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<li id="toc-Electronic_quantum_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electronic_quantum_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Electronic quantum numbers</span> </div> </a> <ul id="toc-Electronic_quantum_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nuclear_quantum_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nuclear_quantum_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Nuclear quantum numbers</span> </div> </a> <ul id="toc-Nuclear_quantum_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connection_to_symmetry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Connection_to_symmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Connection to symmetry</span> </div> </a> <ul id="toc-Connection_to_symmetry-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-General_properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#General_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>General properties</span> </div> </a> <ul id="toc-General_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electron_in_a_hydrogen-like_atom" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Electron_in_a_hydrogen-like_atom"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Electron in a hydrogen-like atom</span> </div> </a> <button aria-controls="toc-Electron_in_a_hydrogen-like_atom-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 Electron in a hydrogen-like atom subsection</span> </button> <ul id="toc-Electron_in_a_hydrogen-like_atom-sublist" class="vector-toc-list"> <li id="toc-Principal_quantum_number" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Principal_quantum_number"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Principal quantum number</span> </div> </a> <ul id="toc-Principal_quantum_number-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Azimuthal_quantum_number" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Azimuthal_quantum_number"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Azimuthal quantum number</span> </div> </a> <ul id="toc-Azimuthal_quantum_number-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Magnetic_quantum_number" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Magnetic_quantum_number"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Magnetic quantum number</span> </div> </a> <ul id="toc-Magnetic_quantum_number-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spin_magnetic_quantum_number" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spin_magnetic_quantum_number"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Spin magnetic quantum number</span> </div> </a> <ul id="toc-Spin_magnetic_quantum_number-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_Aufbau_principle_and_Hund's_Rules" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Aufbau_principle_and_Hund's_Rules"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>The Aufbau principle and Hund's Rules</span> </div> </a> <ul id="toc-The_Aufbau_principle_and_Hund's_Rules-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Spin-orbit_coupled_systems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Spin-orbit_coupled_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Spin-orbit coupled systems</span> </div> </a> <ul id="toc-Spin-orbit_coupled_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Atomic_nuclei" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Atomic_nuclei"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Atomic nuclei</span> </div> </a> <ul id="toc-Atomic_nuclei-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elementary_particles" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Elementary_particles"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Elementary particles</span> </div> </a> <ul id="toc-Elementary_particles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiplicative_quantum_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Multiplicative_quantum_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Multiplicative quantum numbers</span> </div> </a> <ul id="toc-Multiplicative_quantum_numbers-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-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Quantum number</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" 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Available in 56 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-56" 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">56 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D9%83%D9%85%D9%8A" title="عدد كمي – Arabic" lang="ar" hreflang="ar" data-title="عدد كمي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/N%C3%BAmberu_cu%C3%A1nticu" title="Númberu cuánticu – Asturian" lang="ast" hreflang="ast" data-title="Númberu cuánticu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8B%E0%A6%AF%E0%A6%BC%E0%A6%BE%E0%A6%A8%E0%A7%8D%E0%A6%9F%E0%A6%BE%E0%A6%AE_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="কোয়ান্টাম সংখ্যা – Bangla" lang="bn" hreflang="bn" data-title="কোয়ান্টাম সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%B0%D0%B2%D1%8B_%D0%BB%D1%96%D0%BA" title="Квантавы лік – Belarusian" lang="be" hreflang="be" data-title="Квантавы лік" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Квантово число – Bulgarian" lang="bg" hreflang="bg" data-title="Квантово число" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kvantni_broj" title="Kvantni broj – Bosnian" lang="bs" hreflang="bs" data-title="Kvantni broj" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_qu%C3%A0ntic" title="Nombre quàntic – Catalan" lang="ca" hreflang="ca" data-title="Nombre quàntic" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BB%D0%B0_%D1%85%D0%B8%D1%81%D0%B5%D0%BF" title="Квантла хисеп – Chuvash" lang="cv" hreflang="cv" data-title="Квантла хисеп" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kvantov%C3%A9_%C4%8D%C3%ADslo" title="Kvantové číslo – Czech" lang="cs" hreflang="cs" data-title="Kvantové číslo" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Quantenzahl" title="Quantenzahl – German" lang="de" hreflang="de" data-title="Quantenzahl" 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/Kvantarv" title="Kvantarv – Estonian" lang="et" hreflang="et" data-title="Kvantarv" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%B2%CE%B1%CE%BD%CF%84%CE%B9%CE%BA%CF%8C%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Κβαντικός αριθμός – Greek" lang="el" hreflang="el" data-title="Κβαντικός αριθμός" 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/N%C3%BAmero_cu%C3%A1ntico" title="Número cuántico – Spanish" lang="es" hreflang="es" data-title="Número cuántico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kvantuma_nombro" title="Kvantuma nombro – Esperanto" lang="eo" hreflang="eo" data-title="Kvantuma nombro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_kuantiko" title="Zenbaki kuantiko – Basque" lang="eu" hreflang="eu" data-title="Zenbaki kuantiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85%DB%8C" title="عدد کوانتومی – Persian" lang="fa" hreflang="fa" data-title="عدد 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/Nombre_quantique" title="Nombre quantique – French" lang="fr" hreflang="fr" data-title="Nombre quantique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Candamuimhir" title="Candamuimhir – Irish" lang="ga" hreflang="ga" data-title="Candamuimhir" 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/%EC%96%91%EC%9E%90%EC%88%98" title="양자수 – Korean" lang="ko" hreflang="ko" data-title="양자수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%94%D5%BE%D5%A1%D5%B6%D5%BF%D5%A1%D5%B5%D5%AB%D5%B6_%D5%A9%D5%BE%D5%A5%D6%80" title="Քվանտային թվեր – Armenian" lang="hy" hreflang="hy" data-title="Քվանտային թվեր" 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%95%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%A3%E0%A5%8D%E0%A4%9F%E0%A4%AE_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="क्वाण्टम संख्या – Hindi" lang="hi" hreflang="hi" data-title="क्वाण्टम संख्या" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Kvantni_brojevi" title="Kvantni brojevi – Croatian" lang="hr" hreflang="hr" data-title="Kvantni brojevi" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_kuantum" title="Bilangan kuantum – Indonesian" lang="id" hreflang="id" data-title="Bilangan kuantum" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_quantico" title="Numero quantico – Italian" lang="it" hreflang="it" data-title="Numero quantico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A7%D7%95%D7%95%D7%A0%D7%98%D7%99" title="מספר קוונטי – Hebrew" lang="he" hreflang="he" data-title="מספר קוונטי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D1%82%D1%8B%D0%BA_%D1%81%D0%B0%D0%BD%D0%B4%D0%B0%D1%80" title="Кванттык сандар – Kyrgyz" lang="ky" hreflang="ky" data-title="Кванттык сандар" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Kvantu_skait%C4%BCi" title="Kvantu skaitļi – Latvian" lang="lv" hreflang="lv" data-title="Kvantu skaitļi" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Kvantumsz%C3%A1m" title="Kvantumszám – Hungarian" lang="hu" hreflang="hu" data-title="Kvantumszám" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%B5%D0%BD_%D0%B1%D1%80%D0%BE%D1%98" title="Квантен број – Macedonian" lang="mk" hreflang="mk" data-title="Квантен број" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%95%E0%B5%8D%E0%B4%B5%E0%B4%BE%E0%B4%A3%E0%B5%8D%E0%B4%9F%E0%B4%82_%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF%E0%B4%95%E0%B5%BE" title="ക്വാണ്ടം സംഖ്യകൾ – Malayalam" lang="ml" hreflang="ml" data-title="ക്വാണ്ടം സംഖ്യകൾ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nombor_kuantum" title="Nombor kuantum – Malay" lang="ms" hreflang="ms" data-title="Nombor kuantum" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kwantumgetal" title="Kwantumgetal – Dutch" lang="nl" hreflang="nl" data-title="Kwantumgetal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%87%8F%E5%AD%90%E6%95%B0" title="量子数 – Japanese" lang="ja" hreflang="ja" data-title="量子数" 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/Kvantetall" title="Kvantetall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Kvantetall" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk 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lang="pt" hreflang="pt" data-title="Número