Hamiltonian (quantum mechanics)

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<span class="vector-toc-numb">2</span> <span>Schrödinger Hamiltonian</span> </div> </a> <button aria-controls="toc-Schrödinger_Hamiltonian-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 Schrödinger Hamiltonian subsection</span> </button> <ul id="toc-Schrödinger_Hamiltonian-sublist" class="vector-toc-list"> <li id="toc-One_particle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#One_particle"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>One particle</span> </div> </a> <ul id="toc-One_particle-sublist" class="vector-toc-list"> <li id="toc-Expectation_value" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Expectation_value"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Expectation value</span> </div> </a> <ul id="toc-Expectation_value-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Many_particles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Many_particles"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Many particles</span> </div> </a> <ul id="toc-Many_particles-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Schrödinger_equation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Schrödinger_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Schrödinger equation</span> </div> </a> <ul id="toc-Schrödinger_equation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dirac_formalism" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dirac_formalism"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Dirac formalism</span> </div> </a> <ul id="toc-Dirac_formalism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Expressions_for_the_Hamiltonian" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Expressions_for_the_Hamiltonian"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Expressions for the Hamiltonian</span> </div> </a> <button aria-controls="toc-Expressions_for_the_Hamiltonian-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 Expressions for the Hamiltonian subsection</span> </button> <ul id="toc-Expressions_for_the_Hamiltonian-sublist" class="vector-toc-list"> <li id="toc-General_forms_for_one_particle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_forms_for_one_particle"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>General forms for one particle</span> </div> </a> <ul id="toc-General_forms_for_one_particle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Free_particle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Free_particle"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Free particle</span> </div> </a> <ul id="toc-Free_particle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Constant-potential_well" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Constant-potential_well"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Constant-potential well</span> </div> </a> <ul id="toc-Constant-potential_well-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simple_harmonic_oscillator" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simple_harmonic_oscillator"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Simple harmonic oscillator</span> </div> </a> <ul id="toc-Simple_harmonic_oscillator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rigid_rotor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rigid_rotor"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Rigid rotor</span> </div> </a> <ul id="toc-Rigid_rotor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electrostatic_(Coulomb)_potential" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electrostatic_(Coulomb)_potential"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Electrostatic (Coulomb) potential</span> </div> </a> <ul id="toc-Electrostatic_(Coulomb)_potential-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electric_dipole_in_an_electric_field" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electric_dipole_in_an_electric_field"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.7</span> <span>Electric dipole in an electric field</span> </div> </a> <ul id="toc-Electric_dipole_in_an_electric_field-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Magnetic_dipole_in_a_magnetic_field" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Magnetic_dipole_in_a_magnetic_field"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.8</span> <span>Magnetic dipole in a magnetic field</span> </div> </a> <ul id="toc-Magnetic_dipole_in_a_magnetic_field-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Charged_particle_in_an_electromagnetic_field" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Charged_particle_in_an_electromagnetic_field"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.9</span> <span>Charged particle in an electromagnetic field</span> </div> </a> <ul id="toc-Charged_particle_in_an_electromagnetic_field-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Energy_eigenket_degeneracy,_symmetry,_and_conservation_laws" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Energy_eigenket_degeneracy,_symmetry,_and_conservation_laws"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Energy eigenket degeneracy, symmetry, and conservation laws</span> </div> </a> <ul id="toc-Energy_eigenket_degeneracy,_symmetry,_and_conservation_laws-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hamilton's_equations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Hamilton's_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Hamilton's equations</span> </div> </a> <ul id="toc-Hamilton's_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-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">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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">Hamiltonian (quantum mechanics)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 38 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-38" 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">38 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/%D9%87%D8%A7%D9%85%D9%84%D8%AA%D9%88%D9%86%D9%8A_(%D9%85%D9%8A%D9%83%D8%A7%D9%86%D9%8A%D9%83%D8%A7_%D8%A7%D9%84%D9%83%D9%85)" title="هاملتوني (ميكانيكا الكم) – Arabic" lang="ar" hreflang="ar" data-title="هاملتوني (ميكانيكا الكم)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D0%BE%D1%80_%D0%BD%D0%B0_%D0%A5%D0%B0%D0%BC%D0%B8%D0%BB%D1%82%D0%BE%D0%BD" title="Оператор на Хамилтон – Bulgarian" lang="bg" hreflang="bg" data-title="Оператор на Хамилтон" 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/Hamiltoni%C3%A0" title="Hamiltonià – Catalan" lang="ca" hreflang="ca" data-title="Hamiltonià" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Hamilton%C5%AFv_oper%C3%A1tor" title="Hamiltonův operátor – Czech" lang="cs" hreflang="cs" data-title="Hamiltonův operátor" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Hamilton-operator" title="Hamilton-operator – Danish" lang="da" hreflang="da" data-title="Hamilton-operator" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Hamiltonoperator" title="Hamiltonoperator – German" lang="de" hreflang="de" data-title="Hamiltonoperator" 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/Hamiltoniaan" title="Hamiltoniaan – Estonian" lang="et" hreflang="et" data-title="Hamiltoniaan" 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%A7%CE%B1%CE%BC%CE%B9%CE%BB%CF%84%CE%BF%CE%BD%CE%B9%CE%B1%CE%BD%CE%AE" title="Χαμιλτονιανή – Greek" lang="el" hreflang="el" data-title="Χαμιλτονιανή" 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/Hamiltoniano_(mec%C3%A1nica_cu%C3%A1ntica)" title="Hamiltoniano (mecánica cuántica) – Spanish" lang="es" hreflang="es" data-title="Hamiltoniano (mecánica cuántica)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%87%D8%A7%D9%85%DB%8C%D9%84%D8%AA%D9%88%D9%86%DB%8C%D9%86_(%D9%85%DA%A9%D8%A7%D9%86%DB%8C%DA%A9_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85%DB%8C)" title="هامیلتونین (مکانیک کوانتومی) – Persian" lang="fa" hreflang="fa" data-title="هامیلتونین (مکانیک کوانتومی)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Op%C3%A9rateur_hamiltonien" title="Opérateur hamiltonien – French" lang="fr" hreflang="fr" data-title="Opérateur hamiltonien" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Hamalt%C3%B3inian" title="Hamaltóinian – Irish" lang="ga" hreflang="ga" data-title="Hamaltóinian" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%B4%EB%B0%80%ED%86%A0%EB%8B%88%EC%96%B8_(%EC%96%91%EC%9E%90%EC%97%AD%ED%95%99)" title="해밀토니언 (양자역학) – Korean" lang="ko" hreflang="ko" data-title="해밀토니언 (양자역학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D5%B4%D5%AB%D5%AC%D5%BF%D5%B8%D5%B6%D5%B5%D5%A1%D5%B6_(%D6%84%D5%BE%D5%A1%D5%B6%D5%BF%D5%A1%D5%B5%D5%AB%D5%B6_%D5%B4%D5%A5%D5%AD%D5%A1%D5%B6%D5%AB%D5%AF%D5%A1)" title="Համիլտոնյան (քվանտային մեխանիկա) – Armenian" lang="hy" hreflang="hy" data-title="Համիլտոնյան (քվանտային մեխանիկա)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B9%E0%A5%88%E0%A4%AE%E0%A4%BF%E0%A4%B2%E0%A5%8D%E0%A4%9F%E0%A4%A8%E0%A5%80_%E0%A4%91%E0%A4%AA%E0%A4%B0%E0%A5%87%E0%A4%9F%E0%A4%B0" title="हैमिल्टनी ऑपरेटर – Hindi" lang="hi" hreflang="hi" data-title="हैमिल्टनी ऑपरेटर" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Operatore_hamiltoniano" title="Operatore hamiltoniano – Italian" lang="it" hreflang="it" data-title="Operatore hamiltoniano" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Hamiltonianas_(kvantin%C4%97_mechanika)" title="Hamiltonianas (kvantinė mechanika) – Lithuanian" lang="lt" hreflang="lt" data-title="Hamiltonianas (kvantinė mechanika)" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Hamilton-oper%C3%A1tor" title="Hamilton-operátor – Hungarian" lang="hu" hreflang="hu" data-title="Hamilton-operátor" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%8F%E3%83%9F%E3%83%AB%E3%83%88%E3%83%8B%E3%82%A2%E3%83%B3" title="ハミルトニアン – Japanese" lang="ja" hreflang="ja" data-title="ハミルトニアン" 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/Hamilton-operator" title="Hamilton-operator – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Hamilton-operator" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Hamilton-operatoren" title="Hamilton-operatoren – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Hamilton-operatoren" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Gamilton_operatori_(kvant_mexanikasi)" title="Gamilton operatori (kvant mexanikasi) – Uzbek" lang="uz" hreflang="uz" data-title="Gamilton operatori (kvant mexanikasi)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B9%E0%A9%88%E0%A8%AE%E0%A8%BF%E0%A8%B2%E0%A8%9F%E0%A9%8B%E0%A8%A8%E0%A9%80%E0%A8%85%E0%A8%A8_(%E0%A8%95%E0%A9%81%E0%A8%86%E0%A8%82%E0%A8%9F%E0%A8%AE_%E0%A8%AE%E0%A8%95%E0%A9%88%E0%A8%A8%E0%A8%BF%E0%A8%95%E0%A8%B8)" title="ਹੈਮਿਲਟੋਨੀਅਨ (ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ) – Punjabi" lang="pa" hreflang="pa" data-title="ਹੈਮਿਲਟੋਨੀਅਨ (ਕੁਆਂਟਮ ਮਕੈਨਿਕਸ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%DB%81%DB%8C%D9%85%D9%84%D9%B9%D9%88%D9%86%DB%8C%D8%A6%D9%86_(%DA%A9%D9%88%D8%A7%D9%86%D9%B9%D9%85_%D9%85%DA%A9%DB%8C%D9%86%DA%A9%D8%B3)" title="ہیملٹونیئن (کوانٹم مکینکس) – Western Punjabi" lang="pnb" hreflang="pnb" data-title="ہیملٹونیئن (کوانٹم مکینکس)" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Operator_Hamiltona" title="Operator Hamiltona – Polish" lang="pl" hreflang="pl" data-title="Operator Hamiltona" 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/Hamiltoniano_(mec%C3%A2nica_qu%C3%A2ntica)" title="Hamiltoniano (mecânica quântica) – Portuguese" lang="pt" hreflang="pt" data-title="Hamiltoniano (mecânica quântica)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Hamiltonian_(mecanic%C4%83_cuantic%C4%83)" title="Hamiltonian (mecanică cuantică) – Romanian" lang="ro" hreflang="ro" data-title="Hamiltonian (mecanică cuantică)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B0%D0%BC%D0%B8%D0%BB%D1%8C%D1%82%D0%BE%D0%BD%D0%B8%D0%B0%D0%BD_(%D0%BA%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%B0%D1%8F_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0)" title="Гамильтониан (квантовая механика) – Russian" lang="ru" hreflang="ru" data-title="Гамильтониан (квантовая механика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Hamiltonov_oper%C3%A1tor_(Hamiltonovej_funkcie)" title="Hamiltonov operátor (Hamiltonovej funkcie) – Slovak" lang="sk" hreflang="sk" data-title="Hamiltonov operátor (Hamiltonovej funkcie)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Hamiltonin_operaattori" title="Hamiltonin operaattori – Finnish" lang="fi" hreflang="fi" data-title="Hamiltonin operaattori" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Hamiltonoperator" title="Hamiltonoperator – Swedish" lang="sv" hreflang="sv" data-title="Hamiltonoperator" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%93%D0%B0%D0%BC%D0%B8%D0%BB%D1%8C%D1%82%D0%BE%D0%BD%D0%B8%D0%B0%D0%BD_(%D0%BA%D0%B2%D0%B0%D0%BD%D1%82_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0%D1%81%D1%8B)" title="Гамильтониан (квант механикасы) – Tatar" lang="tt" hreflang="tt" data-title="Гамильтониан (квант механикасы)" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%AE%E0%B8%A1%E0%B8%B4%E0%B8%A5%E0%B9%82%E0%B8%97%E0%B9%80%E0%B8%99%E0%B8%B5%E0%B8%A2%E0%B8%99_(%E0%B8%81%E0%B8%A5%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C%E0%B8%84%E0%B8%A7%E0%B8%AD%E0%B8%99%E0%B8%95%E0%B8%B1%E0%B8%A1)" title="แฮมิลโทเนียน (กลศาสตร์ควอนตัม) – Thai" lang="th" hreflang="th" data-title="แฮมิลโทเนียน (กลศาสตร์ควอนตัม)" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Hamilton_i%C5%9Flemcisi" title="Hamilton işlemcisi – Turkish" lang="tr" hreflang="tr" data-title="Hamilton işlemcisi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D0%B0%D0%BC%D1%96%D0%BB%D1%8C%D1%82%D0%BE%D0%BD%D1%96%D0%B0%D0%BD" title="Гамільтоніан – Ukrainian" lang="uk" hreflang="uk" data-title="Гамільтоніан" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/To%C3%A1n_t%E1%BB%AD_Hamilton" title="Toán tử Hamilton – Vietnamese" lang="vi" hreflang="vi" data-title="Toán tử Hamilton" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%93%88%E5%AF%86%E9%A0%93%E7%AE%97%E7%AC%A6" title="哈密頓算符 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="哈密頓算符" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%93%88%E5%AF%86%E9%A1%BF%E7%AE%97%E7%AC%A6" title="哈密顿算符 – Chinese" lang="zh" hreflang="zh" data-title="哈密顿算符" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> 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an <a href="/wiki/Operator_(physics)" title="Operator (physics)">operator</a> corresponding to the total energy of that system, including both <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> and <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a>. Its <a href="/wiki/Spectrum_of_an_operator" class="mw-redirect" title="Spectrum of an operator">spectrum</a>, the system's <i>energy spectrum</i> or its set of <i>energy eigenvalues</i>, is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and <a href="/wiki/Time-evolution_operator" class="mw-redirect" title="Time-evolution operator">time-evolution</a> of a system, it is of fundamental importance in most <a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">formulations of quantum theory</a>. </p><p>The Hamiltonian is named after <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a>, who developed a revolutionary reformulation of <a href="/wiki/Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</a>, known as <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a>, which was historically important to the development of quantum physics. Similar to <a href="/wiki/Vector_notation" title="Vector notation">vector notation</a>, it is typically denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb06de5217295d7fbdbf68fb9c5309a513fc99e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.843ex;" alt="{\displaystyle {\hat {H}}}"></span>, where the hat indicates that it is an operator. It can also be written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\check {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">ˇ</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\check {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4374a1032ffce4c6c50cdc7983284fe33d0cce62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.676ex;" alt="{\displaystyle {\check {H}}}"></span>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=1" title="Edit section: Introduction"><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/Operator_(physics)#Operators_in_quantum_mechanics" title="Operator (physics)">Operator (physics) § Operators in quantum mechanics</a></div> <p>The Hamiltonian of a system represents the total <a href="/wiki/Energy" title="Energy">energy</a> of the system; that is, the sum of the kinetic and potential energies of all particles associated with the system. The Hamiltonian takes different forms and can be simplified in some cases by taking into account the concrete characteristics of the system under analysis, such as single or several particles in the system, interaction between particles, kind of potential energy, time varying potential or time independent one. </p> <div class="mw-heading mw-heading2"><h2 id="Schrödinger_Hamiltonian"><span id="Schr.C3.B6dinger_Hamiltonian"></span>Schrödinger Hamiltonian</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=2" title="Edit section: Schrödinger Hamiltonian"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="One_particle">One particle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=3" title="Edit section: One particle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By analogy with <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">classical mechanics</a>, the Hamiltonian is commonly expressed as the sum of <a href="/wiki/Hermitian_operators" class="mw-redirect" title="Hermitian operators">operators</a> corresponding to the <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic</a> and <a href="/wiki/Potential_energy" title="Potential energy">potential</a> energies of a system in the form </p><p><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 {\hat {H}}={\hat {T}}+{\hat {V}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}={\hat {T}}+{\hat {V}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a03340dfe21d33334c68bce72c1aecded285ac0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.211ex; height:3.176ex;" alt="{\displaystyle {\hat {H}}={\hat {T}}+{\hat {V}},}"></span> </p><p>where <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 {\hat {V}}=V=V(\mathbf {r} ,t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>V</mi> <mo>=</mo> <mi>V</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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {V}}=V=V(\mathbf {r} ,t),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f4261339ba2b438016d7433e7c0822ea5fc5de2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.99ex; height:3.343ex;" alt="{\displaystyle {\hat {V}}=V=V(\mathbf {r} ,t),}"></span> is the <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> operator and <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 {\hat {T}}={\frac {\mathbf {\hat {p}} \cdot \mathbf {\hat {p}} }{2m}}={\frac {{\hat {p}}^{2}}{2m}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">p</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">p</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {T}}={\frac {\mathbf {\hat {p}} \cdot \mathbf {\hat {p}} }{2m}}={\frac {{\hat {p}}^{2}}{2m}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec21764075ecf1b046baba152bdf39033e91c67d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.078ex; height:5.843ex;" alt="{\displaystyle {\hat {T}}={\frac {\mathbf {\hat {p}} \cdot \mathbf {\hat {p}} }{2m}}={\frac {{\hat {p}}^{2}}{2m}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2},}"></span> is the <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> operator in which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is the <a href="/wiki/Mass" title="Mass">mass</a> of the particle, the dot denotes the <a href="/wiki/Dot_product" title="Dot product">dot product</a> of vectors, and <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 {\hat {p}}=-i\hbar \nabla ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mi mathvariant="normal">∇</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {p}}=-i\hbar \nabla ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23e7a584d431887c38bd99ae72f677137e7fc17c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.047ex; height:2.509ex;" alt="{\displaystyle {\hat {p}}=-i\hbar \nabla ,}"></span> is the <a href="/wiki/Momentum_operator" title="Momentum operator">momentum operator</a> where 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 \nabla }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \nabla }"></span> is the <a href="/wiki/Del" title="Del">del</a> <a href="/wiki/Operator_(mathematics)" title="Operator (mathematics)">operator</a>. The <a href="/wiki/Dot_product" title="Dot product">dot product</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 \nabla }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \nabla }"></span> with itself is the <a href="/wiki/Laplacian" class="mw-redirect" title="Laplacian">Laplacian</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 \nabla ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4be87ad083e5ead48d92b0c82f2d4e719cb34a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.99ex; height:2.676ex;" alt="{\displaystyle \nabla ^{2}}"></span>. In three dimensions using <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a> the Laplace operator is <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 \nabla ^{2}={\frac {\partial ^{2}}{{\partial x}^{2}}}+{\frac {\partial ^{2}}{{\partial y}^{2}}}+{\frac {\partial ^{2}}{{\partial z}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}={\frac {\partial ^{2}}{{\partial x}^{2}}}+{\frac {\partial ^{2}}{{\partial y}^{2}}}+{\frac {\partial ^{2}}{{\partial z}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd7191c15d9eea5692764581c5d2e7f5af5b3aa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:24.975ex; height:6.676ex;" alt="{\displaystyle \nabla ^{2}={\frac {\partial ^{2}}{{\partial x}^{2}}}+{\frac {\partial ^{2}}{{\partial y}^{2}}}+{\frac {\partial ^{2}}{{\partial z}^{2}}}}"></span> </p><p>Although this is not the technical definition of the <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian in classical mechanics</a>, it is the form it most commonly takes. Combining these yields the form used in the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>: </p><p><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}{\hat {H}}&={\hat {T}}+{\hat {V}}\\[6pt]&={\frac {\mathbf {\hat {p}} \cdot \mathbf {\hat {p}} }{2m}}+V(\mathbf {r} ,t)\\[6pt]&=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} ,t)\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="0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">p</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">p</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>V</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> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>V</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> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\hat {H}}&={\hat {T}}+{\hat {V}}\\[6pt]&={\frac {\mathbf {\hat {p}} \cdot \mathbf {\hat {p}} }{2m}}+V(\mathbf {r} ,t)\\[6pt]&=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} ,t)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f90c9013581f0b1cd47e5b94c5a9c4e621039f7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.171ex; width:24.163ex; height:17.509ex;" alt="{\displaystyle {\begin{aligned}{\hat {H}}&={\hat {T}}+{\hat {V}}\\[6pt]&={\frac {\mathbf {\hat {p}} \cdot \mathbf {\hat {p}} }{2m}}+V(\mathbf {r} ,t)\\[6pt]&=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} ,t)\end{aligned}}}"></span> </p><p>which allows one to apply the Hamiltonian to systems described by a <a href="/wiki/Wave_function" title="Wave function">wave function</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 \Psi (\mathbf {r} ,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (\mathbf {r} ,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03dc9a12e4176aaee6d293a989de7db205510164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.593ex; height:2.843ex;" alt="{\displaystyle \Psi (\mathbf {r} ,t)}"></span>. This is the approach commonly taken in introductory treatments of quantum mechanics, using the formalism of Schrödinger's wave mechanics. </p><p>One can also make substitutions to certain variables to fit specific cases, such as some involving electromagnetic fields. </p> <div class="mw-heading mw-heading4"><h4 id="Expectation_value">Expectation value</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=4" title="Edit section: Expectation value"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It can be shown that the expectation value of the Hamiltonian which gives the energy expectation value will always be greater than or equal to the minimum potential of the system. </p><p>Consider computing the expectation value of kinetic energy: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}KE&=-{\frac {\hbar ^{2}}{2m}}\int _{-\infty }^{+\infty }\psi ^{*}\left({\frac {d^{2}\psi }{dx^{2}}}\right)\,dx\\&=-{\frac {\hbar ^{2}}{2m}}\left({\left[\psi '(x)\psi ^{*}(x)\right]_{-\infty }^{+\infty }}-\int _{-\infty }^{+\infty }\left({\frac {d\psi }{dx}}\right)\left({\frac {d\psi }{dx}}\right)^{*}\,dx\right)\\&={\frac {\hbar ^{2}}{2m}}\int _{-\infty }^{+\infty }\left|{\frac {d\psi }{dx}}\right|^{2}\,dx\geq 0\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> <mi>K</mi> <mi>E</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mrow> <mo>[</mo> <mrow> <msup> <mi>ψ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mrow> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ψ</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ψ</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ψ</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}KE&=-{\frac {\hbar ^{2}}{2m}}\int _{-\infty }^{+\infty }\psi ^{*}\left({\frac {d^{2}\psi }{dx^{2}}}\right)\,dx\\&=-{\frac {\hbar ^{2}}{2m}}\left({\left[\psi '(x)\psi ^{*}(x)\right]_{-\infty }^{+\infty }}-\int _{-\infty }^{+\infty }\left({\frac {d\psi }{dx}}\right)\left({\frac {d\psi }{dx}}\right)^{*}\,dx\right)\\&={\frac {\hbar ^{2}}{2m}}\int _{-\infty }^{+\infty }\left|{\frac {d\psi }{dx}}\right|^{2}\,dx\geq 0\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef90e86bbf49ad67fd2c753e9cc144b73b4271f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.171ex; width:60.03ex; height:19.509ex;" alt="{\displaystyle {\begin{aligned}KE&=-{\frac {\hbar ^{2}}{2m}}\int _{-\infty }^{+\infty }\psi ^{*}\left({\frac {d^{2}\psi }{dx^{2}}}\right)\,dx\\&=-{\frac {\hbar ^{2}}{2m}}\left({\left[\psi '(x)\psi ^{*}(x)\right]_{-\infty }^{+\infty }}-\int _{-\infty }^{+\infty }\left({\frac {d\psi }{dx}}\right)\left({\frac {d\psi }{dx}}\right)^{*}\,dx\right)\\&={\frac {\hbar ^{2}}{2m}}\int _{-\infty }^{+\infty }\left|{\frac {d\psi }{dx}}\right|^{2}\,dx\geq 0\end{aligned}}}"></span> </p><p>Hence the expectation value of kinetic energy is always non-negative. This result can be used to calculate the expectation value of the total energy which is given for a normalized wavefunction as: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=KE+\langle V(x)\rangle =KE+\int _{-\infty }^{+\infty }V(x)|\psi (x)|^{2}\,dx\geq V_{\text{min}}(x)\int _{-\infty }^{+\infty }|\psi (x)|^{2}\,dx\geq V_{\text{min}}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>K</mi> <mi>E</mi> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>K</mi> <mi>E</mi> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <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> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>≥</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>min</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <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> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>≥</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>min</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=KE+\langle V(x)\rangle =KE+\int _{-\infty }^{+\infty }V(x)|\psi (x)|^{2}\,dx\geq V_{\text{min}}(x)\int _{-\infty }^{+\infty }|\psi (x)|^{2}\,dx\geq V_{\text{min}}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b55b414617ba005ff34b7edbc8e79280be01c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:86.391ex; height:6.176ex;" alt="{\displaystyle E=KE+\langle V(x)\rangle =KE+\int _{-\infty }^{+\infty }V(x)|\psi (x)|^{2}\,dx\geq V_{\text{min}}(x)\int _{-\infty }^{+\infty }|\psi (x)|^{2}\,dx\geq V_{\text{min}}(x)}"></span> </p><p>which complete the proof. Similarly, the condition can be generalized to any higher dimensions using <a href="/wiki/Divergence_theorem" title="Divergence theorem">divergence theorem</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Many_particles">Many particles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=5" title="Edit section: Many particles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The formalism can be extended 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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" 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 N}"></span> particles: </p><p><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 {\hat {H}}=\sum _{n=1}^{N}{\hat {T}}_{n}+{\hat {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}=\sum _{n=1}^{N}{\hat {T}}_{n}+{\hat {V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cb90c4aba583a41945f0e47ac8ca722e92a430f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.524ex; height:7.343ex;" alt="{\displaystyle {\hat {H}}=\sum _{n=1}^{N}{\hat {T}}_{n}+{\hat {V}}}"></span> </p><p>where <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 {\hat {V}}=V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N},t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>V</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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {V}}=V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N},t),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edea261cf5a12d518c7f5731e4c0a6f6cfed9786" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.321ex; height:3.343ex;" alt="{\displaystyle {\hat {V}}=V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N},t),}"></span> is the potential energy function, now a function of the spatial configuration of the system and time (a particular set of spatial positions at some instant of time defines a configuration) and <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 {\hat {T}}_{n}={\frac {\mathbf {\hat {p}} _{n}\cdot \mathbf {\hat {p}} _{n}}{2m_{n}}}=-{\frac {\hbar ^{2}}{2m_{n}}}\nabla _{n}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">p</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">p</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {T}}_{n}={\frac {\mathbf {\hat {p}} _{n}\cdot \mathbf {\hat {p}} _{n}}{2m_{n}}}=-{\frac {\hbar ^{2}}{2m_{n}}}\nabla _{n}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b133e89cc4412f38963cc89e72cf26d39c8a9679" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:27.332ex; height:6.009ex;" alt="{\displaystyle {\hat {T}}_{n}={\frac {\mathbf {\hat {p}} _{n}\cdot \mathbf {\hat {p}} _{n}}{2m_{n}}}=-{\frac {\hbar ^{2}}{2m_{n}}}\nabla _{n}^{2}}"></span> is the kinetic energy operator of 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76d467b5e37601f94cd322a7fc4c9cd02984c1e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.154ex; height:2.509ex;" alt="{\displaystyle \nabla _{n}}"></span> is the gradient for 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, 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 \nabla _{n}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{n}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e720008add4c0e71afaad94aadfd8129dc8bc29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.154ex; height:2.843ex;" alt="{\displaystyle \nabla _{n}^{2}}"></span> is the Laplacian for particle <span class="texhtml mvar" style="font-style:italic;">n</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{n}^{2}={\frac {\partial ^{2}}{\partial x_{n}^{2}}}+{\frac {\partial ^{2}}{\partial y_{n}^{2}}}+{\frac {\partial ^{2}}{\partial z_{n}^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{n}^{2}={\frac {\partial ^{2}}{\partial x_{n}^{2}}}+{\frac {\partial ^{2}}{\partial y_{n}^{2}}}+{\frac {\partial ^{2}}{\partial z_{n}^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a299f8c79e960d369d90749768a3a3ad358d94e3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.249ex; height:6.509ex;" alt="{\displaystyle \nabla _{n}^{2}={\frac {\partial ^{2}}{\partial x_{n}^{2}}}+{\frac {\partial ^{2}}{\partial y_{n}^{2}}}+{\frac {\partial ^{2}}{\partial z_{n}^{2}}},}"></span> </p><p>Combining these yields the Schrödinger Hamiltonian for 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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" 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 N}"></span>-particle case: </p><p><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}{\hat {H}}&=\sum _{n=1}^{N}{\hat {T}}_{n}+{\hat {V}}\\[6pt]&=\sum _{n=1}^{N}{\frac {\mathbf {\hat {p}} _{n}\cdot \mathbf {\hat {p}} _{n}}{2m_{n}}}+V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N},t)\\[6pt]&=-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}\nabla _{n}^{2}+V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N},t)\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="0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">p</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">p</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>V</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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mi>V</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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\hat {H}}&=\sum _{n=1}^{N}{\hat {T}}_{n}+{\hat {V}}\\[6pt]&=\sum _{n=1}^{N}{\frac {\mathbf {\hat {p}} _{n}\cdot \mathbf {\hat {p}} _{n}}{2m_{n}}}+V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N},t)\\[6pt]&=-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}\nabla _{n}^{2}+V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N},t)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/926b8fdf3d6b254223215b88e0bdb7f0298d46ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.005ex; width:43.941ex; height:25.176ex;" alt="{\displaystyle {\begin{aligned}{\hat {H}}&=\sum _{n=1}^{N}{\hat {T}}_{n}+{\hat {V}}\\[6pt]&=\sum _{n=1}^{N}{\frac {\mathbf {\hat {p}} _{n}\cdot \mathbf {\hat {p}} _{n}}{2m_{n}}}+V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N},t)\\[6pt]&=-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}\nabla _{n}^{2}+V(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,\mathbf {r} _{N},t)\end{aligned}}}"></span> </p><p>However, complications can arise in the <a href="/wiki/Many-body_problem" title="Many-body problem">many-body problem</a>. Since the potential energy depends on the spatial arrangement of the particles, the kinetic energy will also depend on the spatial configuration to conserve energy. The motion due to any one particle will vary due to the motion of all the other particles in the system. For this reason cross terms for kinetic energy may appear in the Hamiltonian; a mix of the gradients for two particles: </p><p><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 -{\frac {\hbar ^{2}}{2M}}\nabla _{i}\cdot \nabla _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>M</mi> </mrow> </mfrac> </mrow> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2M}}\nabla _{i}\cdot \nabla _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d35e25a85517bcc312df89f1e380272cc8b8654" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.509ex; height:5.676ex;" alt="{\displaystyle -{\frac {\hbar ^{2}}{2M}}\nabla _{i}\cdot \nabla _{j}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> denotes the mass of the collection of particles resulting in this extra kinetic energy. Terms of this form are known as <i>mass polarization terms</i>, and appear in the Hamiltonian of many-electron atoms (see below). </p><p>For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" 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 N}"></span> interacting particles, i.e. particles which interact mutually and constitute a many-body situation, the potential energy function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> is <i>not</i> simply a sum of the separate potentials (and certainly not a product, as this is dimensionally incorrect). The potential energy function can only be written as above: a function of all the spatial positions of each particle. </p><p>For non-interacting particles, i.e. particles which do not interact mutually and move independently, the potential of the system is the sum of the separate potential energy for each particle,<sup id="cite_ref-QuantumPhysics_1-0" class="reference"><a href="#cite_note-QuantumPhysics-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> that is </p><p><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 V=\sum _{i=1}^{N}V(\mathbf {r} _{i},t)=V(\mathbf {r} _{1},t)+V(\mathbf {r} _{2},t)+\cdots +V(\mathbf {r} _{N},t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mi>V</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>V</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> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\sum _{i=1}^{N}V(\mathbf {r} _{i},t)=V(\mathbf {r} _{1},t)+V(\mathbf {r} _{2},t)+\cdots +V(\mathbf {r} _{N},t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74757793f414adf0f7e0d7f2bce464e0506a7dd1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:53.859ex; height:7.343ex;" alt="{\displaystyle V=\sum _{i=1}^{N}V(\mathbf {r} _{i},t)=V(\mathbf {r} _{1},t)+V(\mathbf {r} _{2},t)+\cdots +V(\mathbf {r} _{N},t)}"></span> </p><p>The general form of the Hamiltonian in this case is: </p><p><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}{\hat {H}}&=-{\frac {\hbar ^{2}}{2}}\sum _{i=1}^{N}{\frac {1}{m_{i}}}\nabla _{i}^{2}+\sum _{i=1}^{N}V_{i}\\[6pt]&=\sum _{i=1}^{N}\left(-{\frac {\hbar ^{2}}{2m_{i}}}\nabla _{i}^{2}+V_{i}\right)\\[6pt]&=\sum _{i=1}^{N}{\hat {H}}_{i}\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="0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <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"> <mi>i</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\hat {H}}&=-{\frac {\hbar ^{2}}{2}}\sum _{i=1}^{N}{\frac {1}{m_{i}}}\nabla _{i}^{2}+\sum _{i=1}^{N}V_{i}\\[6pt]&=\sum _{i=1}^{N}\left(-{\frac {\hbar ^{2}}{2m_{i}}}\nabla _{i}^{2}+V_{i}\right)\\[6pt]&=\sum _{i=1}^{N}{\hat {H}}_{i}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb87996553b25d30463ff8555445506dbf44bbe6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.005ex; width:30.467ex; height:25.176ex;" alt="{\displaystyle {\begin{aligned}{\hat {H}}&=-{\frac {\hbar ^{2}}{2}}\sum _{i=1}^{N}{\frac {1}{m_{i}}}\nabla _{i}^{2}+\sum _{i=1}^{N}V_{i}\\[6pt]&=\sum _{i=1}^{N}\left(-{\frac {\hbar ^{2}}{2m_{i}}}\nabla _{i}^{2}+V_{i}\right)\\[6pt]&=\sum _{i=1}^{N}{\hat {H}}_{i}\end{aligned}}}"></span> </p><p>where the sum is taken over all particles and their corresponding potentials; the result is that the Hamiltonian of the system is the sum of the separate Hamiltonians for each particle. This is an idealized situation—in practice the particles are almost always influenced by some potential, and there are many-body interactions. One illustrative example of a two-body interaction where this form would not apply is for electrostatic potentials due to charged particles, because they interact with each other by Coulomb interaction (electrostatic force), as shown below. </p> <div class="mw-heading mw-heading2"><h2 id="Schrödinger_equation"><span id="Schr.C3.B6dinger_equation"></span>Schrödinger equation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=6" title="Edit section: Schrödinger equation"><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/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a></div> <p>The Hamiltonian generates the time evolution of quantum states. 