a particle with spin is described by a <a href="/wiki/Vector_space" title="Vector space">vector</a>-valued wave function with values in <a href="/wiki/Complex_coordinate_space" title="Complex coordinate space"><b>C</b><sup>2<i>S</i>+1</sup></a>. Equivalently, it is represented by a <a href="/wiki/Complex-valued_function" class="mw-redirect" title="Complex-valued function">complex-valued function</a> of four variables: one discrete <a href="/wiki/Quantum_number" title="Quantum number">quantum number</a> variable (for the spin) is added to the usual three continuous variables (for the position in space). </p> <div class="mw-heading mw-heading3"><h3 id="Many-body_states_and_particle_statistics">Many-body states and particle statistics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_state&action=edit&section=13" title="Edit section: Many-body states and particle statistics"><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">Further information: <a href="/wiki/Particle_statistics" title="Particle statistics">Particle statistics</a></div> <p>The quantum state of a system of <i>N</i> particles, each potentially with spin, is described by a complex-valued function with four variables per particle, corresponding to 3 <a href="/wiki/Spatial_coordinates" class="mw-redirect" title="Spatial coordinates">spatial coordinates</a> and <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a>, e.g. <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 (\mathbf {r} _{1},\,m_{1};\;\dots ;\;\mathbf {r} _{N},\,m_{N})\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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>;</mo> <mspace width="thickmathspace" /> <mo>…</mo> <mo>;</mo> <mspace width="thickmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi (\mathbf {r} _{1},\,m_{1};\;\dots ;\;\mathbf {r} _{N},\,m_{N})\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9525fa8a1c296b02d128c93b97318aef17d9cdf6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.608ex; height:2.843ex;" alt="{\displaystyle |\psi (\mathbf {r} _{1},\,m_{1};\;\dots ;\;\mathbf {r} _{N},\,m_{N})\rangle .}"></span> </p><p>Here, the spin variables <i>m<sub>ν</sub></i> assume values from the set <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 \{-S_{\nu },\,-S_{\nu }+1,\,\ldots ,\,S_{\nu }-1,\,S_{\nu }\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>−</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{-S_{\nu },\,-S_{\nu }+1,\,\ldots ,\,S_{\nu }-1,\,S_{\nu }\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb7fc2e7aa05e5a94c833e602db84b67fcf9a092" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.855ex; height:2.843ex;" alt="{\displaystyle \{-S_{\nu },\,-S_{\nu }+1,\,\ldots ,\,S_{\nu }-1,\,S_{\nu }\}}"></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 S_{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1844fd9ce97ecd926b723d24a33cdce1d948b439" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.528ex; height:2.509ex;" alt="{\displaystyle S_{\nu }}"></span> is the spin of <i>ν</i>th particle. <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 S_{\nu }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\nu }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba43b7b2613f771696f8594a02a992a5e085cfa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.789ex; height:2.509ex;" alt="{\displaystyle S_{\nu }=0}"></span> for a particle that does not exhibit spin. </p><p>The treatment of <a href="/wiki/Identical_particles" class="mw-redirect" title="Identical particles">identical particles</a> is very different for <a href="/wiki/Boson" title="Boson">bosons</a> (particles with integer spin) versus <a href="/wiki/Fermion" title="Fermion">fermions</a> (particles with half-integer spin). The above <i>N</i>-particle function must either be symmetrized (in the bosonic case) or anti-symmetrized (in the fermionic case) with respect to the particle numbers. If not all <i>N</i> particles are identical, but some of them are, then the function must be (anti)symmetrized separately over the variables corresponding to each group of identical variables, according to its statistics (bosonic or fermionic). </p><p>Electrons are fermions with <span class="texhtml"><i>S</i> = 1/2</span>, <a href="/wiki/Photon" title="Photon">photons</a> (quanta of light) are bosons with <span class="texhtml"><i>S</i> = 1</span> (although in the <a href="/wiki/Vacuum" title="Vacuum">vacuum</a> they are <a href="/wiki/Massless_particle" title="Massless particle">massless</a> and can't be described with Schrödinger mechanics). </p><p>When symmetrization or anti-symmetrization is unnecessary, <span class="texhtml"><i>N</i></span>-particle spaces of states can be obtained simply by <a href="/wiki/Tensor_product_of_Hilbert_spaces" title="Tensor product of Hilbert spaces">tensor products</a> of one-particle spaces, to which we will return later. </p> <div class="mw-heading mw-heading3"><h3 id="Basis_states_of_one-particle_systems">Basis states of one-particle systems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_state&action=edit&section=14" title="Edit section: Basis states of one-particle systems"><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">See also: <a href="/wiki/Dirac_delta_function#Quantum_mechanics" title="Dirac delta function">Dirac delta function § Quantum mechanics</a></div> <p>A 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> belonging to a <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="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> can always be expressed uniquely as a <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of elements of an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> 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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>. Using <a href="/wiki/Bra%E2%80%93ket_notation#Outer_products" title="Bra–ket notation">bra–ket notation</a>, this means any 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> can be 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 {\begin{aligned}|\psi \rangle &=\sum _{i}c_{i}|{k_{i}}\rangle ,\\&=\sum _{i}|{k_{i}}\rangle \langle k_{i}|\psi \rangle ,\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"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>k</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> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}|\psi \rangle &=\sum _{i}c_{i}|{k_{i}}\rangle ,\\&=\sum _{i}|{k_{i}}\rangle \langle k_{i}|\psi \rangle ,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0762871018a3bf065f0801d9440038beaba297d7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:20.846ex; height:11.176ex;" alt="{\displaystyle {\begin{aligned}|\psi \rangle &=\sum _{i}c_{i}|{k_{i}}\rangle ,\\&=\sum _{i}|{k_{i}}\rangle \langle k_{i}|\psi \rangle ,\end{aligned}}}"></span> with <a href="/wiki/Complex_number" title="Complex number">complex</a> <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}=\langle {k_{i}}|\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <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 c_{i}=\langle {k_{i}}|\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7687a763e39bb55b489f7a51d79c3d51dba3f3e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.885ex; height:2.843ex;" alt="{\displaystyle c_{i}=\langle {k_{i}}|\psi \rangle }"></span> and basis elements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |k_{i}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |k_{i}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cad00dd98cce55753694bedb59339d0161d8987e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.562ex; height:2.843ex;" alt="{\displaystyle |k_{i}\rangle }"></span>. In this case, the <a href="/wiki/Wave_function#Normalization_condition" title="Wave function">normalization condition</a> translates to <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 \rangle =\sum _{i}\langle \psi |{k_{i}}\rangle \langle k_{i}|\psi \rangle =\sum _{i}\left|c_{i}\right|^{2}=1.