quântico" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_cuantic" title="Număr cuantic – Romanian" lang="ro" hreflang="ro" data-title="Număr cuantic" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BE%D0%B2%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Квантовое число – Russian" lang="ru" hreflang="ru" data-title="Квантовое число" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Quantum_number" title="Quantum number – Simple English" lang="en-simple" hreflang="en-simple" data-title="Quantum number" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kvantov%C3%A9_%C4%8D%C3%ADslo" title="Kvantové číslo – Slovak" lang="sk" hreflang="sk" data-title="Kvantové číslo" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kvantno_%C5%A1tevilo" title="Kvantno število – Slovenian" lang="sl" hreflang="sl" data-title="Kvantno število" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D0%BD%D0%B8_%D0%B1%D1%80%D0%BE%D1%98%D0%B5%D0%B2%D0%B8" title="Квантни бројеви – Serbian" lang="sr" hreflang="sr" data-title="Квантни бројеви" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Kvantni_brojevi" title="Kvantni brojevi – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Kvantni brojevi" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kvanttiluku" 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Quantum_numbers&redirect=no" class="mw-redirect" title="Quantum numbers">Quantum numbers</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Notation for conserved quantities in physics and chemistry</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">"Q-number" redirects here. For the Q-theory concept, see <a href="/wiki/Q-analog" title="Q-analog">Q-analog</a>. For the number format, see <a href="/wiki/Q_(number_format)" title="Q (number format)">Q (number format)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Atomic_orbitals_n123_m-eigenstates.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Atomic_orbitals_n123_m-eigenstates.png/220px-Atomic_orbitals_n123_m-eigenstates.png" decoding="async" width="220" height="308" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Atomic_orbitals_n123_m-eigenstates.png/330px-Atomic_orbitals_n123_m-eigenstates.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Atomic_orbitals_n123_m-eigenstates.png/440px-Atomic_orbitals_n123_m-eigenstates.png 2x" data-file-width="2000" data-file-height="2800" /></a><figcaption>Single electron orbitals for hydrogen-like atoms with quantum numbers <span class="texhtml"><i>n</i> = 1, 2, 3</span> (blocks), <span class="texhtml mvar" style="font-style:italic;">ℓ</span> (rows) and <span class="texhtml mvar" style="font-style:italic;">m</span> (columns). The spin <span class="texhtml mvar" style="font-style:italic;">s</span> is not visible, because it has no spatial dependence.</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output 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.sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist nowraplinks" style="width:19.0em;"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1799e4a910c7d26396922a20ef5ceec25ca1871c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.882ex; height:5.509ex;" alt="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }"></span><div class="sidebar-caption" style="font-size:90%;padding-top:0.4em;font-style:italic;"><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a></div></td></tr><tr><td class="sidebar-above hlist nowrap" style="display:block;margin-bottom:0.4em;"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a></li></ul></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Background</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)">Complementarity</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a class="mw-selflink selflink">Quantum number</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">State</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li></ul></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Experiments</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell's inequality</a></li> <li><a href="/wiki/CHSH_inequality" title="CHSH inequality">CHSH inequality</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Leggett_inequality" title="Leggett inequality">Leggett inequality</a></li> <li><a href="/wiki/Leggett%E2%80%93Garg_inequality" title="Leggett–Garg inequality">Leggett–Garg inequality</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li></ul> </div> <ul><li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a> <ul><li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice</a></li></ul></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed-choice</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Formulations</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Overview</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase-space</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Sum-over-histories (path integral)</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective-collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Advanced topics</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Yakir_Aharonov" title="Yakir Aharonov">Aharonov</a></li> <li><a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">Bell</a></li> <li><a href="/wiki/Hans_Bethe" title="Hans Bethe">Bethe</a></li> <li><a href="/wiki/Patrick_Blackett" title="Patrick Blackett">Blackett</a></li> <li><a href="/wiki/Felix_Bloch" title="Felix Bloch">Bloch</a></li> <li><a href="/wiki/David_Bohm" title="David Bohm">Bohm</a></li> <li><a href="/wiki/Niels_Bohr" title="Niels Bohr">Bohr</a></li> <li><a href="/wiki/Max_Born" title="Max Born">Born</a></li> <li><a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Bose</a></li> <li><a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">de Broglie</a></li> <li><a href="/wiki/Arthur_Compton" title="Arthur Compton">Compton</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Clinton_Davisson" title="Clinton Davisson">Davisson</a></li> <li><a href="/wiki/Peter_Debye" title="Peter Debye">Debye</a></li> <li><a href="/wiki/Paul_Ehrenfest" title="Paul Ehrenfest">Ehrenfest</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Hugh_Everett_III" title="Hugh Everett III">Everett</a></li> <li><a href="/wiki/Vladimir_Fock" title="Vladimir Fock">Fock</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a></li> <li><a href="/wiki/Roy_J._Glauber" title="Roy J. Glauber">Glauber</a></li> <li><a href="/wiki/Martin_Gutzwiller" title="Martin Gutzwiller">Gutzwiller</a></li> <li><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a></li> <li><a href="/wiki/Hans_Kramers" title="Hans Kramers">Kramers</a></li> <li><a href="/wiki/Willis_Lamb" title="Willis Lamb">Lamb</a></li> <li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li> <li><a href="/wiki/Max_von_Laue" title="Max von Laue">Laue</a></li> <li><a href="/wiki/Henry_Moseley" title="Henry Moseley">Moseley</a></li> <li><a href="/wiki/Robert_Andrews_Millikan" title="Robert Andrews Millikan">Millikan</a></li> <li><a href="/wiki/Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Onnes</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/Isidor_Isaac_Rabi" title="Isidor Isaac Rabi">Rabi</a></li> <li><a href="/wiki/C._V._Raman" title="C. V. Raman">Raman</a></li> <li><a href="/wiki/Johannes_Rydberg" title="Johannes Rydberg">Rydberg</a></li> <li><a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödinger</a></li> <li><a href="/wiki/Michelle_Simmons" title="Michelle Simmons">Simmons</a></li> <li><a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Sommerfeld</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Wilhelm_Wien" title="Wilhelm Wien">Wien</a></li> <li><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner</a></li> <li><a href="/wiki/Pieter_Zeeman" title="Pieter Zeeman">Zeeman</a></li> <li><a href="/wiki/Anton_Zeilinger" title="Anton Zeilinger">Zeilinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar" style="border-top:1px solid #aaa;padding-top:0.1em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_mechanics" title="Template:Quantum mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics" title="Template talk:Quantum mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics" title="Special:EditPage/Template:Quantum mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum physics</a> and <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>, <b>quantum numbers</b> are quantities that characterize the possible states of the system. To fully specify the state of the electron in a hydrogen atom, four quantum numbers are needed. The traditional set of quantum numbers includes the <a href="/wiki/Principal_quantum_number" title="Principal quantum number">principal</a>, <a href="/wiki/Azimuthal_quantum_number" title="Azimuthal quantum number">azimuthal</a>, magnetic, and <a href="/wiki/Spin_quantum_number" title="Spin quantum number">spin</a> quantum numbers. To describe other systems, different quantum numbers are required. For subatomic particles, one needs to introduce new quantum numbers, such as the <a href="/wiki/Flavour_(particle_physics)" title="Flavour (particle physics)">flavour</a> of <a href="/wiki/Quarks" class="mw-redirect" title="Quarks">quarks</a>, which have no classical correspondence. </p><p>Quantum numbers are closely related to <a href="/wiki/Eigenvalues" class="mw-redirect" title="Eigenvalues">eigenvalues</a> of <a href="/wiki/Observable" title="Observable">observables</a>. When the corresponding observable commutes with the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> of the system, the quantum number is said to be "<a href="/wiki/Good_quantum_number" title="Good quantum number">good</a>", and acts as a <a href="/wiki/Constant_of_motion" title="Constant of motion">constant of motion</a> in the quantum dynamics. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_number&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History of quantum mechanics</a></div> <div class="mw-heading mw-heading3"><h3 id="Electronic_quantum_numbers">Electronic quantum numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_number&action=edit&section=2" title="Edit section: Electronic quantum numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the era of the <a href="/wiki/Old_quantum_theory" title="Old quantum theory">old quantum theory</a>, starting from <a href="/wiki/Max_Planck" title="Max Planck">Max Planck</a>'s proposal of quanta in his model of <a href="/wiki/Blackbody_radiation" class="mw-redirect" title="Blackbody radiation">blackbody radiation</a> (1900) and <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>'s adaptation of the concept to explain the <a href="/wiki/Photoelectric_effect" title="Photoelectric effect">photoelectric effect</a> (1905), and until <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a> published his eigenfunction equation in 1926,<sup id="cite_ref-schrodinger_1-0" class="reference"><a href="#cite_note-schrodinger-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> the concept behind quantum numbers developed based on atomic spectroscopy and theories from classical mechanics with extra ad hoc constraints.<sup id="cite_ref-Whittaker_2-0" class="reference"><a href="#cite_note-Whittaker-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 106">: 106 </span></sup> Many results from atomic spectroscopy had been summarized in the <a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg formula</a> involving differences between two series of energies related by integer steps. The <a href="/wiki/Bohr_atom" class="mw-redirect" title="Bohr atom">model of the atom</a>, first proposed by <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a> in 1913, relied on a single quantum number. Together with Bohr's constraint that radiation absorption is not classical, it was able to explain the <a href="/wiki/Balmer_series" title="Balmer series">Balmer series</a> portion of Rydberg's atomic spectrum formula.<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>As Bohr notes in his subsequent Nobel lecture, the next step was taken by <a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Arnold Sommerfeld</a> in 1915.<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> Sommerfeld's atomic model added a second quantum number and the concept of quantized phase integrals to justify them.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 207">: 207 </span></sup> Sommerfeld's model was still essentially two dimensional, modeling the electron as orbiting in a plane; in 1919 he extended his work to three dimensions using 'space quantization' in place of the quantized phase integrals.<sup id="cite_ref-Kragh2012Bohr_6-0" class="reference"><a href="#cite_note-Kragh2012Bohr-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 152">: 152 </span></sup> <a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Karl Schwarzschild</a> and Sommerfeld's student, <a href="/wiki/Paul_Epstein" title="Paul Epstein">Paul Epstein</a>, independently showed that adding third quantum number gave a complete account for the <a href="/wiki/Stark_effect" title="Stark effect">Stark effect</a> results. </p><p>A consequence of space quantization was that the electron's orbital interaction with an external magnetic field would be quantized. This seemed to be confirmed when the results of the <a href="/wiki/Stern-Gerlach" class="mw-redirect" title="Stern-Gerlach">Stern-Gerlach</a> experiment reported quantized results for silver atoms in an inhomogeneous magnetic field. The confirmation would turn out to be premature: more quantum numbers would be needed.<sup id="cite_ref-FriedrichHerschbach_7-0" class="reference"><a href="#cite_note-FriedrichHerschbach-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The fourth and fifth quantum numbers of the atomic era arose from attempts to understand the <a href="/wiki/Zeeman_effect" title="Zeeman effect">Zeeman effect</a>. Like the Stern-Gerlach experiment, the Zeeman effect reflects the interaction of atoms with a magnetic field; in a weak field the experimental results were called "anomalous", they diverged from any theory at the time. <a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Wolfgang Pauli</a>'s solution to this issue was to introduce another quantum number taking only two possible values, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \hbar /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm \hbar /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1872f04411a1e6d0f1c2970ea0308d026ad78cce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.44ex; height:2.843ex;" alt="{\displaystyle \pm \hbar /2}"></span>.<sup id="cite_ref-Giulini_8-0" class="reference"><a href="#cite_note-Giulini-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> This would ultimately become the quantized values of the projection of <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a>, an intrinsic angular momentum quantum of the electron. In 1927 Ronald Fraser demonstrated that the quantization in the Stern-Gerlach experiment was due to the magnetic moment associated with the electron spin rather than its orbital angular momentum.<sup id="cite_ref-FriedrichHerschbach_7-1" class="reference"><a href="#cite_note-FriedrichHerschbach-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Pauli's success in developing the arguments for a spin quantum number without relying on classical models set the stage for the development of quantum numbers for elementary particles in the remainder of the 20th century.<sup id="cite_ref-Giulini_8-1" class="reference"><a href="#cite_note-Giulini-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Bohr, with his <a href="/wiki/Aufbau_principle" title="Aufbau principle">Aufbau</a> or "building up" principle, and Pauli with his <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">exclusion principle</a> connected the atom's electronic quantum numbers in to a framework for predicting the properties of atoms.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> When Schrödinger published his <a href="/wiki/Schrodinger_equation" class="mw-redirect" title="Schrodinger equation">wave equation</a> and calculated the energy levels of hydrogen, these two principles carried over to become the basis of atomic physics. </p> <div class="mw-heading mw-heading3"><h3 id="Nuclear_quantum_numbers">Nuclear quantum numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_number&action=edit&section=3" title="Edit section: Nuclear quantum numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>With successful models of the atom, the attention of physics turned to models of the nucleus. Beginning with Heisenberg's initial model of proton-neutron binding in 1932, <a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Eugene Wigner</a> introduced <a href="/wiki/Isospin" title="Isospin">isospin</a> in 1937, the first 'internal' quantum number unrelated to a symmetry in real space-time.<sup id="cite_ref-Brown1987_10-0" class="reference"><a href="#cite_note-Brown1987-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 45">: 45 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Connection_to_symmetry">Connection to symmetry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_number&action=edit&section=4" title="Edit section: Connection to symmetry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As quantum mechanics developed, abstraction increased and models based on symmetry and invariance played increasing roles. Two years before his work on the quantum wave equation, Schrödinger applied the symmetry ideas originated by <a href="/wiki/Emmy_Noether" title="Emmy Noether">Emmy Noether</a> and <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> to the electromagnetic field.<sup id="cite_ref-Baggott40_11-0" class="reference"><a href="#cite_note-Baggott40-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 198">: 198 </span></sup> As <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a> developed in the 1930s and 1940s, <a href="/wiki/Group_theory" title="Group theory">group theory</a> became an important tool. By 1953 <a href="/wiki/Chen_Ning_Yang" class="mw-redirect" title="Chen Ning Yang">Chen Ning Yang</a> had become obsessed with the idea that group theory could be applied to connect the conserved quantum numbers of nuclear collisions to symmetries in a field theory of nucleons.<sup id="cite_ref-Baggott40_11-1" class="reference"><a href="#cite_note-Baggott40-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 202">: 202 </span></sup> With <a href="/wiki/Robert_Mills_(physicist)" title="Robert Mills (physicist)">Robert Mills</a>, Yang developed a <a href="/wiki/Non-abelian_gauge_theory" class="mw-redirect" title="Non-abelian gauge theory">non-abelian gauge theory</a> based on the conservation of the nuclear <a href="/wiki/Isospin" title="Isospin">isospin</a> quantum numbers. </p> <div class="mw-heading mw-heading2"><h2 id="General_properties">General properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_number&action=edit&section=5" title="Edit section: General properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Good quantum numbers correspond to <a href="/wiki/Eigenvalues" class="mw-redirect" title="Eigenvalues">eigenvalues</a> of <a href="/wiki/Operator_(quantum_mechanics)" class="mw-redirect" title="Operator (quantum mechanics)">operators</a> that commute with the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>, quantities that can be known with precision at the same time as the system's energy. Specifically, observables that <a href="/wiki/Commutator" title="Commutator">commute</a> with the Hamiltonian are <a href="/wiki/Simultaneously_diagonalizable" class="mw-redirect" title="Simultaneously diagonalizable">simultaneously diagonalizable</a> with it and so the eigenvalues <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> and the energy (eigenvalues of the Hamiltonian) are not limited by an <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty relation</a> arising from non-commutativity. Together, a specification of all of the quantum numbers of a quantum system fully characterize a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> state of the system, and can in principle be <a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">measured</a> together. Many observables have discrete <a href="/wiki/Spectrum_of_an_operator" class="mw-redirect" title="Spectrum of an operator">spectra (sets of eigenvalues)</a> in quantum mechanics, so the quantities can only be measured in discrete values. In particular, this leads to quantum numbers that take values in <a