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 \left|\psi (t)\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\psi (t)\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff277e550e74f3cd25a806b3684cee56703135d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.714ex; height:2.843ex;" alt="{\displaystyle \left|\psi (t)\right\rangle }"></span> is the state of the system at 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>, then </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\left|\psi (t)\right\rangle =i\hbar {d \over \ dt}\left|\psi (t)\right\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mrow> <mo>|</mo> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mtext> </mtext> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>|</mo> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\left|\psi (t)\right\rangle =i\hbar {d \over \ dt}\left|\psi (t)\right\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da1c183e37fcdc61d6bbe4f2ba1fb0cb1150c8d6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.979ex; height:5.509ex;" alt="{\displaystyle H\left|\psi (t)\right\rangle =i\hbar {d \over \ dt}\left|\psi (t)\right\rangle .}"></span> </p><p>This equation is the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>. It takes the same form as the <a href="/wiki/Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a>, which is one of the reasons <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> is also called the Hamiltonian. Given the state at some initial 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=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}"></span>), we can solve it to obtain the state at any subsequent time. In particular, 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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> is independent of time, then </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\psi (t)\right\rangle =e^{-iHt/\hbar }\left|\psi (0)\right\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>H</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <mrow> <mo>|</mo> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>⟩</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\psi (t)\right\rangle =e^{-iHt/\hbar }\left|\psi (0)\right\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00cb75fed234e146c13f2de0aea9f4831df45cfd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.23ex; height:3.343ex;" alt="{\displaystyle \left|\psi (t)\right\rangle =e^{-iHt/\hbar }\left|\psi (0)\right\rangle .}"></span> </p><p>The <a href="/wiki/Matrix_exponential" title="Matrix exponential">exponential</a> operator on the right hand side of the Schrödinger equation is usually defined by the corresponding <a href="/wiki/Exponential_function#Formal_definition" title="Exponential function">power series</a> 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>. One might notice that taking polynomials or power series of <a href="/wiki/Unbounded_operator" title="Unbounded operator">unbounded operators</a> that are not defined everywhere may not make mathematical sense. Rigorously, to take functions of unbounded operators, a <a href="/wiki/Functional_calculus" title="Functional calculus">functional calculus</a> is required. In the case of the exponential function, the <a href="/wiki/Continuous_functional_calculus" title="Continuous functional calculus">continuous</a>, or just the <a href="/wiki/Holomorphic_functional_calculus" title="Holomorphic functional calculus">holomorphic functional calculus</a> suffices. We note again, however, that for common calculations the physicists' formulation is quite sufficient. </p><p>By the *-<a href="/wiki/Homomorphism" title="Homomorphism">homomorphism</a> property of the functional calculus, the operator </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=e^{-iHt/\hbar }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>H</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=e^{-iHt/\hbar }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce5a34ca8fe0fc3c15e0ca34746c5e08c4eb168e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.841ex; height:2.843ex;" alt="{\displaystyle U=e^{-iHt/\hbar }}"></span> </p><p>is a <a href="/wiki/Unitary_operator" title="Unitary operator">unitary operator</a>. It is the <i><a href="/wiki/Time_evolution" title="Time evolution">time evolution</a> operator</i> or <i><a href="/wiki/Propagator" title="Propagator">propagator</a></i> of a closed quantum system. If the Hamiltonian is time-independent, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{U(t)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{U(t)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74787c4f2d2087a3a15dd7003e3c044849bd8528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.756ex; height:2.843ex;" alt="{\displaystyle \{U(t)\}}"></span> form a <a href="/wiki/Stone%27s_theorem_on_one-parameter_unitary_groups" title="Stone's theorem on one-parameter unitary groups">one parameter unitary group</a> (more than a <a href="/wiki/C0_semigroup" class="mw-redirect" title="C0 semigroup">semigroup</a>); this gives rise to the physical principle of <a href="/wiki/Detailed_balance" title="Detailed balance">detailed balance</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Dirac_formalism">Dirac formalism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=7" title="Edit section: Dirac formalism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>However, in the <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">more general formalism</a> of <a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a>, the Hamiltonian is typically implemented as an operator on a <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> in the following way: </p><p>The eigenkets (<a href="/wiki/Eigenvectors" class="mw-redirect" title="Eigenvectors">eigenvectors</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>, denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|a\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mi>a</mi> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|a\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ab685af4d9718658221dfd3c397329938a99df5" 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 \left|a\right\rangle }"></span>, provide an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> for the Hilbert space. The spectrum of allowed energy levels of the system is given by the set of eigenvalues, denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{E_{a}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{E_{a}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b80e913280012d768475d11d2c752dbb1dd972ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.142ex; height:2.843ex;" alt="{\displaystyle \{E_{a}\}}"></span>, solving the equation: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\left|a\right\rangle =E_{a}\left|a\right\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mrow> <mo>|</mo> <mi>a</mi> <mo>⟩</mo> </mrow> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mrow> <mo>|</mo> <mi>a</mi> <mo>⟩</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\left|a\right\rangle =E_{a}\left|a\right\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af3c0bea9af0117671f937057a9e6631d078e3ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.35ex; height:2.843ex;" alt="{\displaystyle H\left|a\right\rangle =E_{a}\left|a\right\rangle .}"></span> </p><p>Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> is a <a href="/wiki/Hermitian_operator" class="mw-redirect" title="Hermitian operator">Hermitian operator</a>, the energy is always a <a href="/wiki/Real_number" title="Real number">real number</a>. </p><p>From a mathematically rigorous point of view, care must be taken with the above assumptions. Operators on infinite-dimensional Hilbert spaces need not have eigenvalues (the set of eigenvalues does not necessarily coincide with the <a href="/wiki/Spectrum_of_an_operator" class="mw-redirect" title="Spectrum of an operator">spectrum of an operator</a>). However, all routine quantum mechanical calculations can be done using the physical formulation.<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (December 2011)">clarification needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Expressions_for_the_Hamiltonian">Expressions for the Hamiltonian</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=8" title="Edit section: Expressions for the Hamiltonian"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Following are expressions for the Hamiltonian in a number of situations.<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> Typical ways to classify the expressions are the number of particles, number of dimensions, and the nature of the potential energy function—importantly space and time dependence. Masses are denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>, and charges by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="General_forms_for_one_particle">General forms for one particle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=9" title="Edit section: General forms for one particle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Free_particle">Free particle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=10" title="Edit section: Free particle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The particle is not bound by any potential energy, so the potential is zero and this Hamiltonian is the simplest. For one dimension: </p><p><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 {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a20bf2204c159d4bf3465228de3974958e415ac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.547ex; height:6.009ex;" alt="{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}}"></span> </p><p>and in higher dimensions: </p><p><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 {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bb8b3f0c2fccdb923fb2d91632b12443e40bd60" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.999ex; height:5.676ex;" alt="{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Constant-potential_well">Constant-potential well</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=11" title="Edit section: Constant-potential well"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a particle in a region of constant potential <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=V_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=V_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d6e788c814e46b41d15a1d88c866d999b67d0b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.295ex; height:2.509ex;" alt="{\displaystyle V=V_{0}}"></span> (no dependence on space or time), in one dimension, the Hamiltonian is: </p><p><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 {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+V_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+V_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ebc8f44d7a0037b7a176cc9698ce53d2ca695e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.797ex; height:6.009ex;" alt="{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+V_{0}}"></span> </p><p>in three dimensions </p><p><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 {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e03f6c6584cde6223c18e0d52c3f48ff69a8a46" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.249ex; height:5.676ex;" alt="{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V_{0}}"></span> </p><p>This applies to the elementary "<a href="/wiki/Particle_in_a_box" title="Particle in a box">particle in a box</a>" problem, and <a href="/wiki/Step_potential" title="Step potential">step potentials</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Simple_harmonic_oscillator">Simple harmonic oscillator</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=12" title="Edit section: Simple harmonic oscillator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a <a href="/wiki/Simple_harmonic_oscillator" class="mw-redirect" title="Simple harmonic oscillator">simple harmonic oscillator</a> in one dimension, the potential varies with position (but not time), according to: </p><p><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 V={\frac {k}{2}}x^{2}={\frac {m\omega ^{2}}{2}}x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {k}{2}}x^{2}={\frac {m\omega ^{2}}{2}}x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/351cf2b8fd9e9f2f138f9577dbc5a79f690f86f3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.176ex; height:5.676ex;" alt="{\displaystyle V={\frac {k}{2}}x^{2}={\frac {m\omega ^{2}}{2}}x^{2}}"></span> </p><p>where the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span>, effective <a href="/wiki/Spring_constant" class="mw-redirect" title="Spring constant">spring constant</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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>, and mass <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> of the oscillator satisfy: </p><p><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 \omega ^{2}={\frac {k}{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>m</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{2}={\frac {k}{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e98436665d031f5893adfa65e5382576098b81" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.475ex; height:5.343ex;" alt="{\displaystyle \omega ^{2}={\frac {k}{m}}}"></span> </p><p>so the Hamiltonian is: </p><p><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 {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {m\omega ^{2}}{2}}x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {m\omega ^{2}}{2}}x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/113fbf59cc852bfe9b2e835f38535093210f563c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:26.148ex; height:6.009ex;" alt="{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {m\omega ^{2}}{2}}x^{2}}"></span> </p><p>For three dimensions, this becomes </p><p><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 {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+{\frac {m\omega ^{2}}{2}}r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+{\frac {m\omega ^{2}}{2}}r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1422b298a5d9ec2145f1c7bfe14dd3137e147891" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.319ex; height:5.676ex;" alt="{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+{\frac {m\omega ^{2}}{2}}r^{2}}"></span> </p><p>where the three-dimensional position 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 \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span> using Cartesian coordinates 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 (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </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/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}"></span>, its magnitude is </p><p><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 r^{2}=\mathbf {r} \cdot \mathbf {r} =|\mathbf {r} |^{2}=x^{2}+y^{2}+z^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r^{2}=\mathbf {r} \cdot \mathbf {r} =|\mathbf {r} |^{2}=x^{2}+y^{2}+z^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed80ac062e68a8cde089fc5b43926b2bc69bb394" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.156ex; height:3.343ex;" alt="{\displaystyle r^{2}=\mathbf {r} \cdot \mathbf {r} =|\mathbf {r} |^{2}=x^{2}+y^{2}+z^{2}}"></span> </p><p>Writing the Hamiltonian out in full shows it is simply the sum of the one-dimensional Hamiltonians in each direction: </p><p><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}{\hat {H}}&=-{\frac {\hbar ^{2}}{2m}}\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}\right)+{\frac {m\omega ^{2}}{2}}\left(x^{2}+y^{2}+z^{2}\right)\\[6pt]&=\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {m\omega ^{2}}{2}}x^{2}\right)+\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {m\omega ^{2}}{2}}y^{2}\right)+\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial z^{2}}}+{\frac {m\omega ^{2}}{2}}z^{2}\right)\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="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\hat {H}}&=-{\frac {\hbar ^{2}}{2m}}\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}\right)+{\frac {m\omega ^{2}}{2}}\left(x^{2}+y^{2}+z^{2}\right)\\[6pt]&=\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {m\omega ^{2}}{2}}x^{2}\right)+\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {m\omega ^{2}}{2}}y^{2}\right)+\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial z^{2}}}+{\frac {m\omega ^{2}}{2}}z^{2}\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb5266b5f61047e72a2f39c47b66faf660ebd659" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:83.999ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}{\hat {H}}&=-{\frac {\hbar ^{2}}{2m}}\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}\right)+{\frac {m\omega ^{2}}{2}}\left(x^{2}+y^{2}+z^{2}\right)\\[6pt]&=\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {m\omega ^{2}}{2}}x^{2}\right)+\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {m\omega ^{2}}{2}}y^{2}\right)+\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial z^{2}}}+{\frac {m\omega ^{2}}{2}}z^{2}\right)\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Rigid_rotor">Rigid rotor</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=13" title="Edit section: Rigid rotor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a <a href="/wiki/Rigid_rotor" title="Rigid rotor">rigid rotor</a>—i.e., system of particles which can rotate freely about any axes, not bound in any potential (such as free molecules with negligible vibrational <a href="/wiki/Degrees_of_freedom_(physics_and_chemistry)" title="Degrees of freedom (physics and chemistry)">degrees of freedom</a>, say due to <a href="/wiki/Double_bond" title="Double bond">double</a> or <a href="/wiki/Triple_bond" title="Triple bond">triple</a> <a href="/wiki/Chemical_bond" title="Chemical bond">chemical bonds</a>), the Hamiltonian is: </p><p><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 {\hat {H}}=-{\frac {\hbar ^{2}}{2I_{xx}}}{\hat {J}}_{x}^{2}-{\frac {\hbar ^{2}}{2I_{yy}}}{\hat {J}}_{y}^{2}-{\frac {\hbar ^{2}}{2I_{zz}}}{\hat {J}}_{z}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>J</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>J</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>J</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2I_{xx}}}{\hat {J}}_{x}^{2}-{\frac {\hbar ^{2}}{2I_{yy}}}{\hat {J}}_{y}^{2}-{\frac {\hbar ^{2}}{2I_{zz}}}{\hat {J}}_{z}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/541ae78c8003211be8460aef6b8b6a23ffa79367" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.215ex; height:6.343ex;" alt="{\displaystyle {\hat {H}}=-{\frac {\hbar ^{2}}{2I_{xx}}}{\hat {J}}_{x}^{2}-{\frac {\hbar ^{2}}{2I_{yy}}}{\hat {J}}_{y}^{2}-{\frac {\hbar ^{2}}{2I_{zz}}}{\hat {J}}_{z}^{2}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{xx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{xx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23d06dfedcfe6aa332c557ec72c913472732de65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.136ex; height:2.509ex;" alt="{\displaystyle I_{xx}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{yy}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{yy}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/198e446f7fc446d3d3b304e92b140035f47cc49a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.889ex; height:2.843ex;" alt="{\displaystyle I_{yy}}"></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 I_{zz}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{zz}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/168645154f8c026d4d6ca46925d83bf5bce57939" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.794ex; height:2.509ex;" alt="{\displaystyle I_{zz}}"></span> are the <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a> components (technically the diagonal elements of the <a href="/wiki/Moment_of_inertia#Moment_of_inertia_tensor" title="Moment of inertia">moment of inertia tensor</a>), and <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {J}}_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>J</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {J}}_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f1a3fa2d259fac5e64ae442669d54829b792109" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.995ex; height:3.176ex;" alt="{\displaystyle {\hat {J}}_{x}}"></span>,</span> <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {J}}_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>J</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {J}}_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbe1f5b78d7c9f8a33b121abba4145e750c1c78f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.872ex; height:3.509ex;" alt="{\displaystyle {\hat {J}}_{y}}"></span>,</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 {\hat {J}}_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>J</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {J}}_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a693bfe172f6b45b3faba47ab326bd20bd116789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.824ex; height:3.176ex;" alt="{\displaystyle {\hat {J}}_{z}}"></span> are the total <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> operators (components), about 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>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></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 z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> axes respectively. </p> <div class="mw-heading mw-heading3"><h3 id="Electrostatic_(Coulomb)_potential"><span id="Electrostatic_.28Coulomb.29_potential"></span>Electrostatic (Coulomb) potential</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=14" title="Edit section: Electrostatic (Coulomb) potential"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Coulomb_potential_energy" class="mw-redirect" title="Coulomb potential energy">Coulomb potential energy</a> for two point charges <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9daa41f6e8f78ea6bb5711d7ac97901ce564b94e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.009ex;" alt="{\displaystyle q_{1}}"></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 q_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd2d05084feb02b8ba29b0673440fb673b102589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.009ex;" alt="{\displaystyle q_{2}}"></span> (i.e., those that have no spatial extent independently), in three dimensions, is (in <a href="/wiki/SI_units" class="mw-redirect" title="SI units">SI units</a>—rather than <a href="/wiki/Gaussian_units" title="Gaussian units">Gaussian units</a> which are frequently used in <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a>): </p><p><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 V={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}|\mathbf {r} |}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}|\mathbf {r} |}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cad6ba1fa8fc61d4b3fea42fa475672e0bee0f0b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.75ex; height:5.676ex;" alt="{\displaystyle V={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}|\mathbf {r} |}}}"></span> </p><p>However, this is only the potential for one point charge due to another. If there are many charged particles, each charge has a potential energy due to every other point charge (except itself). For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" 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 N}"></span> charges, the potential energy of charge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0d567ac2d170501680d2efa4c1d71d6a8569ef1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.947ex; height:2.343ex;" alt="{\displaystyle q_{j}}"></span> due to all other charges is (see also <a href="/wiki/Electric_potential_energy#Electrostatic_potential_energy_stored_in_a_system_of_point_charges" title="Electric potential energy">Electrostatic potential energy stored in a configuration of discrete point charges</a>):<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p><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 V_{j}={\frac {1}{2}}\sum _{i\neq j}q_{i}\phi (\mathbf {r} _{i})={\frac {1}{8\pi \varepsilon _{0}}}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mrow> </munder> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>8</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{j}={\frac {1}{2}}\sum _{i\neq j}q_{i}\phi (\mathbf {r} _{i})={\frac {1}{8\pi \varepsilon _{0}}}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b28d0fc0db9998c7bd98ee1ac2f456fd5455ca27" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:40.004ex; height:6.843ex;" alt="{\displaystyle V_{j}={\frac {1}{2}}\sum _{i\neq j}q_{i}\phi (\mathbf {r} _{i})={\frac {1}{8\pi \varepsilon _{0}}}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (\mathbf {r} _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (\mathbf {r} _{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6364aefe582c86979ff3a378b13b7588f70c815a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.096ex; height:2.843ex;" alt="{\displaystyle \phi (\mathbf {r} _{i})}"></span> is the electrostatic potential of charge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0d567ac2d170501680d2efa4c1d71d6a8569ef1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.947ex; height:2.343ex;" alt="{\displaystyle q_{j}}"></span> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed603561819ebd007acd75a0931d3ba401ad677a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.902ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{i}}"></span>. The total potential of the system is then the sum 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 j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>: </p><p><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 V={\frac {1}{8\pi \varepsilon _{0}}}\sum _{j=1}^{N}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>8</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {1}{8\pi \varepsilon _{0}}}\sum _{j=1}^{N}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e68baeb6689948cc17906471f2a4ab9a1fa97516" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:27.109ex; height:7.843ex;" alt="{\displaystyle V={\frac {1}{8\pi \varepsilon _{0}}}\sum _{j=1}^{N}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}}"></span> </p><p>so the Hamiltonian is: </p><p><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}{\hat {H}}&=-{\frac {\hbar ^{2}}{2}}\sum _{j=1}^{N}{\frac {1}{m_{j}}}\nabla _{j}^{2}+{\frac {1}{8\pi \varepsilon _{0}}}\sum _{j=1}^{N}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}\\&=\sum _{j=1}^{N}\left(-{\frac {\hbar ^{2}}{2m_{j}}}\nabla _{j}^{2}+{\frac {1}{8\pi \varepsilon _{0}}}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}\right)\\\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"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mfrac> </mrow> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>8</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>8</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>≠</mo> <mi>j</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\hat {H}}&=-{\frac {\hbar ^{2}}{2}}\sum _{j=1}^{N}{\frac {1}{m_{j}}}\nabla _{j}^{2}+{\frac {1}{8\pi \varepsilon _{0}}}\sum _{j=1}^{N}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}\\&=\sum _{j=1}^{N}\left(-{\frac {\hbar ^{2}}{2m_{j}}}\nabla _{j}^{2}+{\frac {1}{8\pi \varepsilon _{0}}}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}\right)\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29ad26451abe12bca208b3d8be82ed2e10ff8c25" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:46.903ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}{\hat {H}}&=-{\frac {\hbar ^{2}}{2}}\sum _{j=1}^{N}{\frac {1}{m_{j}}}\nabla _{j}^{2}+{\frac {1}{8\pi \varepsilon _{0}}}\sum _{j=1}^{N}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}\\&=\sum _{j=1}^{N}\left(-{\frac {\hbar ^{2}}{2m_{j}}}\nabla _{j}^{2}+{\frac {1}{8\pi \varepsilon _{0}}}\sum _{i\neq j}{\frac {q_{i}q_{j}}{|\mathbf {r} _{i}-\mathbf {r} _{j}|}}\right)\\\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Electric_dipole_in_an_electric_field">Electric dipole in an electric field</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=15" title="Edit section: Electric dipole in an electric field"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For an <a href="/wiki/Electric_dipole_moment" title="Electric dipole moment">electric dipole moment</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 {d} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {d} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec3b626fc045b6ff579316e29978fccfed884c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {d} }"></span> constituting charges of magnitude <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>, in a uniform, <a href="/wiki/Electrostatic_field" class="mw-redirect" title="Electrostatic field">electrostatic field</a> (time-independent) <span class="mwe-math-element"><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 {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} }"></span>, positioned in one place, the potential is: </p><p><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 V=-\mathbf {\hat {d}} \cdot \mathbf {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">d</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=-\mathbf {\hat {d}} \cdot \mathbf {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/badea711f617c9622426a1ac698cdd4c5d1000d6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.615ex; height:3.009ex;" alt="{\displaystyle V=-\mathbf {\hat {d}} \cdot \mathbf {E} }"></span> </p><p>the dipole moment itself is the operator </p><p><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 \mathbf {\hat {d}} =q\mathbf {\hat {r}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">d</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {d}} =q\mathbf {\hat {r}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee2de2111312ee9566dfc9eeeb63b61548ecbc0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.99ex; height:3.176ex;" alt="{\displaystyle \mathbf {\hat {d}} =q\mathbf {\hat {r}} }"></span> </p><p>Since the particle is stationary, there is no translational kinetic energy of the dipole, so the Hamiltonian of the dipole is just the potential energy: </p><p><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 {\hat {H}}=-\mathbf {\hat {d}} \cdot \mathbf {E} =-q\mathbf {\hat {r}} \cdot \mathbf {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">d</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mo>−</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}=-\mathbf {\hat {d}} \cdot \mathbf {E} =-q\mathbf {\hat {r}} \cdot \mathbf {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b1209643d630d7aff79b8333091f49902907d8b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.641ex; height:3.176ex;" alt="{\displaystyle {\hat {H}}=-\mathbf {\hat {d}} \cdot \mathbf {E} =-q\mathbf {\hat {r}} \cdot \mathbf {E} }"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Magnetic_dipole_in_a_magnetic_field">Magnetic dipole in a magnetic field</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=16" title="Edit section: Magnetic dipole in a magnetic field"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a magnetic dipole moment <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\mu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\mu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1aee7d7b4a36d96dfb35bfee9c7751bba1fdfbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.646ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {\mu }}}"></span> in a uniform, magnetostatic field (time-independent) <span class="mwe-math-element"><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 {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cafb0ef39b0f5ffa23c170aa7f7b4e718327c4d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.901ex; height:2.176ex;" alt="{\displaystyle \mathbf {B} }"></span>, positioned in one place, the potential is: </p><p><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 V=-{\boldsymbol {\mu }}\cdot \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=-{\boldsymbol {\mu }}\cdot \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6936db20c8d8c40568d28dfd246108f7303f956b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.92ex; height:2.509ex;" alt="{\displaystyle V=-{\boldsymbol {\mu }}\cdot \mathbf {B} }"></span> </p><p>Since the particle is stationary, there is no translational kinetic energy of the dipole, so the Hamiltonian of the dipole is just the potential energy: </p><p><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 {\hat {H}}=-{\boldsymbol {\mu }}\cdot \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}=-{\boldsymbol {\mu }}\cdot \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bc074e4989efa6a7a498eb5c0d8c97f8623e9bd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.196ex; height:3.176ex;" alt="{\displaystyle {\hat {H}}=-{\boldsymbol {\mu }}\cdot \mathbf {B} }"></span> </p><p>For a <a href="/wiki/Spin-1/2" title="Spin-1/2">spin-<style data-mw-deduplicate="TemplateStyles:r1154941027">'"`UNIQ--templatestyles-00000069-QINU`"'</style><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span></a> particle, the corresponding spin magnetic moment is:<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p><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 {\boldsymbol {\mu }}_{S}={\frac {g_{s}e}{2m}}\mathbf {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mi>e</mi> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\mu }}_{S}={\frac {g_{s}e}{2m}}\mathbf {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58cb9b9e0665e7e65f7f8d6b2f20a710bb174964" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.561ex; height:4.843ex;" alt="{\displaystyle {\boldsymbol {\mu }}_{S}={\frac {g_{s}e}{2m}}\mathbf {S} }"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ddd218a9b5b346d0dd85e51359c7cab5adf8697" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.112ex; height:2.009ex;" alt="{\displaystyle g_{s}}"></span> is the "spin <a href="/wiki/G-factor_(physics)" title="G-factor (physics)">g-factor</a>" (not to be confused with the <a href="/wiki/Gyromagnetic_ratio" title="Gyromagnetic ratio">gyromagnetic ratio</a>), <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <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/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> is the electron charge, <span class="mwe-math-element"><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 {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac8a515de34f0af7d15de46f73bf674950d444a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {S} }"></span> is the <a href="/wiki/Spin_(physics)#Pauli_matrices_and_spin_operators" title="Spin (physics)">spin operator</a> vector, whose components are the <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a>, hence </p><p><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 {\hat {H}}={\frac {g_{s}e}{2m}}\mathbf {S} \cdot \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mi>e</mi> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}={\frac {g_{s}e}{2m}}\mathbf {S} \cdot \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdc3361e472b3024e3379994b9dcbbdda647dd9c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.267ex; height:4.843ex;" alt="{\displaystyle {\hat {H}}={\frac {g_{s}e}{2m}}\mathbf {S} \cdot \mathbf {B} }"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Charged_particle_in_an_electromagnetic_field">Charged particle in an electromagnetic field</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=17" title="Edit section: Charged particle in an electromagnetic field"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a particle with mass <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> and charge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> in an electromagnetic field, described by the <a href="/wiki/Scalar_potential" title="Scalar potential">scalar potential</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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> and <a href="/wiki/Vector_potential" title="Vector potential">vector potential</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 {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span>, there are two parts to the Hamiltonian to substitute for.<sup id="cite_ref-QuantumPhysics_1-1" class="reference"><a href="#cite_note-QuantumPhysics-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The canonical momentum operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {p}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">p</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {p}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df42593e56d14c2122d2f0a4e9bb809d529252c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.676ex;" alt="{\displaystyle \mathbf {\hat {p}} }"></span>, which includes a contribution from 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 \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span> field and fulfils the <a href="/wiki/Canonical_commutation_relation" title="Canonical commutation relation">canonical commutation relation</a>, must be quantized; </p><p><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 \mathbf {\hat {p}} =m{\dot {\mathbf {r} }}+q\mathbf {A} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">p</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {p}} =m{\dot {\mathbf {r} }}+q\mathbf {A} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/434eafd536115f1f91a0ad2a0f88e1bc9de9d9cd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.363ex; height:2.676ex;" alt="{\displaystyle \mathbf {\hat {p}} =m{\dot {\mathbf {r} }}+q\mathbf {A} ,}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\dot {\mathbf {r} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\dot {\mathbf {r} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c597201e71e52f5a1514f59aaf8d7263659f413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.203ex; height:2.176ex;" alt="{\displaystyle m{\dot {\mathbf {r} }}}"></span> is the <a href="/wiki/Kinetic_momentum" class="mw-redirect" title="Kinetic momentum">kinetic momentum</a>. The quantization prescription reads </p><p><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 \mathbf {\hat {p}} =-i\hbar \nabla ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">p</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mi mathvariant="normal">∇</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/705ca0bef90d1e04e0fb35a6b66d119628b22cdd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.084ex; height:2.676ex;" alt="{\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla ,}"></span> </p><p>so the corresponding kinetic energy operator is </p><p><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 {\hat {T}}={\frac {1}{2}}m{\dot {\mathbf {r} }}\cdot {\dot {\mathbf {r} }}={\frac {1}{2m}}\left(\mathbf {\hat {p}} -q\mathbf {A} \right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">p</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {T}}={\frac {1}{2}}m{\dot {\mathbf {r} }}\cdot {\dot {\mathbf {r} }}={\frac {1}{2m}}\left(\mathbf {\hat {p}} -q\mathbf {A} \right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66aec84c1cf707c2effa9c4fdefd116a82df8644" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.331ex; height:5.176ex;" alt="{\displaystyle {\hat {T}}={\frac {1}{2}}m{\dot {\mathbf {r} }}\cdot {\dot {\mathbf {r} }}={\frac {1}{2m}}\left(\mathbf {\hat {p}} -q\mathbf {A} \right)^{2}}"></span> </p><p>and the potential energy, which is due to 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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> field, is given by </p><p><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 {\hat {V}}=q\phi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>q</mi> <mi>ϕ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {V}}=q\phi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b4edac075fc991d0ce3e44528c7775ce4cd9d1b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.988ex; height:3.176ex;" alt="{\displaystyle {\hat {V}}=q\phi .