}"> <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>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>k</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> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msup> <mrow> <mo>|</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi |\psi \rangle =\sum _{i}\langle \psi |{k_{i}}\rangle \langle k_{i}|\psi \rangle =\sum _{i}\left|c_{i}\right|^{2}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91047ad51c96f7b614c5ad23d1178f0a02349c94" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.799ex; height:5.509ex;" alt="{\displaystyle \langle \psi |\psi \rangle =\sum _{i}\langle \psi |{k_{i}}\rangle \langle k_{i}|\psi \rangle =\sum _{i}\left|c_{i}\right|^{2}=1.}"></span> In physical terms, <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> has been expressed as a <a href="/wiki/Quantum_superposition" title="Quantum superposition">quantum superposition</a> of the "basis 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 |{k_{i}}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{k_{i}}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9297bd0b56581e9e15f73e18b9c22db73815924c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.562ex; height:2.843ex;" alt="{\displaystyle |{k_{i}}\rangle }"></span>, i.e., the <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenstates</a> of an observable. In particular, if said observable is measured on the 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>, 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 |c_{i}|^{2}=|\langle {k_{i}}|\psi \rangle |^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |c_{i}|^{2}=|\langle {k_{i}}|\psi \rangle |^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30b7edb370eba6f05cf70be30e38cba14c9f31c3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.228ex; height:3.343ex;" alt="{\displaystyle |c_{i}|^{2}=|\langle {k_{i}}|\psi \rangle |^{2},}"></span> is the probability that the result of the measurement 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 k_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f29138ed3ad54ffce527daccadc49c520459b0b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.011ex; height:2.509ex;" alt="{\displaystyle k_{i}}"></span>.<sup id="cite_ref-sakurai_5-1" class="reference"><a href="#cite_note-sakurai-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 22">: 22 </span></sup> </p><p>In general, the expression for probability always consist of a relation between the quantum state and a <a href="/wiki/Decomposition_of_spectrum_(functional_analysis)#Quantum_mechanics" title="Decomposition of spectrum (functional analysis)">portion of the spectrum</a> of the dynamical variable (i.e. <a href="/wiki/Random_variable" title="Random variable">random variable</a>) being observed.<sup id="cite_ref-jauch_18-0" class="reference"><a href="#cite_note-jauch-18"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 98">: 98 </span></sup><sup id="cite_ref-ballentine_19-0" class="reference"><a href="#cite_note-ballentine-19"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 53">: 53 </span></sup> For example, the situation above describes the discrete case as <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f29138ed3ad54ffce527daccadc49c520459b0b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.011ex; height:2.509ex;" alt="{\displaystyle k_{i}}"></span> belong to the <a href="/wiki/Point_spectrum" class="mw-redirect" title="Point spectrum">point spectrum</a>. Likewise, the <a href="/wiki/Wave_function" title="Wave function">wave function</a> is just the <a href="/wiki/Eigenfunction" title="Eigenfunction">eigenfunction</a> of the <a href="/wiki/Hamiltonian_operator" class="mw-redirect" title="Hamiltonian operator">Hamiltonian operator</a> with corresponding eigenvalue(s) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span>; the energy of the system. </p><p>An example of the continuous case is given by the <a href="/wiki/Position_operator" title="Position operator">position operator</a>. The probability measure for a system in state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> is given by:<sup id="cite_ref-landsman_20-0" class="reference"><a href="#cite_note-landsman-20"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> <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 \mathrm {Pr} (x\in B|\psi )=\int _{B\subset \mathbb {R} }|\psi (x)|^{2}dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∈</mo> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo 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>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Pr} (x\in B|\psi )=\int _{B\subset \mathbb {R} }|\psi (x)|^{2}dx,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ccb9ccc9583f375fc1e843d91fc119b9e2cb30b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.313ex; height:5.676ex;" alt="{\displaystyle \mathrm {Pr} (x\in B|\psi )=\int _{B\subset \mathbb {R} }|\psi (x)|^{2}dx,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi (x)|^{2}}"> <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>x</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi (x)|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99516be05fd9132a16e5f2dcd0cb03bd632eab12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7ex; height:3.343ex;" alt="{\displaystyle |\psi (x)|^{2}}"></span> is the probability density function for finding a particle at a given position. These examples emphasize the distinction in charactertistics between the state and the observable. That is, whereas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> is a pure state belonging to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>, the <a href="/wiki/Dirac_delta_function#Quantum_mechanics" title="Dirac delta function">(generalized) eigenvectors</a> of the position operator do <i>not</i>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Pure_states_vs._bound_states">Pure states vs. bound states</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_state&action=edit&section=15" title="Edit section: Pure states vs. bound states"><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">See also: <a href="/wiki/Decomposition_of_spectrum_(functional_analysis)#Quantum_mechanics" title="Decomposition of spectrum (functional analysis)">Decomposition of spectrum (functional analysis) § Quantum mechanics</a></div> <p>Though closely related, pure states are not the same as bound states belonging to the <a href="/wiki/Spectrum_(functional_analysis)#Point_spectrum" title="Spectrum (functional analysis)">pure point spectrum</a> of an observable with no quantum uncertainty. A particle is said to be in a <b><a href="/wiki/Bound_state" title="Bound state">bound state</a></b> if it remains localized in a bounded region of space for all times. A pure 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 |\phi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/312d43de853a9e6ca74888e63394fc8081f56a43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.937ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle }"></span> is called a bound state <i>if and only if</i> for every <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 \varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04ec3670b50384a3ce48aca42e7cc5131a06b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}"></span> there is a <a href="/wiki/Compact_set" class="mw-redirect" title="Compact set">compact set</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\subset \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\subset \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1321442944ba52b596166488c200d2a284bea5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.897ex; height:2.676ex;" alt="{\displaystyle K\subset \mathbb {R} ^{3}}"></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 \int _{K}|\phi (\mathbf {r} ,t)|^{2}\,\mathrm {d} ^{3}\mathbf {r} \geq 1-\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>≥</mo> <mn>1</mn> <mo>−</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{K}|\phi (\mathbf {r} ,t)|^{2}\,\mathrm {d} ^{3}\mathbf {r} \geq 1-\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09b2439469f5a5995af7eeb1a2cb34db597fe1ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.912ex; height:5.676ex;" alt="{\displaystyle \int _{K}|\phi (\mathbf {r} ,t)|^{2}\,\mathrm {d} ^{3}\mathbf {r} \geq 1-\varepsilon }"></span> for all <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\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/592bced0c39b10fc90e74c6a66223abfbfb029de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.358ex; height:2.176ex;" alt="{\displaystyle t\in \mathbb {R} }"></span>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The integral represents the probability that a particle is found in a bounded region <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> at any time <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>. If the probability remains arbitrarily close to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> then the particle is said to remain in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>. </p><p>For example, <a href="/wiki/Wave_function#Normalization_condition" title="Wave function">non-normalizable</a> solutions of the <a href="/wiki/Free_particle" title="Free particle">free Schrödinger equation</a> can be expressed as functions that are normalizable, using <a href="/wiki/Wave_packets" class="mw-redirect" title="Wave packets">wave packets</a>. These wave packets belong to the pure point spectrum of a corresponding <a href="/wiki/Projection_(linear_algebra)#Orthogonal_projections" title="Projection (linear algebra)">projection operator</a> which, mathematically speaking, constitutes an observable.<sup id="cite_ref-ballentine_19-1" class="reference"><a href="#cite_note-ballentine-19"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 48">: 48 </span></sup> However, they are not bound states. </p> <div class="mw-heading mw-heading3"><h3 id="Superposition_of_pure_states">Superposition of pure states</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_state&action=edit&section=16" title="Edit section: Superposition of pure states"><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/Quantum_superposition" title="Quantum superposition">Quantum superposition</a></div> <p>As mentioned above, quantum states may be <a href="/wiki/Superposition_principle" title="Superposition principle">superposed</a>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\alpha \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f42032e642ee1c9d27adb318d34c7cc85f7a95d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.039ex; height:2.843ex;" alt="{\displaystyle |\alpha \rangle }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\beta \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>β</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\beta \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db6da67f5321b389fcaf855bfab45d09e7f0d923" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.883ex; height:2.843ex;" alt="{\displaystyle |\beta \rangle }"></span> are two kets corresponding to quantum states, the ket <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 c_{\alpha }|\alpha \rangle +c_{\beta }|\beta \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>β</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{\alpha }|\alpha \rangle +c_{\beta }|\beta \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8cb3714e7780728dd605fcc6b63142b50ad17f4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.235ex; height:3.009ex;" alt="{\displaystyle c_{\alpha }|\alpha \rangle +c_{\beta }|\beta \rangle }"></span> is also a quantum state of the same system. Both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dcb7aadbcfa0ea1be48b6d5e135843344acbc3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.291ex; height:2.009ex;" alt="{\displaystyle c_{\alpha }}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4f6b06b639ac71e21089730ad64b557a7e96a4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.181ex; height:2.343ex;" alt="{\displaystyle c_{\beta }}"></span> can be complex numbers; their relative amplitude and relative phase will influence the resulting quantum state. </p><p>Writing the superposed state using <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 c_{\alpha }=A_{\alpha }e^{i\theta _{\alpha }}\ \ c_{\beta }=A_{\beta }e^{i\theta _{\beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mrow> </msup> <mtext> </mtext> <mtext> </mtext> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{\alpha }=A_{\alpha }e^{i\theta _{\alpha }}\ \ c_{\beta }=A_{\beta }e^{i\theta _{\beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd580519acbe1c47388e1659d0caf1c065b61d64" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.03ex; height:3.343ex;" alt="{\displaystyle c_{\alpha }=A_{\alpha }e^{i\theta _{\alpha }}\ \ c_{\beta }=A_{\beta }e^{i\theta _{\beta }}}"></span> and defining the norm of the state 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 |c_{\alpha }|^{2}+|c_{\beta }|^{2}=A_{\alpha }^{2}+A_{\beta }^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |c_{\alpha }|^{2}+|c_{\beta }|^{2}=A_{\alpha }^{2}+A_{\beta }^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01340e4d5cac8c329e1f4e27002a62d673fbc3b3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:28.152ex; height:3.843ex;" alt="{\displaystyle |c_{\alpha }|^{2}+|c_{\beta }|^{2}=A_{\alpha }^{2}+A_{\beta }^{2}=1}"></span> and extracting the common factors gives: <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 e^{i\theta _{\alpha }}\left(A_{\alpha }|\alpha \rangle +{\sqrt {1-A_{\alpha }^{2}}}e^{i\theta _{\beta }-i\theta _{\alpha }}|\beta \rangle \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>−</mo> <mi>i</mi> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>β</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\theta _{\alpha }}\left(A_{\alpha }|\alpha \rangle +{\sqrt {1-A_{\alpha }^{2}}}e^{i\theta _{\beta }-i\theta _{\alpha }}|\beta \rangle \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4767d42a7aadce220600b3202dc8001bedb1399e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.198ex; height:4.843ex;" alt="{\displaystyle e^{i\theta _{\alpha }}\left(A_{\alpha }|\alpha \rangle +{\sqrt {1-A_{\alpha }^{2}}}e^{i\theta _{\beta }-i\theta _{\alpha }}|\beta \rangle \right)}"></span> The overall phase factor in front has no physical effect.