href="/wiki/Discrete_mathematics" title="Discrete mathematics">discrete sets of integers</a> or <a href="/wiki/Half-integers" class="mw-redirect" title="Half-integers">half-integers</a>; although they could approach <a href="/wiki/Infinity" title="Infinity">infinity</a> in some cases. </p><p>The tally of quantum numbers varies from system to system and has no universal answer. Hence these parameters must be found for each system to be analyzed. A quantized system requires at least one quantum number. The dynamics (i.e. time evolution) of any quantum system are described by a <a href="/wiki/Quantum_operator" class="mw-redirect" title="Quantum operator">quantum operator</a> in the form of a <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>, <span class="texhtml mvar" style="font-style:italic;">H</span>. There is one quantum number of the system corresponding to the system's energy; i.e., one of the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of the Hamiltonian. There is also one quantum number for each <a href="/wiki/Linear_independence" title="Linear independence">linearly independent</a> operator <span class="texhtml mvar" style="font-style:italic;">O</span> that <a href="/wiki/Commutivity" class="mw-redirect" title="Commutivity">commutes</a> with the Hamiltonian. A <a href="/wiki/Complete_set_of_commuting_observables" title="Complete set of commuting observables">complete set of commuting observables</a> (CSCO) that commute with the Hamiltonian characterizes the system with all its quantum numbers. There is a one-to-one relationship between the quantum numbers and the operators of the CSCO, with each quantum number taking one of the eigenvalues of its corresponding operator. As a result of the different <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> that may be arbitrarily chosen to form a complete set of commuting operators, different sets of quantum numbers may be used for the description of the same system in different situations. </p> <div class="mw-heading mw-heading2"><h2 id="Electron_in_a_hydrogen-like_atom">Electron in a hydrogen-like atom</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_number&action=edit&section=6" title="Edit section: Electron in a hydrogen-like atom"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Four quantum numbers can describe an electron energy level in a <a href="/wiki/Hydrogen-like_atom" title="Hydrogen-like atom">hydrogen-like atom</a> completely: </p> <ul><li><a href="/wiki/Principal_quantum_number" title="Principal quantum number">Principal quantum number</a> (<span class="texhtml mvar" style="font-style:italic;">n</span>)</li> <li><a href="/wiki/Azimuthal_quantum_number" title="Azimuthal quantum number">Azimuthal quantum number</a> (<span class="texhtml mvar" style="font-style:italic;">ℓ</span>)</li> <li><a href="/wiki/Magnetic_quantum_number" title="Magnetic quantum number">Magnetic quantum number</a> (<span class="texhtml mvar" style="font-style:italic;">m<sub>ℓ</sub></span>)</li> <li><a href="/wiki/Spin_quantum_number" title="Spin quantum number">Spin quantum number</a> (<span class="texhtml mvar" style="font-style:italic;">m<sub>s</sub></span>)</li></ul> <p>These quantum numbers are also used in the classical description of nuclear particle states (e.g. protons and neutrons).<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2024)">citation needed</span></a></i>]</sup> A quantum description of <a href="/wiki/Molecular_orbitals" class="mw-redirect" title="Molecular orbitals">molecular orbitals</a> requires other quantum numbers, because the symmetries of the molecular system are different. </p> <div class="mw-heading mw-heading3"><h3 id="Principal_quantum_number">Principal quantum number</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_number&action=edit&section=7" title="Edit section: Principal quantum number"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The principal quantum number describes the <a href="/wiki/Electron_shell" title="Electron shell">electron shell</a> of an electron. The value of <span class="texhtml mvar" style="font-style:italic;">n</span> ranges from 1 to the shell containing the outermost electron of that atom, that is<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1,2,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1,2,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6fe6c942f7596c0d3ade1a83faab2e290d0927" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.609ex; height:2.509ex;" alt="{\displaystyle n=1,2,\ldots }"></span> </p><p>For example, in <a href="/wiki/Caesium" title="Caesium">caesium</a> (Cs), the outermost <a href="/wiki/Valence_(chemistry)" title="Valence (chemistry)">valence</a> electron is in the shell with energy level 6, so an electron in caesium can have an <span class="texhtml mvar" style="font-style:italic;">n</span> value from 1 to 6. The average distance between the electron and the nucleus increases with <span class="texhtml mvar" style="font-style:italic;">n</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Azimuthal_quantum_number">Azimuthal quantum number</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_number&action=edit&section=8" title="Edit section: Azimuthal quantum number"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The azimuthal quantum number, also known as the <i>orbital angular momentum quantum number</i>, describes the <a href="/wiki/Electron_shell#Subshells" title="Electron shell">subshell</a>, and gives the magnitude of the orbital <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> through the relation </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 L^{2}=\hbar ^{2}\ell (\ell +1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ℓ</mi> <mo stretchy="false">(</mo> <mi>ℓ</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}=\hbar ^{2}\ell (\ell +1).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24c879dc6bcfe68c45e8d58807c61f1243f33b99" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.51ex; height:3.176ex;" alt="{\displaystyle L^{2}=\hbar ^{2}\ell (\ell +1).}"></span> </p><p>In chemistry and spectroscopy, <span class="texhtml"><i>ℓ</i> = 0</span> is called s orbital, <span class="texhtml"><i>ℓ</i> = 1</span>, p orbital, <span class="texhtml"><i>ℓ</i> = 2</span>, d orbital, and <span class="texhtml"><i>ℓ</i> = 3</span>, f orbital. </p><p>The value of <span class="texhtml mvar" style="font-style:italic;">ℓ</span> ranges from 0 to <span class="texhtml"><i>n</i> − 1</span>, so the first p orbital (<span class="texhtml"><i>ℓ</i> = 1</span>) appears in the second electron shell (<span class="texhtml"><i>n</i> = 2</span>), the first d orbital (<span class="texhtml"><i>ℓ</i> = 2</span>) appears in the third shell (<span class="texhtml"><i>n</i> = 3</span>), and so on:<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p><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 \ell =0,1,2,\ldots ,n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell =0,1,2,\ldots ,n-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5fab5295ff42f6b6e4c4b9a5c17506475fd2ba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.199ex; height:2.509ex;" alt="{\displaystyle \ell =0,1,2,\ldots ,n-1}"></span> </p><p>A quantum number beginning in <span class="texhtml"><i>n</i> = 3,<i>ℓ</i> = 0</span>, describes an electron in the s orbital of the third electron shell of an atom. In chemistry, this quantum number is very important, since it specifies the shape of an <a href="/wiki/Atomic_orbital" title="Atomic orbital">atomic orbital</a> and strongly influences <a href="/wiki/Chemical_bond" title="Chemical bond">chemical bonds</a> and <a href="/wiki/Bond_angle" class="mw-redirect" title="Bond angle">bond angles</a>. The azimuthal quantum number can also denote the number of angular nodes present in an orbital. For example, for p orbitals, <span class="texhtml"><i>ℓ</i> = 1</span> and thus the amount of angular nodes in a p orbital is 1. </p> <div class="mw-heading mw-heading3"><h3 id="Magnetic_quantum_number">Magnetic quantum number</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_number&action=edit&section=9" title="Edit section: Magnetic quantum number"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Magnetic_quantum_number" title="Magnetic quantum number">magnetic quantum number</a> describes the specific <a href="/wiki/Atomic_orbital" title="Atomic orbital">orbital</a> within the subshell, and yields the <i>projection</i> of the orbital <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> <i>along a specified axis</i>: </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 L_{z}=m_{\ell }\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{z}=m_{\ell }\hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc11894b68e9b681459955536d924c9b42fd5568" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.948ex; height:2.509ex;" alt="{\displaystyle L_{z}=m_{\ell }\hbar }"></span> </p><p>The values of <span class="texhtml mvar" style="font-style:italic;">m<sub>ℓ</sub></span> range from <span class="texhtml">−<i>ℓ</i></span> to <span class="texhtml mvar" style="font-style:italic;">ℓ</span>, with integer intervals.