}"></span> </p><p>Casting all of these into the Hamiltonian gives </p><p><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 {\hat {H}}={\frac {1}{2m}}\left(-i\hbar \nabla -q\mathbf {A} \right)^{2}+q\phi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mi mathvariant="normal">∇</mi> <mo>−</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>q</mi> <mi>ϕ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}={\frac {1}{2m}}\left(-i\hbar \nabla -q\mathbf {A} \right)^{2}+q\phi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3284a7250dbd774ada7992c9fbbd91e67d43d899" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.789ex; height:5.176ex;" alt="{\displaystyle {\hat {H}}={\frac {1}{2m}}\left(-i\hbar \nabla -q\mathbf {A} \right)^{2}+q\phi .}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Energy_eigenket_degeneracy,_symmetry,_and_conservation_laws"><span id="Energy_eigenket_degeneracy.2C_symmetry.2C_and_conservation_laws"></span>Energy eigenket degeneracy, symmetry, and conservation laws</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=18" title="Edit section: Energy eigenket degeneracy, symmetry, and conservation laws"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In many systems, two or more energy eigenstates have the same energy. A simple example of this is a free particle, whose energy eigenstates have wavefunctions that are propagating plane waves. The energy of each of these plane waves is inversely proportional to the square of its <a href="/wiki/Wavelength" title="Wavelength">wavelength</a>. A wave propagating in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> direction is a different state from one propagating in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> direction, but if they have the same wavelength, then their energies will be the same. When this happens, the states are said to be <i>degenerate</i>. </p><p>It turns out that <a href="/wiki/Degenerate_energy_levels" title="Degenerate energy levels">degeneracy</a> occurs whenever a nontrivial <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary operator</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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> <a href="/wiki/Commutation_relation" class="mw-redirect" title="Commutation relation">commutes</a> with the Hamiltonian. To see this, suppose that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/022d80b506eff10c19feb8b5b70bc8c72baa1fa1" 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 |a\rangle }"></span> is an energy eigenket. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U|a\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U|a\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae0c3c4c6e7778a3e1a174a32f9591340fc0d10e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.564ex; height:2.843ex;" alt="{\displaystyle U|a\rangle }"></span> is an energy eigenket with the same eigenvalue, since </p><p><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 UH|a\rangle =UE_{a}|a\rangle =E_{a}(U|a\rangle )=H\;(U|a\rangle ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>U</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle UH|a\rangle =UE_{a}|a\rangle =E_{a}(U|a\rangle )=H\;(U|a\rangle ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab44a1343e93399290c0245cd589f4469c546ea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.223ex; height:2.843ex;" alt="{\displaystyle UH|a\rangle =UE_{a}|a\rangle =E_{a}(U|a\rangle )=H\;(U|a\rangle ).}"></span> </p><p>Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> is nontrivial, at least one pair 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 |a\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/022d80b506eff10c19feb8b5b70bc8c72baa1fa1" 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 |a\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 U|a\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U|a\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae0c3c4c6e7778a3e1a174a32f9591340fc0d10e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.564ex; height:2.843ex;" alt="{\displaystyle U|a\rangle }"></span> must represent distinct states. Therefore, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> has at least one pair of degenerate energy eigenkets. In the case of the free particle, the unitary operator which produces the symmetry is the <a href="/wiki/Rotation_operator_(quantum_mechanics)" title="Rotation operator (quantum mechanics)">rotation operator</a>, which rotates the wavefunctions by some angle while otherwise preserving their shape. </p><p>The existence of a symmetry operator implies the existence of a <a href="/wiki/Conservation_law_(physics)" class="mw-redirect" title="Conservation law (physics)">conserved</a> observable. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> be the Hermitian generator 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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=I-i\varepsilon G+O(\varepsilon ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mi>I</mi> <mo>−</mo> <mi>i</mi> <mi>ε</mi> <mi>G</mi> <mo>+</mo> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=I-i\varepsilon G+O(\varepsilon ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d4f107974ad9b0956495069bae8b66131e03b1a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.167ex; height:3.176ex;" alt="{\displaystyle U=I-i\varepsilon G+O(\varepsilon ^{2})}"></span> </p><p>It is straightforward to show that 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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> commutes with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>, then so does <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [H,G]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>H</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [H,G]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a0488ee00fe98e52930b366705f78bf7ec7e7a5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.479ex; height:2.843ex;" alt="{\displaystyle [H,G]=0}"></span> </p><p>Therefore, </p><p><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 {\frac {\partial }{\partial t}}\langle \psi (t)|G|\psi (t)\rangle ={\frac {1}{i\hbar }}\langle \psi (t)|[G,H]|\psi (t)\rangle =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">[</mo> <mi>G</mi> <mo>,</mo> <mi>H</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial t}}\langle \psi (t)|G|\psi (t)\rangle ={\frac {1}{i\hbar }}\langle \psi (t)|[G,H]|\psi (t)\rangle =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32552e107d9aefd6219289bfa4b1b7d03b696bd0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:44.844ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial }{\partial t}}\langle \psi (t)|G|\psi (t)\rangle ={\frac {1}{i\hbar }}\langle \psi (t)|[G,H]|\psi (t)\rangle =0.}"></span> </p><p>In obtaining this result, we have used the Schrödinger equation, as well as its <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">dual</a>, </p><p><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 (t)|H=-i\hbar {d \over \ dt}\langle \psi (t)|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mtext> </mtext> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi (t)|H=-i\hbar {d \over \ dt}\langle \psi (t)|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14782b9f4c93d02bef46f0b54b65d2706dd486ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.625ex; height:5.509ex;" alt="{\displaystyle \langle \psi (t)|H=-i\hbar {d \over \ dt}\langle \psi (t)|.}"></span> </p><p>Thus, the <a href="/wiki/Expected_value" title="Expected value">expected value</a> of the observable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> is conserved for any state of the system. In the case of the free particle, the conserved quantity is the <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Hamilton's_equations"><span id="Hamilton.27s_equations"></span>Hamilton's equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=19" title="Edit section: Hamilton's equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a>'s equations in classical <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a> have a direct analogy in quantum mechanics. Suppose we have a set of 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 \left\{\left|n\right\rangle \right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mo>|</mo> <mi>n</mi> <mo>⟩</mo> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\left|n\right\rangle \right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f6583921deccf0138e7a01258dd4616d7b3103" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.271ex; height:2.843ex;" alt="{\displaystyle \left\{\left|n\right\rangle \right\}}"></span>, which need not necessarily be eigenstates of the energy. For simplicity, we assume that they are discrete, and that they are orthonormal, i.e., </p><p><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 n'|n\rangle =\delta _{nn'}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>n</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <msup> <mi>n</mi> <mo>′</mo> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle n'|n\rangle =\delta _{nn'}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dee702d319e18cbfed5cb266a2f139ad89e51d3b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.797ex; height:3.009ex;" alt="{\displaystyle \langle n'|n\rangle =\delta _{nn'}}"></span> </p><p>Note that these basis states are assumed to be independent of time. We will assume that the Hamiltonian is also independent of time. </p><p>The instantaneous state of the system at 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>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\psi \left(t\right)\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>ψ</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\psi \left(t\right)\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c56e2ffd124d44e11e8aba5e1f065eb023e52cbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.101ex; height:2.843ex;" alt="{\displaystyle \left|\psi \left(t\right)\right\rangle }"></span>, can be expanded in terms of these basis states: </p><p><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}a_{n}(t)|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> <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>a</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> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi (t)\rangle =\sum _{n}a_{n}(t)|n\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed6fd75eb3cc7fd3366b672a14123ea57545e092" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.598ex; height:5.509ex;" alt="{\displaystyle |\psi (t)\rangle =\sum _{n}a_{n}(t)|n\rangle }"></span> </p><p>where </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}(t)=\langle n|\psi (t)\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}(t)=\langle n|\psi (t)\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32a6fe28958032b2ed3fe34513d747eff4c697dd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.855ex; height:2.843ex;" alt="{\displaystyle a_{n}(t)=\langle n|\psi (t)\rangle .}"></span> </p><p>The coefficients <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a70b15e372decf709e90bc086b532b9bfb95add6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.097ex; height:2.843ex;" alt="{\displaystyle a_{n}(t)}"></span> are <a href="/wiki/Complex_number" title="Complex number">complex</a> variables. We can treat them as coordinates which specify the state of the system, like the position and momentum coordinates which specify a classical system. Like classical coordinates, they are generally not constant in time, and their time dependence gives rise to the time dependence of the system as a whole. </p><p>The expectation value of the Hamiltonian of this state, which is also the mean energy, is </p><p><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 H(t)\rangle \mathrel {\stackrel {\mathrm {def} }{=}} \langle \psi (t)|H|\psi (t)\rangle =\sum _{nn'}a_{n'}^{*}a_{n}\langle n'|H|n\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-REL"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <msup> <mi>n</mi> <mo>′</mo> </msup> </mrow> </munder> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>n</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle H(t)\rangle \mathrel {\stackrel {\mathrm {def} }{=}} \langle \psi (t)|H|\psi (t)\rangle =\sum _{nn'}a_{n'}^{*}a_{n}\langle n'|H|n\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ea125a2fb72183b3cba8979e90d363b1300fd5f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:44.468ex; height:6.343ex;" alt="{\displaystyle \langle H(t)\rangle \mathrel {\stackrel {\mathrm {def} }{=}} \langle \psi (t)|H|\psi (t)\rangle =\sum _{nn'}a_{n'}^{*}a_{n}\langle n'|H|n\rangle }"></span> </p><p>where the last step was obtained by expanding <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\psi \left(t\right)\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>ψ</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\psi \left(t\right)\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c56e2ffd124d44e11e8aba5e1f065eb023e52cbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.101ex; height:2.843ex;" alt="{\displaystyle \left|\psi \left(t\right)\right\rangle }"></span> in terms of the basis states. </p><p>Each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a70b15e372decf709e90bc086b532b9bfb95add6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.097ex; height:2.843ex;" alt="{\displaystyle a_{n}(t)}"></span> actually corresponds to <i>two</i> independent degrees of freedom, since the variable has a real part and an imaginary part. We now perform the following trick: instead of using the real and imaginary parts as the independent variables, we use <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a70b15e372decf709e90bc086b532b9bfb95add6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.097ex; height:2.843ex;" alt="{\displaystyle a_{n}(t)}"></span> and its <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</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 a_{n}^{*}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}^{*}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9056510733515007a63fe617fa9cba8a0bd27aa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.097ex; height:2.843ex;" alt="{\displaystyle a_{n}^{*}(t)}"></span>. With this choice of independent variables, we can calculate the <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivative</a> </p><p><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 {\frac {\partial \langle H\rangle }{\partial a_{n'}^{*}}}=\sum _{n}a_{n}\langle n'|H|n\rangle =\langle n'|H|\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo fence="false" stretchy="false">⟨</mo> <mi>H</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>n</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>n</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \langle H\rangle }{\partial a_{n'}^{*}}}=\sum _{n}a_{n}\langle n'|H|n\rangle =\langle n'|H|\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cd9ae85b159f6a10ecf062f29316120aac6a14c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.814ex; height:6.843ex;" alt="{\displaystyle {\frac {\partial \langle H\rangle }{\partial a_{n'}^{*}}}=\sum _{n}a_{n}\langle n'|H|n\rangle =\langle n'|H|\psi \rangle }"></span> </p><p>By applying <a href="/wiki/Schr%C3%B6dinger%27s_equation" class="mw-redirect" title="Schrödinger's equation">Schrödinger's equation</a> and using the orthonormality of the basis states, this further reduces to </p><p><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 {\frac {\partial \langle H\rangle }{\partial a_{n'}^{*}}}=i\hbar {\frac {\partial a_{n'}}{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo fence="false" stretchy="false">⟨</mo> <mi>H</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mo>′</mo> </msup> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \langle H\rangle }{\partial a_{n'}^{*}}}=i\hbar {\frac {\partial a_{n'}}{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f68b51d6e24175cc7e793bbab504e1f7fa215e30" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:16.369ex; height:6.676ex;" alt="{\displaystyle {\frac {\partial \langle H\rangle }{\partial a_{n'}^{*}}}=i\hbar {\frac {\partial a_{n'}}{\partial t}}}"></span> </p><p>Similarly, one can show that </p><p><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 {\frac {\partial \langle H\rangle }{\partial a_{n}}}=-i\hbar {\frac {\partial a_{n}^{*}}{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo fence="false" stretchy="false">⟨</mo> <mi>H</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \langle H\rangle }{\partial a_{n}}}=-i\hbar {\frac {\partial a_{n}^{*}}{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/283f7a98afabfafa44fa4129f7c778b3c36a2308" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.645ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial \langle H\rangle }{\partial a_{n}}}=-i\hbar {\frac {\partial a_{n}^{*}}{\partial t}}}"></span> </p><p>If we define "conjugate momentum" variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3144ebd68015a67d92ab797a63d232c65ead26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.544ex; height:2.009ex;" alt="{\displaystyle \pi _{n}}"></span> by </p><p><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 \pi _{n}(t)=i\hbar a_{n}^{*}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{n}(t)=i\hbar a_{n}^{*}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1199754809517f3d2a072422791ae967dcfc429f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.497ex; height:2.843ex;" alt="{\displaystyle \pi _{n}(t)=i\hbar a_{n}^{*}(t)}"></span> </p><p>then the above equations become </p><p><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 {\frac {\partial \langle H\rangle }{\partial \pi _{n}}}={\frac {\partial a_{n}}{\partial t}},\quad {\frac {\partial \langle H\rangle }{\partial a_{n}}}=-{\frac {\partial \pi _{n}}{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo fence="false" stretchy="false">⟨</mo> <mi>H</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo fence="false" stretchy="false">⟨</mo> <mi>H</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \langle H\rangle }{\partial \pi _{n}}}={\frac {\partial a_{n}}{\partial t}},\quad {\frac {\partial \langle H\rangle }{\partial a_{n}}}=-{\frac {\partial \pi _{n}}{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc0c5359abc0c8fbc36f7e40d5c9ef6d457235c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.716ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial \langle H\rangle }{\partial \pi _{n}}}={\frac {\partial a_{n}}{\partial t}},\quad {\frac {\partial \langle H\rangle }{\partial a_{n}}}=-{\frac {\partial \pi _{n}}{\partial t}}}"></span> </p><p>which is precisely the form of Hamilton's equations, with 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 a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span>s as the generalized coordinates, 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 \pi _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3144ebd68015a67d92ab797a63d232c65ead26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.544ex; height:2.009ex;" alt="{\displaystyle \pi _{n}}"></span>s as the conjugate momenta, 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 \langle H\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>H</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle H\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20097765034f46c439c5cc029e8971db1662e5bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.873ex; height:2.843ex;" alt="{\displaystyle \langle H\rangle }"></span> taking the place of the classical Hamiltonian. </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=Hamiltonian_(quantum_mechanics)&action=edit&section=20" 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: 20em;"> <ul><li><a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></li> <li><a href="/wiki/Two-state_quantum_system" title="Two-state quantum system">Two-state quantum system</a></li> <li><a href="/wiki/Operator_(physics)" title="Operator (physics)">Operator (physics)</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear algebra</a></li> <li><a href="/wiki/Conservation_of_energy" title="Conservation of energy">Conservation of energy</a></li> <li><a href="/wiki/Potential_theory" title="Potential theory">Potential theory</a></li> <li><a href="/wiki/Many-body_problem" title="Many-body problem">Many-body problem</a></li> <li><a href="/wiki/Electrostatics" title="Electrostatics">Electrostatics</a></li> <li><a href="/wiki/Electric_field" title="Electric field">Electric field</a></li> <li><a href="/wiki/Magnetic_field" title="Magnetic field">Magnetic field</a></li> <li><a href="/wiki/Lieb%E2%80%93Thirring_inequality" title="Lieb–Thirring inequality">Lieb–Thirring inequality</a></li></ul> </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=Hamiltonian_(quantum_mechanics)&action=edit&section=21" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-QuantumPhysics-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-QuantumPhysics_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-QuantumPhysics_1-1"><sup><i><b>b</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="CITEREFResnickEisberg1985" class="citation book cs1">Resnick, R.; Eisberg, R. (1985). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/quantumphysicsof00eisb"><i>Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles</i></a></span> (2nd ed.). John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-87373-X" title="Special:BookSources/0-471-87373-X"><bdi>0-471-87373-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Physics+of+Atoms%2C+Molecules%2C+Solids%2C+Nuclei+and+Particles&rft.edition=2nd&rft.pub=John+Wiley+%26+Sons&rft.date=1985&rft.isbn=0-471-87373-X&rft.aulast=Resnick&rft.aufirst=R.&rft.au=Eisberg%2C+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumphysicsof00eisb&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHamiltonian+%28quantum+mechanics%29" 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="CITEREFAtkins1974" class="citation book cs1">Atkins, P. W. (1974). <i>Quanta: A Handbook of Concepts</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-855493-1" title="Special:BookSources/0-19-855493-1"><bdi>0-19-855493-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quanta%3A+A+Handbook+of+Concepts&rft.pub=Oxford+University+Press&rft.date=1974&rft.isbn=0-19-855493-1&rft.aulast=Atkins&rft.aufirst=P.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHamiltonian+%28quantum+mechanics%29" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrantPhillips2008" class="citation book cs1">Grant, I. S.; Phillips, W. R. (2008). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/electromagnetism0000gran"><i>Electromagnetism</i></a></span>. Manchester Physics Series (2nd ed.). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-92712-9" title="Special:BookSources/978-0-471-92712-9"><bdi>978-0-471-92712-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electromagnetism&rft.series=Manchester+Physics+Series&rft.edition=2nd&rft.date=2008&rft.isbn=978-0-471-92712-9&rft.aulast=Grant&rft.aufirst=I.+S.&rft.au=Phillips%2C+W.+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felectromagnetism0000gran&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHamiltonian+%28quantum+mechanics%29" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBransdenJoachain1983" class="citation book cs1">Bransden, B. H.; Joachain, C. J. (1983). <i>Physics of Atoms and Molecules</i>. Longman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-582-44401-2" title="Special:BookSources/0-582-44401-2"><bdi>0-582-44401-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+of+Atoms+and+Molecules&rft.pub=Longman&rft.date=1983&rft.isbn=0-582-44401-2&rft.aulast=Bransden&rft.aufirst=B.+H.&rft.au=Joachain%2C+C.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHamiltonian+%28quantum+mechanics%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hamiltonian_(quantum_mechanics)&action=edit&section=22" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" 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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_mechanics" 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 class="mw-selflink selflink">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Formulations</a></li> <li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Equations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Weyl_equation" title="Weyl equation">Weyl</a></li> <li><a href="/wiki/Majorana_equation" title="Majorana equation">Majorana</a></li> <li><a href="/wiki/Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell test</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li> <li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed choice</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_nanoscience" class="mw-redirect" title="Quantum nanoscience">Science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_biology" title="Quantum biology">Quantum biology</a></li> <li><a href="/wiki/Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li> <li><a href="/wiki/Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li> <li><a href="/wiki/Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li> <li><a href="/wiki/Quantum_mind" title="Quantum mind">Quantum mind</a></li> <li><a href="/wiki/Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li> <li><a href="/wiki/Quantum_spacetime" title="Quantum spacetime">Quantum spacetime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li> <li><a href="/wiki/Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">Quantum bus</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a> <ul><li><a href="/wiki/Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li> <li><a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">Quantum electronics</a></li> <li><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li> <li><a href="/wiki/Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li> <li><a href="/wiki/Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li> <li><a href="/wiki/Quantum_machine" title="Quantum machine">Quantum machine</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li> <li><a href="/wiki/Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li> <li><a href="/wiki/Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">Quantum network</a></li> <li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li> <li><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Extensions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li> <li><a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a> <ul><li><a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></li></ul></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat_in_popular_culture" title="Schrödinger's cat in popular culture">in popular culture</a></li></ul></li> <li><a href="/wiki/Wigner%27s_friend" title="Wigner's friend">Wigner's friend</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Quantum_mysticism" title="Quantum mysticism">Quantum mysticism</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, 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Hamiltonian (quantum mechanics) - Wikipedia