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 108">: 108 </span></sup> Only the relative phase affects the physical nature of the superposition. </p><p>One example of superposition is the <a href="/wiki/Double-slit_experiment" title="Double-slit experiment">double-slit experiment</a>, in which superposition leads to <a href="/wiki/Interference_(wave_propagation)#Quantum_interference" class="mw-redirect" title="Interference (wave propagation)">quantum interference</a>. Another example of the importance of relative phase is <a href="/wiki/Rabi_oscillation" class="mw-redirect" title="Rabi oscillation">Rabi oscillations</a>, where the relative phase of two states varies in time due to the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>. The resulting superposition ends up oscillating back and forth between two different states. </p> <div class="mw-heading mw-heading3"><h3 id="Mixed_states">Mixed states</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_state&action=edit&section=17" title="Edit section: Mixed states"><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/Density_matrix" title="Density matrix">Density matrix</a></div> <p>A <i>pure quantum state</i> is a state which can be described by a single ket vector, as described above. A <i>mixed quantum state</i> is a <a href="/wiki/Statistical_ensemble" class="mw-redirect" title="Statistical ensemble">statistical ensemble</a> of pure states (see <i><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></i>).<sup id="cite_ref-peres_3-2" class="reference"><a href="#cite_note-peres-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 73">: 73 </span></sup> </p><p>Mixed states arise in quantum mechanics in two different situations: first, when the preparation of the system is not fully known, and thus one must deal with a <a href="/wiki/Statistical_ensemble" class="mw-redirect" title="Statistical ensemble">statistical ensemble</a> of possible preparations; and second, when one wants to describe a physical system which is <a href="/wiki/Quantum_entanglement" title="Quantum entanglement">entangled</a> with another, as its state cannot be described by a pure state. In the first case, there could theoretically be another person who knows the full history of the system, and therefore describe the same system as a pure state; in this case, the density matrix is simply used to represent the limited knowledge of a quantum state. In the second case, however, the existence of quantum entanglement theoretically prevents the existence of complete knowledge about the subsystem, and it's impossible for any person to describe the subsystem of an entangled pair as a pure state. </p><p>Mixed states inevitably arise from pure states when, for a composite quantum system <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1}\otimes H_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊗</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{1}\otimes H_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61259d98a32a113165094a93d7e779f665d2928e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.811ex; height:2.509ex;" alt="{\displaystyle H_{1}\otimes H_{2}}"></span> with an <a href="/wiki/Quantum_entanglement" title="Quantum entanglement">entangled</a> state on it, the part <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa4324515cc7343ee952e3840a1bb1aa8c7f74c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{2}}"></span> is inaccessible to the observer.<sup id="cite_ref-peres_3-3" class="reference"><a href="#cite_note-peres-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages: 121–122">: 121–122 </span></sup> The state of the part <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d4d9a872a55b209f2eb7cc23a71e5e1541bd1f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{1}}"></span> is expressed then as the <a href="/wiki/Partial_trace" title="Partial trace">partial trace</a> over <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa4324515cc7343ee952e3840a1bb1aa8c7f74c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{2}}"></span>. </p><p>A mixed state <i>cannot</i> be described with a single ket vector.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages: 691–692">: 691–692 </span></sup> Instead, it is described by its associated <i>density matrix</i> (or <i>density operator</i>), usually denoted <i>ρ</i>. Density matrices can describe both mixed <i>and</i> pure states, treating them on the same footing. Moreover, a mixed quantum state on a given quantum system described by 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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> can be always represented as the partial trace of a pure quantum state (called a <a href="/wiki/Purification_of_quantum_state" class="mw-redirect" title="Purification of quantum state">purification</a>) on a larger bipartite system <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\otimes K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>⊗</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\otimes K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92b81ae1b368244915c6606efc3b6013378bf524" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.97ex; height:2.343ex;" alt="{\displaystyle H\otimes K}"></span> for a sufficiently large 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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>. </p><p>The density matrix describing a mixed state is defined to be an operator 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 \rho =\sum _{s}p_{s}|\psi _{s}\rangle \langle \psi _{s}|}"> <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>s</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</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>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =\sum _{s}p_{s}|\psi _{s}\rangle \langle \psi _{s}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6206b4d2f339b1715f96832b45ea3a3ab09d722e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.351ex; height:5.509ex;" alt="{\displaystyle \rho =\sum _{s}p_{s}|\psi _{s}\rangle \langle \psi _{s}|}"></span> where <span class="texhtml"><i>p</i><sub><i>s</i></sub></span> is the fraction of the ensemble in each pure 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 _{s}\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>s</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi _{s}\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961c57c2d16414e7de2d1fe53ff82a100b2234b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.715ex; height:2.843ex;" alt="{\displaystyle |\psi _{s}\rangle .}"></span> The density matrix can be thought of as a way of using the one-particle <a href="/wiki/Formalism_(mathematics)" class="mw-redirect" title="Formalism (mathematics)">formalism</a> to describe the behavior of many similar particles by giving a probability distribution (or ensemble) of states that these particles can be found in. </p><p>A simple criterion for checking whether a density matrix is describing a pure or mixed state is that the <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> of <span class="texhtml"><i>ρ</i><sup>2</sup></span> is equal to 1 if the state is pure, and less than 1 if the state is mixed.