<sup id="cite_ref-FOOTNOTEEisbergResnick1985_14-0" class="reference"><a href="#cite_note-FOOTNOTEEisbergResnick1985-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (February 2019)">page needed</span></a></i>]</sup> </p><p>The s subshell (<span class="texhtml"><i>ℓ</i> = 0</span>) contains only one orbital, and therefore the <span class="texhtml">m<sub>ℓ</sub></span> of an electron in an s orbital will always be 0. The p subshell (<span class="texhtml"><i>ℓ</i> = 1</span>) contains three orbitals, so the <span class="texhtml mvar" style="font-style:italic;">m<sub>ℓ</sub></span> of an electron in a p orbital will be −1, 0, or 1. The d subshell (<span class="texhtml"><i>ℓ</i> = 2</span>) contains five orbitals, with <span class="texhtml mvar" style="font-style:italic;">m<sub>ℓ</sub></span> values of −2, −1, 0, 1, and 2. </p> <div class="mw-heading mw-heading3"><h3 id="Spin_magnetic_quantum_number">Spin magnetic quantum number</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_number&action=edit&section=10" title="Edit section: Spin magnetic quantum number"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Spin_magnetic_quantum_number" class="mw-redirect" title="Spin magnetic quantum number">spin magnetic quantum number</a> describes the intrinsic <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin angular momentum</a> of the electron within each orbital and gives the projection of the spin angular momentum <span class="texhtml mvar" style="font-style:italic;">S</span> along the specified axis: </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 S_{z}=m_{s}\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{z}=m_{s}\hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/632f17003411ed93a47a81fa17742580363505dc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.875ex; height:2.509ex;" alt="{\displaystyle S_{z}=m_{s}\hbar }"></span> </p><p>In general, the values of <span class="texhtml mvar" style="font-style:italic;">m<sub>s</sub></span> range from <span class="texhtml">−<i>s</i></span> to <span class="texhtml mvar" style="font-style:italic;">s</span>, where <span class="texhtml mvar" style="font-style:italic;">s</span> is the spin quantum number, associated with the magnitude of particle's intrinsic spin angular momentum:<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<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 m_{s}=-s,-s+1,-s+2,\cdots ,s-2,s-1,s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>s</mi> <mo>,</mo> <mo>−</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mi>s</mi> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>s</mi> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mi>s</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{s}=-s,-s+1,-s+2,\cdots ,s-2,s-1,s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/086812aa4daeafd0ee35356f0769114f995b8c57" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:43.435ex; height:2.509ex;" alt="{\displaystyle m_{s}=-s,-s+1,-s+2,\cdots ,s-2,s-1,s}"></span> </p><p>An electron state has spin number <span class="texhtml"><i>s</i> = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>, consequently <span class="texhtml mvar" style="font-style:italic;">m<sub>s</sub></span> will be +<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> ("spin up") or -<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> "spin down" states. Since electron are <a href="/wiki/Fermions" class="mw-redirect" title="Fermions">fermions</a> they obey the <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a>: each electron state must have different quantum numbers. Therefore, every orbital will be occupied with at most two electrons, one for each spin state. </p> <div class="mw-heading mw-heading3"><h3 id="The_Aufbau_principle_and_Hund's_Rules"><span id="The_Aufbau_principle_and_Hund.27s_Rules"></span>The Aufbau principle and Hund's Rules</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_number&action=edit&section=11" title="Edit section: The Aufbau principle and Hund's Rules"><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 articles: <a href="/wiki/Aufbau_principle" title="Aufbau principle">Aufbau principle</a> and <a href="/wiki/Hund%27s_rules" title="Hund's rules">Hund's rules</a></div> <p>A multi-electron atom can be modeled qualitatively as a hydrogen like atom with higher nuclear charge and correspondingly more electrons. The occupation of the electron states in such an atom can be predicted by the Aufbau principle and Hund's empirical rules for the quantum numbers. The Aufbau principle fills orbitals based on their principal and azimuthal quantum numbers (lowest <span class="texhtml"><i>n</i> + l</span> first, with lowest <span class="texhtml mvar" style="font-style:italic;">n</span> breaking ties; Hund's rule favors unpaired electrons in the outermost orbital). These rules are empirical but they can be related to electron physics.<sup id="cite_ref-Jolly_16-0" class="reference"><a href="#cite_note-Jolly-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 10">: 10 </span></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 260">: 260 </span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Spin-orbit_coupled_systems">Spin-orbit coupled systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_number&action=edit&section=12" title="Edit section: Spin-orbit coupled systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When one takes the <a href="/wiki/Spin%E2%80%93orbit_interaction" title="Spin–orbit interaction">spin–orbit interaction</a> into consideration, the <span class="texhtml mvar" style="font-style:italic;">L</span> and <span class="texhtml mvar" style="font-style:italic;">S</span> operators no longer <a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">commute</a> with the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>, and the eigenstates of the system no longer have well-defined orbital angular momentum and spin. Thus another set of quantum numbers should be used. This set includes<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Atkins_1977_19-0" class="reference"><a href="#cite_note-Atkins_1977-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <ol><li>The <a href="/wiki/Total_angular_momentum_quantum_number" title="Total angular momentum quantum number">total angular momentum quantum number</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=|\ell \pm s|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ℓ</mi> <mo>±</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=|\ell \pm s|,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43c69ce77c1bb8c849dfe2c7ea7533b32056d92a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:10.924ex; height:2.843ex;" alt="{\displaystyle j=|\ell \pm s|,}"></span> which gives the total angular momentum through the relation <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 J^{2}=\hbar ^{2}j(j+1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>j</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J^{2}=\hbar ^{2}j(j+1).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82126cef0a71d61356380137fa57562bc288ec50" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.43ex; height:3.176ex;" alt="{\displaystyle J^{2}=\hbar ^{2}j(j+1).}"></span></li> <li>The <a href="/wiki/Azimuthal_quantum_number#Total_angular_momentum_of_an_electron_in_the_atom" title="Azimuthal quantum number">projection of the total angular momentum</a> along a specified axis:<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 m_{j}=-j,-j+1,-j+2,\cdots ,j-2,j-1,j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>j</mi> <mo>,</mo> <mo>−</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mi>j</mi> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>j</mi> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{j}=-j,-j+1,-j+2,\cdots ,j-2,j-1,j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2059cab6e8d9318290d2db56c4f7fa83dd93d7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.547ex; height:2.843ex;" alt="{\displaystyle m_{j}=-j,-j+1,-j+2,\cdots ,j-2,j-1,j}"></span> analogous to the above and satisfies both <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 m_{j}=m_{\ell }+m_{s},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{j}=m_{\ell }+m_{s},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b5fd940009821c1b9495965b012ef7713281ea4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.538ex; height:2.676ex;" alt="{\displaystyle m_{j}=m_{\ell }+m_{s},}"></span> 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 |m_{\ell }+m_{s}|\leq j.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>j</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |m_{\ell }+m_{s}|\leq j.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ef7fe67dfc161d3345d9782a1d8ca4cabb87ecc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.839ex; height:2.843ex;" alt="{\displaystyle |m_{\ell }+m_{s}|\leq j.}"></span></li> <li><a href="/wiki/Parity_(physics)" title="Parity (physics)">Parity</a><br />This is the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> under reflection: positive (+1) for states which came from even <span class="texhtml mvar" style="font-style:italic;">ℓ</span> and negative (−1) for states which came from odd <span class="texhtml mvar" style="font-style:italic;">ℓ</span>. The former is also known as <b>even parity</b> and the latter as <b>odd parity</b>, and is given by<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=(-1)^{\ell }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=(-1)^{\ell }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4cd0d6891223948a97d4732bebe783e7f3c4f2b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.189ex; height:3.176ex;" alt="{\displaystyle P=(-1)^{\ell }.}"></span></li></ol> <p>For example, consider the following 8 states, defined by their quantum numbers: </p> <dl><dd><table style="border: none; border-spacing: 1em 0" class="wikitable"> <tbody><tr> <th> </th> <th><span class="texhtml mvar" style="font-style:italic;">n</span> </th> <th><span class="texhtml mvar" style="font-style:italic;">ℓ</span> </th> <th><span class="texhtml mvar" style="font-style:italic;">m<sub>ℓ</sub></span> </th> <th><span class="texhtml mvar" style="font-style:italic;">m<sub>s</sub></span> </th> <td rowspan="9" style="border:0px;"> </td> <th><span class="texhtml"><i>ℓ</i> + <i>s</i></span> </th> <th><span class="texhtml"><i>ℓ</i> − <i>s</i></span> </th> <th><span class="texhtml"><i>m<sub>ℓ</sub></i> + <i>m<sub>s</sub></i></span> </th></tr> <tr align="right"> <th>(1) </th> <td>2</td> <td>1</td> <td>1</td> <td>+<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td><s><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></s></td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </td></tr> <tr align="right"> <th>(2) </th> <td>2</td> <td>1</td> <td>1</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </td></tr> <tr align="right"> <th>(3) </th> <td>2</td> <td>1</td> <td>0</td> <td>+<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </td></tr> <tr align="right"> <th>(4) </th> <td>2</td> <td>1</td> <td>0</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </td></tr> <tr align="right"> <th>(5) </th> <td>2</td> <td>1</td> <td>−1</td> <td>+<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </td></tr> <tr align="right"> <th>(6) </th> <td>2</td> <td>1</td> <td>−1</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td><s><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></s></td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </td></tr> <tr align="right"> <th>(7) </th> <td>2</td> <td>0</td> <td>0</td> <td>+<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </td></tr> <tr align="right"> <th>(8) </th> <td>2</td> <td>0</td> <td>0</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> </td></tr></tbody></table></dd></dl> <p>The <a href="/wiki/Quantum_state" title="Quantum state">quantum states</a> in the system can be described as linear combination of these 8 states. However, in the presence of <a href="/wiki/Spin%E2%80%93orbit_interaction" title="Spin–orbit interaction">spin–orbit interaction</a>, if one wants to describe the same system by 8 states that are <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvectors</a> of the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> (i.e. each represents a state that does not mix with others over time), we should consider the following 8 states: </p> <dl><dd><table class="wikitable"> <tbody><tr> <th><span class="texhtml"><i>j</i></span></th> <th><span class="texhtml"><i>m<sub>j</sub></i></span></th> <th>parity</th> <th> </th></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td align="right"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td align="right">odd</td> <td>coming from state (1) above </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td align="right"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td align="right">odd</td> <td>coming from states (2) and (3) above </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td align="right">−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td align="right">odd</td> <td>coming from states (4) and (5) above </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td align="right">−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td align="right">odd</td> <td>coming from state (6) above </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td align="right"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td align="right">odd</td> <td>coming from states (2) and (3) above </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td align="right">−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td align="right">odd</td> <td>coming from states (4) and (5) above </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td align="right"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td align="right">even</td> <td>coming from state (7) above </td></tr> <tr> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td align="right">−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></td> <td align="right">even</td> <td>coming from state (8) above </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Atomic_nuclei">Atomic nuclei</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_number&action=edit&section=13" title="Edit section: Atomic nuclei"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Atomic_nucleus" title="Atomic nucleus">nuclei</a>, the entire assembly of <a href="/wiki/Proton" title="Proton">protons</a> and <a href="/wiki/Neutron" title="Neutron">neutrons</a> (<a href="/wiki/Nucleon" title="Nucleon">nucleons</a>) has a resultant <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> due to the angular momenta of each nucleon, usually denoted <span class="texhtml mvar" style="font-style:italic;">I</span>. If the total angular momentum of a neutron is <span class="texhtml"><i>j</i><sub>n</sub> = <i>ℓ</i> + <i>s</i></span> and for a proton is <span class="texhtml"><i>j</i><sub>p</sub> = <i>ℓ</i> + <i>s</i></span> (where <span class="texhtml mvar" style="font-style:italic;">s</span> for protons and neutrons happens to be <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> again (<i>see note</i>)), then the <b>nuclear angular momentum quantum numbers</b> <span class="texhtml mvar" style="font-style:italic;">I</span> are 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 I=|j_{n}-j_{p}|,|j_{n}-j_{p}|+1,|j_{n}-j_{p}|+2,\cdots ,(j_{n}+j_{p})-2,(j_{n}+j_{p})-1,(j_{n}+j_{p})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=|j_{n}-j_{p}|,|j_{n}-j_{p}|+1,|j_{n}-j_{p}|+2,\cdots ,(j_{n}+j_{p})-2,(j_{n}+j_{p})-1,(j_{n}+j_{p})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e236f978bcea857940b12804225cf3443f95e19e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:81.11ex; height:3.009ex;" alt="{\displaystyle I=|j_{n}-j_{p}|,|j_{n}-j_{p}|+1,|j_{n}-j_{p}|+2,\cdots ,(j_{n}+j_{p})-2,(j_{n}+j_{p})-1,(j_{n}+j_{p})}"></span> </p><p><i>Note: </i>The orbital angular momenta of the nuclear (and atomic) states are all integer multiples of ħ while the intrinsic angular momentum of the neutron and proton are half-integer multiples. It should be immediately apparent that the combination of the intrinsic spins of the nucleons with their orbital motion will always give half-integer values for the total spin, <span class="texhtml mvar" style="font-style:italic;">I</span>, of any odd-A nucleus and integer values for any even-A nucleus. </p><p>Parity with the number <span class="texhtml mvar" style="font-style:italic;">I</span> is used to label nuclear angular momentum states, examples for some isotopes of <a href="/wiki/Hydrogen" title="Hydrogen">hydrogen</a> (H), <a href="/wiki/Carbon" title="Carbon">carbon</a> (C), and <a href="/wiki/Sodium" title="Sodium">sodium</a> (Na) are;<sup id="cite_ref-Krane_1988_20-0" class="reference"><a href="#cite_note-Krane_1988-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><table> <tbody><tr> <td style="text-align:right;"><span style="white-space:nowrap;"><a href="/wiki/Hydrogen-1" class="mw-redirect" title="Hydrogen-1"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>H<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></a></span></td> <td><span class="texhtml mvar" style="font-style:italic;">I</span> = (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>)<sup>+</sup></td> <td> </td> <td style="text-align:right;"><span style="white-space:nowrap;"><a href="/wiki/Carbon-9" class="mw-redirect" title="Carbon-9"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">9</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">6</sub></span></span>C<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></a></span></td> <td><span class="texhtml mvar" style="font-style:italic;">I</span> = (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>)<sup>−</sup></td> <td> </td> <td style="text-align:right;"><span style="white-space:nowrap;"><a href="/wiki/Sodium-20" class="mw-redirect" title="Sodium-20"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">20</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">11</sub></span></span>Na<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></a></span></td> <td><span class="texhtml mvar" style="font-style:italic;">I</span> = 2<sup>+</sup> </td></tr> <tr> <td style="text-align:right;"><span style="white-space:nowrap;"><a href="/wiki/Hydrogen-2" class="mw-redirect" title="Hydrogen-2"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>H<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></a></span></td> <td><span class="texhtml mvar" style="font-style:italic;">I</span> = 1<sup>+</sup></td> <td> </td> <td style="text-align:right;"><span style="white-space:nowrap;"><a href="/wiki/Carbon-10" class="mw-redirect" title="Carbon-10"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">10</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">6</sub></span></span>C<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></a></span></td> <td><span class="texhtml mvar" style="font-style:italic;">I</span> = 0<sup>+</sup></td> <td> </td> <td style="text-align:right;"><span style="white-space:nowrap;"><a href="/wiki/Sodium-21" class="mw-redirect" title="Sodium-21"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">21</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">11</sub></span></span>Na<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></a></span></td> <td><span class="texhtml mvar" style="font-style:italic;">I</span> = (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>)<sup>+</sup> </td></tr> <tr> <td style="text-align:right;"><span style="white-space:nowrap;"><a href="/wiki/Hydrogen-3" class="mw-redirect" title="Hydrogen-3"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">3</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">1</sub></span></span>H<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></a></span></td> <td><span class="texhtml mvar" style="font-style:italic;">I</span> = (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>)<sup>+</sup></td> <td> </td> <td style="text-align:right;"><span style="white-space:nowrap;"><a href="/wiki/Carbon-11" class="mw-redirect" title="Carbon-11"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">11</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">6</sub></span></span>C<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></a></span></td> <td><span class="texhtml mvar" style="font-style:italic;">I</span> = (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>)<sup>−</sup></td> <td> </td> <td style="text-align:right;"><span style="white-space:nowrap;"><a href="/wiki/Sodium-22" class="mw-redirect" title="Sodium-22"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">22</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">11</sub></span></span>Na<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></a></span></td> <td><span class="texhtml mvar" style="font-style:italic;">I</span> = 3<sup>+</sup> </td></tr> <tr> <td></td> <td></td> <td> </td> <td style="text-align:right;"><span style="white-space:nowrap;"><a href="/wiki/Carbon-12" title="Carbon-12"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">12</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">6</sub></span></span>C<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></a></span></td> <td><span class="texhtml mvar" style="font-style:italic;">I</span> = 0<sup>+</sup></td> <td> </td> <td style="text-align:right;"><span style="white-space:nowrap;"><a href="/wiki/Sodium-23" class="mw-redirect" title="Sodium-23"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">23</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">11</sub></span></span>Na<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></a></span></td> <td><span class="texhtml mvar" style="font-style:italic;">I</span> = (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>)<sup>+</sup> </td></tr> <tr> <td></td> <td></td> <td> </td> <td