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>d<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Another, equivalent, criterion is that the <a href="/wiki/Von_Neumann_entropy" title="Von Neumann entropy">von Neumann entropy</a> is 0 for a pure state, and strictly positive for a mixed state. </p><p><span class="anchor" id="expectation"></span>The rules for measurement in quantum mechanics are particularly simple to state in terms of density matrices. For example, the ensemble average (<a href="/wiki/Expectation_value_(quantum_mechanics)" title="Expectation value (quantum mechanics)">expectation value</a>) of a measurement corresponding to an observable <span class="texhtml"><i>A</i></span> is given by <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 A\rangle =\sum _{s}p_{s}\langle \psi _{s}|A|\psi _{s}\rangle =\sum _{s}\sum _{i}p_{s}a_{i}|\langle \alpha _{i}|\psi _{s}\rangle |^{2}=\operatorname {tr} (\rho A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</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> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle A\rangle =\sum _{s}p_{s}\langle \psi _{s}|A|\psi _{s}\rangle =\sum _{s}\sum _{i}p_{s}a_{i}|\langle \alpha _{i}|\psi _{s}\rangle |^{2}=\operatorname {tr} (\rho A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f68fa039dd1e306e2c09418991b16f1df4e479f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:56.507ex; height:5.509ex;" alt="{\displaystyle \langle A\rangle =\sum _{s}p_{s}\langle \psi _{s}|A|\psi _{s}\rangle =\sum _{s}\sum _{i}p_{s}a_{i}|\langle \alpha _{i}|\psi _{s}\rangle |^{2}=\operatorname {tr} (\rho A)}"></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 |\alpha _{i}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\alpha _{i}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0894b0cce03f69124a81e6f240e13af70918a098" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.839ex; height:2.843ex;" alt="{\displaystyle |\alpha _{i}\rangle }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"></span> are eigenkets and eigenvalues, respectively, for the operator <span class="texhtml"><i>A</i></span>, and "<span class="texhtml">tr</span>" denotes trace.<sup id="cite_ref-peres_3-4" class="reference"><a href="#cite_note-peres-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 73">: 73 </span></sup> It is important to note that two types of averaging are occurring, one (over <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>) being the usual expected value of the observable when the quantum is in 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 _{s}\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>s</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi _{s}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a988c8553e53ea04e11a1661a10c4d576ca86613" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.068ex; height:2.843ex;" alt="{\displaystyle |\psi _{s}\rangle }"></span>, and the other (over <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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>) being a statistical (said <i>incoherent</i>) average with the probabilities <span class="texhtml"><i>p<sub>s</sub></i></span> that the quantum is in those states. </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_generalizations">Mathematical generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_state&action=edit&section=18" title="Edit section: Mathematical generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>States can be formulated in terms of observables, rather than as vectors in a vector space. These are <a href="/wiki/State_(functional_analysis)" title="State (functional analysis)">positive normalized linear functionals</a> on a <a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a>, or sometimes other classes of algebras of observables. See <i><a href="/wiki/State_(functional_analysis)" title="State (functional analysis)">State on a C*-algebra</a></i> and <i><a href="/wiki/Gelfand%E2%80%93Naimark%E2%80%93Segal_construction" title="Gelfand–Naimark–Segal construction">Gelfand–Naimark–Segal construction</a></i> for more details. </p> <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=Quantum_state&action=edit&section=19" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 26em;"> <ul><li><a href="/wiki/Atomic_electron_transition" title="Atomic electron transition">Atomic electron transition</a></li> <li><a href="/wiki/Bloch_sphere" title="Bloch sphere">Bloch sphere</a></li> <li><a href="/wiki/Greenberger%E2%80%93Horne%E2%80%93Zeilinger_state" title="Greenberger–Horne–Zeilinger state">Greenberger–Horne–Zeilinger state</a></li> <li><a href="/wiki/Ground_state" title="Ground state">Ground state</a></li> <li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction to quantum mechanics</a></li> <li><a href="/wiki/No-cloning_theorem" title="No-cloning theorem">No-cloning theorem</a></li> <li><a href="/wiki/Orthonormal_basis" title="Orthonormal basis">Orthonormal basis</a></li> <li><a href="/wiki/PBR_theorem" class="mw-redirect" title="PBR theorem">PBR theorem</a></li> <li><a href="/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">Quantum harmonic oscillator</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum logic gate</a></li> <li><a href="/wiki/Stationary_state" title="Stationary state">Stationary state</a></li> <li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Wave function collapse</a></li> <li><a href="/wiki/W_state" title="W state">W state</a></li></ul> </div> <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=Quantum_state&action=edit&section=20" 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 reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">To avoid misunderstandings: Here we mean that <span class="texhtml"><i>Q</i>(<i>t</i>)</span> and <span class="texhtml"><i>P</i>(<i>t</i>)</span> are measured in the same state, but <i>not</i> in the same run of the experiment.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">i.e. separated by a zero delay. One can think of it as stopping the time, then making the two measurements one after the other, then resuming the time. Thus, the measurements occurred at the same time, but it is still possible to tell which was first.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">For concreteness' sake, suppose that <span class="texhtml"><i>A</i> = <i>Q</i>(<i>t</i><sub>1</sub>)</span> and <span class="texhtml"><i>B</i> = <i>P</i>(<i>t</i><sub>2</sub>)</span> in the above example, with <span class="texhtml"><i>t</i><sub>2</sub> > <i>t</i><sub>1</sub> > 0</span>.</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">Note that this criterion works when the density matrix is normalized so that the trace of <span class="texhtml"><i>ρ</i></span> is 1, as it is for the standard definition given in this section. Occasionally a density matrix will be normalized differently, in which case the criterion 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} (\rho ^{2})=(\operatorname {Tr} \rho )^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>Tr</mi> <mo>⁡</mo> <mi>ρ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Tr} (\rho ^{2})=(\operatorname {Tr} \rho )^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9382335bcc60573a93191a6def6665201f70dfd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.796ex; height:3.176ex;" alt="{\displaystyle \operatorname {Tr} (\rho ^{2})=(\operatorname {Tr} \rho )^{2}}"></span></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=Quantum_state&action=edit&section=21" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-messiah-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-messiah_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-messiah_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-messiah_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-messiah_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-messiah_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-messiah_1-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-messiah_1-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-messiah_1-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-messiah_1-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-messiah_1-9"><sup><i><b>j</b></i></sup></a> <a href="#cite_ref-messiah_1-10"><sup><i><b>k</b></i></sup></a> <a href="#cite_ref-messiah_1-11"><sup><i><b>l</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMessiah1966" class="citation book cs1">Messiah, Albert (1966). <i>Quantum Mechanics</i>. North Holland, John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0486409244" title="Special:BookSources/0486409244"><bdi>0486409244</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics&rft.pub=North+Holland%2C+John+Wiley+%26+Sons&rft.date=1966&rft.isbn=0486409244&rft.aulast=Messiah&rft.aufirst=Albert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohen-TannoudjiDiuLaloë1977" class="citation book cs1">Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (1977). <i>Quantum Mechanics</i>. Wiley. pp. <span class="nowrap">231–</span>235.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E231-%3C%2Fspan%3E235&rft.pub=Wiley&rft.date=1977&rft.aulast=Cohen-Tannoudji&rft.aufirst=Claude&rft.au=Diu%2C+Bernard&rft.au=Lalo%C3%AB%2C+Franck&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></span> </li> <li id="cite_note-peres-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-peres_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-peres_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-peres_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-peres_3-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-peres_3-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeres1995" class="citation book cs1"><a href="/wiki/Asher_Peres" title="Asher Peres">Peres, Asher</a> (1995). <i><a href="/wiki/Quantum_Theory:_Concepts_and_Methods" title="Quantum Theory: Concepts and Methods">Quantum Theory: Concepts and Methods</a></i>. Kluwer Academic Publishers. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7923-2549-4" title="Special:BookSources/0-7923-2549-4"><bdi>0-7923-2549-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Theory%3A+Concepts+and+Methods&rft.pub=Kluwer+Academic+Publishers&rft.date=1995&rft.isbn=0-7923-2549-4&rft.aulast=Peres&rft.aufirst=Asher&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></span> </li> <li id="cite_note-Whittaker-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Whittaker_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhittaker1989" class="citation book cs1">Whittaker, Sir Edmund (1989-01-01). <i>A History of the Theories of Aether and Electricity</i>. Vol. 2. Courier Dover Publications. p. 87. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-26126-3" title="Special:BookSources/0-486-26126-3"><bdi>0-486-26126-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+the+Theories+of+Aether+and+Electricity&rft.pages=87&rft.pub=Courier+Dover+Publications&rft.date=1989-01-01&rft.isbn=0-486-26126-3&rft.aulast=Whittaker&rft.aufirst=Sir+Edmund&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></span> </li> <li id="cite_note-sakurai-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-sakurai_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-sakurai_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSakuraiNapolitano2020" class="citation book cs1">Sakurai, J. J.; Napolitano, Jim (2020). <i>Modern Quantum Mechanics</i>. Cambridge University Press. <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/2020mqm..book.....S">2020mqm..book.....S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2F9781108587280">10.1017/9781108587280</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-108-58728-0" title="Special:BookSources/978-1-108-58728-0"><bdi>978-1-108-58728-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Quantum+Mechanics&rft.pub=Cambridge+University+Press&rft.date=2020&rft_id=info%3Adoi%2F10.1017%2F9781108587280&rft_id=info%3Abibcode%2F2020mqm..book.....S&rft.isbn=978-1-108-58728-0&rft.aulast=Sakurai&rft.aufirst=J.+J.&rft.au=Napolitano%2C+Jim&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></span> </li> <li id="cite_note-rieffel-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-rieffel_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRieffelPolak2011" class="citation book cs1"><a href="/wiki/Eleanor_Rieffel" title="Eleanor Rieffel">Rieffel, Eleanor G.</a>; Polak, Wolfgang H. (2011-03-04). <a href="/wiki/Quantum_Computing:_A_Gentle_Introduction" title="Quantum Computing: A Gentle Introduction"><i>Quantum Computing: A Gentle Introduction</i></a>. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-01506-6" title="Special:BookSources/978-0-262-01506-6"><bdi>978-0-262-01506-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Computing%3A+A+Gentle+Introduction&rft.pub=MIT+Press&rft.date=2011-03-04&rft.isbn=978-0-262-01506-6&rft.aulast=Rieffel&rft.aufirst=Eleanor+G.&rft.au=Polak%2C+Wolfgang+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></span> </li> <li id="cite_note-holevo-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-holevo_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHolevo2001" class="citation book cs1"><a href="/wiki/Alexander_Holevo" title="Alexander Holevo">Holevo, Alexander S.</a> (2001). <i>Statistical Structure of Quantum Theory</i>. Lecture Notes in Physics. Springer. p. 15. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-42082-7" title="Special:BookSources/3-540-42082-7"><bdi>3-540-42082-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/318268606">318268606</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistical+Structure+of+Quantum+Theory&rft.series=Lecture+Notes+in+Physics&rft.pages=15&rft.pub=Springer&rft.date=2001&rft_id=info%3Aoclcnum%2F318268606&rft.isbn=3-540-42082-7&rft.aulast=Holevo&rft.aufirst=Alexander+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKirkpatrick2006" class="citation journal cs1">Kirkpatrick, K. A. (February 2006). "The Schrödinger–HJW Theorem". <i><a href="/wiki/Foundations_of_Physics_Letters" class="mw-redirect" title="Foundations of Physics Letters">Foundations of Physics Letters</a></i>. <b>19</b> (1): <span class="nowrap">95–</span>102. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0305068">quant-ph/0305068</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006FoPhL..19...95K">2006FoPhL..19...95K</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%2Fs10702-006-1852-1">10.1007/s10702-006-1852-1</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/0894-9875">0894-9875</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:15995449">15995449</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+Letters&rft.atitle=The+Schr%C3%B6dinger%E2%80%93HJW+Theorem&rft.volume=19&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E95-%3C%2Fspan%3E102&rft.date=2006-02&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15995449%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2006FoPhL..19...95K&rft_id=info%3Aarxiv%2Fquant-ph%2F0305068&rft.issn=0894-9875&rft_id=info%3Adoi%2F10.1007%2Fs10702-006-1852-1&rft.aulast=Kirkpatrick&rft.aufirst=K.