style="text-align:right;"><span style="white-space:nowrap;"><a href="/wiki/Carbon-13" title="Carbon-13"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">13</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">6</sub></span></span>C<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></a></span></td> <td><span class="texhtml mvar" style="font-style:italic;">I</span> = (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>)<sup>−</sup></td> <td> </td> <td style="text-align:right;"><span style="white-space:nowrap;"><a href="/wiki/Sodium-24" class="mw-redirect" title="Sodium-24"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">24</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">11</sub></span></span>Na<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></a></span></td> <td><span class="texhtml mvar" style="font-style:italic;">I</span> = 4<sup>+</sup> </td></tr> <tr> <td></td> <td></td> <td> </td> <td style="text-align:right;"><span style="white-space:nowrap;"><a href="/wiki/Carbon-14" title="Carbon-14"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">14</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">6</sub></span></span>C<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></a></span></td> <td><span class="texhtml mvar" style="font-style:italic;">I</span> = 0<sup>+</sup></td> <td> </td> <td style="text-align:right;"><span style="white-space:nowrap;"><a href="/wiki/Sodium-25" class="mw-redirect" title="Sodium-25"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">25</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">11</sub></span></span>Na<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></a></span></td> <td><span class="texhtml mvar" style="font-style:italic;">I</span> = (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>)<sup>+</sup> </td></tr> <tr> <td></td> <td></td> <td> </td> <td style="text-align:right;"><span style="white-space:nowrap;"><a href="/wiki/Carbon-15" class="mw-redirect" title="Carbon-15"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">15</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">6</sub></span></span>C<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></a></span></td> <td><span class="texhtml mvar" style="font-style:italic;">I</span> = (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>)<sup>+</sup></td> <td> </td> <td style="text-align:right;"><span style="white-space:nowrap;"><a href="/wiki/Sodium-26" class="mw-redirect" title="Sodium-26"><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.0em;font-size:80%;text-align:right"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">26</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">11</sub></span></span>Na<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:0.8em;line-height:1.0em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"></sub></span></span></a></span></td> <td><span class="texhtml mvar" style="font-style:italic;">I</span> = 3<sup>+</sup> </td></tr> </tbody></table></dd></dl> <p>The reason for the unusual fluctuations in <span class="texhtml mvar" style="font-style:italic;">I</span>, even by differences of just one nucleon, are due to the odd and even numbers of protons and neutrons – pairs of nucleons have a total angular momentum of zero (just like electrons in orbitals), leaving an odd or even number of unpaired nucleons. The property of nuclear spin is an important factor for the operation of <a href="/wiki/NMR" class="mw-redirect" title="NMR">NMR</a> spectroscopy in <a href="/wiki/Organic_chemistry" title="Organic chemistry">organic chemistry</a>,<sup id="cite_ref-Atkins_1977_19-1" class="reference"><a href="#cite_note-Atkins_1977-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/MRI" class="mw-redirect" title="MRI">MRI</a> in <a href="/wiki/Nuclear_medicine" title="Nuclear medicine">nuclear medicine</a>,<sup id="cite_ref-Krane_1988_20-1" class="reference"><a href="#cite_note-Krane_1988-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> due to the <a href="/wiki/Nuclear_magnetic_moment" title="Nuclear magnetic moment">nuclear magnetic moment</a> interacting with an external <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Elementary_particles">Elementary particles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_number&action=edit&section=14" title="Edit section: Elementary particles"><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">For a more complete description of the quantum states of elementary particles, see <a href="/wiki/Standard_model" class="mw-redirect" title="Standard model">Standard model</a> and <a href="/wiki/Flavour_(particle_physics)" title="Flavour (particle physics)">Flavour (particle physics)</a>.</div> <p><a href="/wiki/Elementary_particle" title="Elementary particle">Elementary particles</a> contain many quantum numbers which are usually said to be intrinsic to them. However, it should be understood that the elementary particles are <a href="/wiki/Quantum_state" title="Quantum state">quantum states</a> of the <a href="/wiki/Standard_model" class="mw-redirect" title="Standard model">standard model</a> of <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a>, and hence the quantum numbers of these particles bear the same relation to the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> of this model as the quantum numbers of the <a href="/wiki/Bohr_atom" class="mw-redirect" title="Bohr atom">Bohr atom</a> does to its <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>. In other words, each quantum number denotes a symmetry of the problem. It is more useful in <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a> to distinguish between spacetime and internal symmetries. </p><p>Typical quantum numbers related to <a href="/wiki/Spacetime_symmetries" title="Spacetime symmetries">spacetime symmetries</a> are spin (related to rotational symmetry), the <a href="/wiki/Parity_(physics)" title="Parity (physics)">parity</a>, <a href="/wiki/C-parity" class="mw-redirect" title="C-parity">C-parity</a> and <a href="/wiki/T-parity" class="mw-redirect" title="T-parity">T-parity</a> (related to the <a href="/wiki/Poincar%C3%A9_symmetry" class="mw-redirect" title="Poincaré symmetry">Poincaré symmetry</a> of <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>). Typical <b>internal symmetries</b><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. (August 2016)">clarification needed</span></a></i>]</sup> are <a href="/wiki/Lepton_number" title="Lepton number">lepton number</a> and <a href="/wiki/Baryon_number" title="Baryon number">baryon number</a> or the <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a>. (For a full list of quantum numbers of this kind see the article on <a href="/wiki/Flavour_(particle_physics)" title="Flavour (particle physics)">flavour</a>.) </p> <div class="mw-heading mw-heading2"><h2 id="Multiplicative_quantum_numbers">Multiplicative quantum numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_number&action=edit&section=15" title="Edit section: Multiplicative quantum numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most conserved quantum numbers are additive, so in an elementary particle reaction, the <i>sum</i> of the quantum numbers should be the same before and after the reaction. However, some, usually called a <i>parity</i>, are multiplicative; i.e., their <i>product</i> is conserved. All <a href="/wiki/Multiplicative_quantum_number" title="Multiplicative quantum number">multiplicative quantum numbers</a> belong to a symmetry (like parity) in which applying the symmetry transformation twice is equivalent to doing nothing (<a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involution</a>). </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_number&action=edit&section=16" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid 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style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_number&action=edit&section=17" 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 mw-references-columns"><ol class="references"> <li id="cite_note-schrodinger-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-schrodinger_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSchrödinger,_Erwin1926" class="citation journal cs1">Schrödinger, Erwin (1926). "Quantisation as an Eigenvalue Problem". <i>Annalen der Physik</i>. <b>81</b> (18): 109–139. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1926AnP...386..109S">1926AnP...386..109S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19263861802">10.1002/andp.19263861802</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annalen+der+Physik&rft.atitle=Quantisation+as+an+Eigenvalue+Problem&rft.volume=81&rft.issue=18&rft.pages=109-139&rft.date=1926&rft_id=info%3Adoi%2F10.1002%2Fandp.19263861802&rft_id=info%3Abibcode%2F1926AnP...386..109S&rft.au=Schr%C3%B6dinger%2C+Erwin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span></span> </li> <li id="cite_note-Whittaker-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Whittaker_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhittaker1989" class="citation book cs1">Whittaker, Edmund T. (1989). <i>A history of the theories of aether & electricity. 2: The modern theories, 1900 - 1926</i> (Repr ed.). New York: Dover Publ. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-26126-3" title="Special:BookSources/978-0-486-26126-3"><bdi>978-0-486-26126-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+history+of+the+theories+of+aether+%26+electricity.+2%3A+The+modern+theories%2C+1900+-+1926&rft.place=New+York&rft.edition=Repr&rft.pub=Dover+Publ&rft.date=1989&rft.isbn=978-0-486-26126-3&rft.aulast=Whittaker&rft.aufirst=Edmund+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" 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="CITEREFHeilbron2013" class="citation journal cs1">Heilbron, John L. (June 2013). <a rel="nofollow" class="external text" href="https://www.nature.com/articles/498027a">"The path to the quantum atom"</a>. <i>Nature</i>. <b>498</b> (7452): 27–30. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F498027a">10.1038/498027a</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0028-0836">0028-0836</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/23739408">23739408</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=The+path+to+the+quantum+atom&rft.volume=498&rft.issue=7452&rft.pages=27-30&rft.date=2013-06&rft.issn=0028-0836&rft_id=info%3Apmid%2F23739408&rft_id=info%3Adoi%2F10.1038%2F498027a&rft.aulast=Heilbron&rft.aufirst=John+L.