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20190923005726/xbeams.chem.yale.edu/~batista/vaa/node4.html">"Statistical Mixture of States"</a>. Archived from <a rel="nofollow" class="external text" href="http://xbeams.chem.yale.edu/~batista/vaa/node4.html">the original</a> on September 23, 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">November 9,</span> 2021</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Statistical+Mixture+of+States&rft_id=http%3A%2F%2Fxbeams.chem.yale.edu%2F~batista%2Fvaa%2Fnode4.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120115220044/http://electron6.phys.utk.edu/qm1/modules/m6/statistical.htm">"The Density Matrix"</a>. Archived from <a rel="nofollow" class="external text" href="http://electron6.phys.utk.edu/qm1/modules/m6/statistical.htm">the original</a> on January 15, 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">January 24,</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Density+Matrix&rft_id=http%3A%2F%2Felectron6.phys.utk.edu%2Fqm1%2Fmodules%2Fm6%2Fstatistical.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg, W.</a> (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, <i>Z. Phys.</i> <b>43</b>: 172–198. Translation as <a rel="nofollow" class="external text" href="https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19840008978.pdf">'The actual content of quantum theoretical kinematics and mechanics'</a>. Also translated as 'The physical content of quantum kinematics and mechanics' at pp. 62–84 by editors John Wheeler and Wojciech Zurek, in <i>Quantum Theory and Measurement</i> (1983), Princeton University Press, Princeton NJ.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="/wiki/Niels_Bohr" title="Niels Bohr">Bohr, N.</a> (1927/1928). The quantum postulate and the recent development of atomic theory, <a rel="nofollow" class="external text" href="http://www.nature.com/nature/journal/v121/n3050/abs/121580a0.html"><i>Nature</i> Supplement April 14 1928, <b>121</b>: 580–590</a>.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDirac1981" class="citation book cs1">Dirac, Paul Adrien Maurice (1981). <i>The Principles of Quantum Mechanics</i>. Oxford: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-852011-5" title="Special:BookSources/978-0-19-852011-5"><bdi>978-0-19-852011-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Principles+of+Quantum+Mechanics&rft.place=Oxford&rft.pub=Oxford+University+Press&rft.date=1981&rft.isbn=978-0-19-852011-5&rft.aulast=Dirac&rft.aufirst=Paul+Adrien+Maurice&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></span> </li> <li id="cite_note-Gottfried_(2013)-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-Gottfried_(2013)_17-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGottfriedYan2003" class="citation book cs1"><a href="/wiki/Kurt_Gottfried" title="Kurt Gottfried">Gottfried, Kurt</a>; <a href="/wiki/Tung-Mow_Yan" title="Tung-Mow Yan">Yan, Tung-Mow</a> (2003). <i>Quantum Mechanics: Fundamentals</i> (2nd, illustrated ed.). Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387955766" title="Special:BookSources/9780387955766"><bdi>9780387955766</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics%3A+Fundamentals&rft.edition=2nd%2C+illustrated&rft.pub=Springer&rft.date=2003&rft.isbn=9780387955766&rft.aulast=Gottfried&rft.aufirst=Kurt&rft.au=Yan%2C+Tung-Mow&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></span> </li> <li id="cite_note-jauch-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-jauch_18-0">^</a></b></span> <span class="reference-text"><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/978-0-201-03298-7" title="Special:BookSources/978-0-201-03298-7"><bdi>978-0-201-03298-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Quantum+Mechanics&rft.place=Reading%2C+Mass.&rft.pub=Addison-Wesley&rft.date=1968&rft.isbn=978-0-201-03298-7&rft.aulast=Jauch&rft.aufirst=Josef+Maria&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></span> </li> <li id="cite_note-ballentine-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-ballentine_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ballentine_19-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBallentine2014" class="citation book cs1">Ballentine, Leslie E (2014). <i>Quantum Mechanics: A Modern Development</i> (2nd ed.). World Scientific Publishing Company. <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%2F9038">10.1142/9038</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4578-60-8" title="Special:BookSources/978-981-4578-60-8"><bdi>978-981-4578-60-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics%3A+A+Modern+Development&rft.edition=2nd&rft.pub=World+Scientific+Publishing+Company&rft.date=2014&rft_id=info%3Adoi%2F10.1142%2F9038&rft.isbn=978-981-4578-60-8&rft.aulast=Ballentine&rft.aufirst=Leslie+E&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></span> </li> <li id="cite_note-landsman-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-landsman_20-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandsman2009" class="citation book cs1">Landsman, Nicolaas P. (2009). "Born Rule and its Interpretation". <a rel="nofollow" class="external text" href="https://www.math.ru.nl/~landsman/Born.pdf"><i>Compendium of Quantum Physics</i></a> <span class="cs1-format">(PDF)</span>. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. <span class="nowrap">64–</span>70. <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%2F978-3-540-70626-7_20">10.1007/978-3-540-70626-7_20</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-70622-9" title="Special:BookSources/978-3-540-70622-9"><bdi>978-3-540-70622-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Born+Rule+and+its+Interpretation&rft.btitle=Compendium+of+Quantum+Physics&rft.place=Berlin%2C+Heidelberg&rft.pages=%3Cspan+class%3D%22nowrap%22%3E64-%3C%2Fspan%3E70&rft.pub=Springer+Berlin+Heidelberg&rft.date=2009&rft_id=info%3Adoi%2F10.1007%2F978-3-540-70626-7_20&rft.isbn=978-3-540-70622-9&rft.aulast=Landsman&rft.aufirst=Nicolaas+P.&rft_id=https%3A%2F%2Fwww.math.ru.nl%2F~landsman%2FBorn.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall2013" class="citation book cs1">Hall, B. C. (2013). "Chapter 6: Perspectives on the Spectral Theorem". <i>Quantum Theory for Mathematicians</i>. Graduate Texts in Mathematics. Vol. 267. Springer. <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/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1461471158" title="Special:BookSources/978-1461471158"><bdi>978-1461471158</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+6%3A+Perspectives+on+the+Spectral+Theorem&rft.btitle=Quantum+Theory+for+Mathematicians&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=2013&rft_id=info%3Abibcode%2F2013qtm..book.....H&rft.isbn=978-1461471158&rft.aulast=Hall&rft.aufirst=B.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlanchardBrüning2015" class="citation book cs1">Blanchard, Philippe; Brüning, Erwin (2015). <i>Mathematical Methods in Physics</i>. Birkhäuser. p. 431. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-14044-5" title="Special:BookSources/978-3-319-14044-5"><bdi>978-3-319-14044-5</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+Physics&rft.pages=431&rft.pub=Birkh%C3%A4user&rft.date=2015&rft.isbn=978-3-319-14044-5&rft.aulast=Blanchard&rft.aufirst=Philippe&rft.au=Br%C3%BCning%2C+Erwin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSusskindFriedmanSusskind2014" class="citation book cs1">Susskind, Leonard; Friedman, Art; Susskind, Leonard (2014). <i>Quantum mechanics: the theoretical minimum; [what you need to know to start doing physics]</i>. The theoretical minimum / Leonard Susskind and George Hrabovsky. New York, NY: Basic Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-465-06290-4" title="Special:BookSources/978-0-465-06290-4"><bdi>978-0-465-06290-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+mechanics%3A+the+theoretical+minimum%3B+%5Bwhat+you+need+to+know+to+start+doing+physics%5D&rft.place=New+York%2C+NY&rft.series=The+theoretical+minimum+%2F+Leonard+Susskind+and+George+Hrabovsky&rft.pub=Basic+Books&rft.date=2014&rft.isbn=978-0-465-06290-4&rft.aulast=Susskind&rft.aufirst=Leonard&rft.au=Friedman%2C+Art&rft.au=Susskind%2C+Leonard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZwiebach2022" class="citation book cs1"><a href="/wiki/Barton_Zwiebach" title="Barton Zwiebach">Zwiebach, Barton</a> (2022). <i>Mastering Quantum Mechanics: Essentials, Theory, and Applications</i>. Cambridge, Mass: <a href="/wiki/MIT_Press" title="MIT Press">MIT Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-04613-8" title="Special:BookSources/978-0-262-04613-8"><bdi>978-0-262-04613-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mastering+Quantum+Mechanics%3A+Essentials%2C+Theory%2C+and+Applications&rft.place=Cambridge%2C+Mass&rft.pub=MIT+Press&rft.date=2022&rft.isbn=978-0-262-04613-8&rft.aulast=Zwiebach&rft.aufirst=Barton&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=kl-pMd9Qx04C&pg=PA39">Blum, <i>Density matrix theory and applications</i>, page 39</a>.</span> </li> </ol></div></div> <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=Quantum_state&action=edit&section=22" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The concept of quantum states, in particular the content of the section <a href="#Formalism_in_quantum_physics">Formalism in quantum physics</a> above, is covered in most standard textbooks on quantum mechanics. </p><p>For a discussion of conceptual aspects and a comparison with classical states, see: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIsham1995" class="citation book cs1">Isham, Chris J (1995). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/lecturesonquantu0000isha"><i>Lectures on Quantum Theory: Mathematical and Structural Foundations</i></a></span>. <a href="/wiki/Imperial_College_Press" title="Imperial College Press">Imperial College Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-86094-001-9" title="Special:BookSources/978-1-86094-001-9"><bdi>978-1-86094-001-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Quantum+Theory%3A+Mathematical+and+Structural+Foundations&rft.pub=Imperial+College+Press&rft.date=1995&rft.isbn=978-1-86094-001-9&rft.aulast=Isham&rft.aufirst=Chris+J&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flecturesonquantu0000isha&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span></li></ul> <p>For a more detailed coverage of mathematical aspects, see: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBratteliRobinson1987" class="citation book cs1"><a href="/wiki/Ola_Bratteli" title="Ola Bratteli">Bratteli, Ola</a>; <a href="/wiki/Derek_W._Robinson" title="Derek W. Robinson">Robinson, Derek W</a> (1987). <i>Operator Algebras and Quantum Statistical Mechanics 1</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-17093-8" title="Special:BookSources/978-3-540-17093-8"><bdi>978-3-540-17093-8</bdi></a>. 2nd edition.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Operator+Algebras+and+Quantum+Statistical+Mechanics+1&rft.pub=Springer&rft.date=1987&rft.isbn=978-3-540-17093-8&rft.aulast=Bratteli&rft.aufirst=Ola&rft.au=Robinson%2C+Derek+W&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span> In particular, see Sec. 2.3.</li></ul> <p>For a discussion of purifications of mixed quantum states, see Chapter 2 of John Preskill's lecture notes for <a rel="nofollow" class="external text" href="http://www.theory.caltech.edu/~preskill/ph229/">Physics 219 </a> at Caltech. </p><p>For a discussion of geometric aspects see: </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBengtsson_IŻyczkowski_K2006" class="citation book cs1">Bengtsson I; <a href="/wiki/Karol_%C5%BByczkowski" title="Karol Życzkowski">Życzkowski K</a> (2006). <a href="/wiki/Geometry_of_Quantum_States" title="Geometry of Quantum States"><i>Geometry of Quantum States</i></a>. Cambridge: Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+of+Quantum+States&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2006&rft.au=Bengtsson+I&rft.au=%C5%BByczkowski+K&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+state" class="Z3988"></span>, <a rel="nofollow" class="external text" href="http://chaos.if.uj.edu.pl/~karol/geometry.htm">second, revised edition (2017)</a></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 .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output 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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 class="mw-selflink selflink">Quantum state</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Formulations</a></li> <li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Equations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Weyl_equation" title="Weyl equation">Weyl</a></li> <li><a href="/wiki/Majorana_equation" title="Majorana equation">Majorana</a></li> <li><a href="/wiki/Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell test</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li> <li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed choice</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_nanoscience" class="mw-redirect" title="Quantum nanoscience">Science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_biology" title="Quantum biology">Quantum biology</a></li> <li><a href="/wiki/Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li> <li><a href="/wiki/Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li> <li><a href="/wiki/Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li> <li><a href="/wiki/Quantum_mind" title="Quantum mind">Quantum mind</a></li> <li><a href="/wiki/Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li> <li><a href="/wiki/Quantum_spacetime" title="Quantum spacetime">Quantum spacetime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li> <li><a href="/wiki/Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">Quantum bus</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a> <ul><li><a href="/wiki/Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li> <li><a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">Quantum electronics</a></li> <li><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li> <li><a href="/wiki/Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li> <li><a href="/wiki/Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li> <li><a href="/wiki/Quantum_machine" title="Quantum machine">Quantum machine</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li> <li><a href="/wiki/Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li> <li><a href="/wiki/Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">Quantum network</a></li> <li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li> <li><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Extensions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li> <li><a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a> <ul><li><a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></li></ul></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li></ul> 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