&rft_id=https%3A%2F%2Fwww.nature.com%2Farticles%2F498027a&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" 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"><a rel="nofollow" class="external text" href="https://www.nobelprize.org/prizes/physics/1922/bohr/lecture/">Niels Bohr – Nobel Lecture</a>. NobelPrize.org. Nobel Prize Outreach AB 2024. Sun. 25 Feb 2024.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEckertEckertArtin2013" class="citation book cs1">Eckert, Michael; Eckert, Michael; Artin, Tom (2013). <i>Arnold Sommerfeld: science, life and turbulent times 1868-1951</i>. New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4614-7461-6" title="Special:BookSources/978-1-4614-7461-6"><bdi>978-1-4614-7461-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Arnold+Sommerfeld%3A+science%2C+life+and+turbulent+times+1868-1951&rft.place=New+York&rft.pub=Springer&rft.date=2013&rft.isbn=978-1-4614-7461-6&rft.aulast=Eckert&rft.aufirst=Michael&rft.au=Eckert%2C+Michael&rft.au=Artin%2C+Tom&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span></span> </li> <li id="cite_note-Kragh2012Bohr-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kragh2012Bohr_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKragh2012" class="citation book cs1">Kragh, Helge (2012-05-17). <a rel="nofollow" class="external text" href="http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199654987.001.0001/acprof-9780199654987"><i>Niels Bohr and the Quantum Atom: The Bohr Model of Atomic Structure 1913–1925</i></a>. Oxford University Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Facprof%3Aoso%2F9780199654987.003.0004">10.1093/acprof:oso/9780199654987.003.0004</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-965498-7" title="Special:BookSources/978-0-19-965498-7"><bdi>978-0-19-965498-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Niels+Bohr+and+the+Quantum+Atom%3A+The+Bohr+Model+of+Atomic+Structure+1913%E2%80%931925&rft.pub=Oxford+University+Press&rft.date=2012-05-17&rft_id=info%3Adoi%2F10.1093%2Facprof%3Aoso%2F9780199654987.003.0004&rft.isbn=978-0-19-965498-7&rft.aulast=Kragh&rft.aufirst=Helge&rft_id=http%3A%2F%2Fwww.oxfordscholarship.com%2Fview%2F10.1093%2Facprof%3Aoso%2F9780199654987.001.0001%2Facprof-9780199654987&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span></span> </li> <li id="cite_note-FriedrichHerschbach-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-FriedrichHerschbach_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FriedrichHerschbach_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFriedrichHerschbach2003" class="citation journal cs1">Friedrich, Bretislav; Herschbach, Dudley (2003-12-01). <a rel="nofollow" class="external text" href="https://pubs.aip.org/physicstoday/article/56/12/53/632269/Stern-and-Gerlach-How-a-Bad-Cigar-Helped-Reorient">"Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics"</a>. <i>Physics Today</i>. <b>56</b> (12): 53–59. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003PhT....56l..53F">2003PhT....56l..53F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1650229">10.1063/1.1650229</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9228">0031-9228</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=Stern+and+Gerlach%3A+How+a+Bad+Cigar+Helped+Reorient+Atomic+Physics&rft.volume=56&rft.issue=12&rft.pages=53-59&rft.date=2003-12-01&rft.issn=0031-9228&rft_id=info%3Adoi%2F10.1063%2F1.1650229&rft_id=info%3Abibcode%2F2003PhT....56l..53F&rft.aulast=Friedrich&rft.aufirst=Bretislav&rft.au=Herschbach%2C+Dudley&rft_id=https%3A%2F%2Fpubs.aip.org%2Fphysicstoday%2Farticle%2F56%2F12%2F53%2F632269%2FStern-and-Gerlach-How-a-Bad-Cigar-Helped-Reorient&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span></span> </li> <li id="cite_note-Giulini-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Giulini_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Giulini_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGiulini2008" class="citation journal cs1">Giulini, Domenico (2008-09-01). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/S1355219808000269">"Electron spin or "classically non-describable two-valuedness"<span class="cs1-kern-right"></span>"</a>. <i>Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics</i>. <b>39</b> (3): 557–578. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0710.3128">0710.3128</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008SHPMP..39..557G">2008SHPMP..39..557G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.shpsb.2008.03.005">10.1016/j.shpsb.2008.03.005</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/11858%2F00-001M-0000-0013-13C8-1">11858/00-001M-0000-0013-13C8-1</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1355-2198">1355-2198</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Studies+in+History+and+Philosophy+of+Science+Part+B%3A+Studies+in+History+and+Philosophy+of+Modern+Physics&rft.atitle=Electron+spin+or+%22classically+non-describable+two-valuedness%22&rft.volume=39&rft.issue=3&rft.pages=557-578&rft.date=2008-09-01&rft_id=info%3Ahdl%2F11858%2F00-001M-0000-0013-13C8-1&rft_id=info%3Abibcode%2F2008SHPMP..39..557G&rft_id=info%3Aarxiv%2F0710.3128&rft.issn=1355-2198&rft_id=info%3Adoi%2F10.1016%2Fj.shpsb.2008.03.005&rft.aulast=Giulini&rft.aufirst=Domenico&rft_id=https%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fpii%2FS1355219808000269&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKragh2012" class="citation book cs1">Kragh, Helge (2012-05-17). <a rel="nofollow" class="external text" href="http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199654987.001.0001/acprof-9780199654987"><i>Niels Bohr and the Quantum Atom: The Bohr Model of Atomic Structure 1913–1925</i></a>. Oxford University Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Facprof%3Aoso%2F9780199654987.003.0007">10.1093/acprof:oso/9780199654987.003.0007</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-965498-7" title="Special:BookSources/978-0-19-965498-7"><bdi>978-0-19-965498-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Niels+Bohr+and+the+Quantum+Atom%3A+The+Bohr+Model+of+Atomic+Structure+1913%E2%80%931925&rft.pub=Oxford+University+Press&rft.date=2012-05-17&rft_id=info%3Adoi%2F10.1093%2Facprof%3Aoso%2F9780199654987.003.0007&rft.isbn=978-0-19-965498-7&rft.aulast=Kragh&rft.aufirst=Helge&rft_id=http%3A%2F%2Fwww.oxfordscholarship.com%2Fview%2F10.1093%2Facprof%3Aoso%2F9780199654987.001.0001%2Facprof-9780199654987&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span></span> </li> <li id="cite_note-Brown1987-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Brown1987_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrown1988" class="citation book cs1">Brown, L.M. (1988). <a rel="nofollow" class="external text" href="https://archive.org/details/festivalfestschr0000unse/page/40/mode/2up?q=isospin">"Remarks on the history of isospin"</a>. In Winter, Klaus; Telegdi, Valentine L. (eds.). <i>Festi-Val: Festschrift for Val Telegdi; essays in physics in honour of his 65th birthday; [a symposium ... was held at CERN, Geneva on 6 July 1987]</i>. Amsterdam: North-Holland Physics Publ. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-87099-5" title="Special:BookSources/978-0-444-87099-5"><bdi>978-0-444-87099-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Remarks+on+the+history+of+isospin&rft.btitle=Festi-Val%3A+Festschrift+for+Val+Telegdi%3B+essays+in+physics+in+honour+of+his+65th+birthday%3B+%5Ba+symposium+...+was+held+at+CERN%2C+Geneva+on+6+July+1987%5D&rft.place=Amsterdam&rft.pub=North-Holland+Physics+Publ&rft.date=1988&rft.isbn=978-0-444-87099-5&rft.aulast=Brown&rft.aufirst=L.M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffestivalfestschr0000unse%2Fpage%2F40%2Fmode%2F2up%3Fq%3Disospin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span></span> </li> <li id="cite_note-Baggott40-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-Baggott40_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Baggott40_11-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaggott2013" class="citation book cs1">Baggott, J. E. (2013). <i>The quantum story: a history in 40 moments</i> (Impression: 3 ed.). Oxford: Oxford Univ. Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-956684-6" title="Special:BookSources/978-0-19-956684-6"><bdi>978-0-19-956684-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+quantum+story%3A+a+history+in+40+moments&rft.place=Oxford&rft.edition=Impression%3A+3&rft.pub=Oxford+Univ.+Press&rft.date=2013&rft.isbn=978-0-19-956684-6&rft.aulast=Baggott&rft.aufirst=J.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeiser1987" class="citation book cs1">Beiser, A. (1987). <i>Concepts of Modern Physics</i> (4th ed.). McGraw-Hill (International). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-100144-1" title="Special:BookSources/0-07-100144-1"><bdi>0-07-100144-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Concepts+of+Modern+Physics&rft.edition=4th&rft.pub=McGraw-Hill+%28International%29&rft.date=1987&rft.isbn=0-07-100144-1&rft.aulast=Beiser&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (November 2019)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAtkins1977" class="citation book cs1">Atkins, P. W. (1977). <i>Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry</i>. Vol. 1. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-855129-0" title="Special:BookSources/0-19-855129-0"><bdi>0-19-855129-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Molecular+Quantum+Mechanics+Parts+I+and+II%3A+An+Introduction+to+Quantum+Chemistry&rft.pub=Oxford+University+Press&rft.date=1977&rft.isbn=0-19-855129-0&rft.aulast=Atkins&rft.aufirst=P.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (February 2019)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-FOOTNOTEEisbergResnick1985-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEEisbergResnick1985_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFEisbergResnick1985">Eisberg & Resnick 1985</a>.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPelegPniniZaarurHecht2010" class="citation book cs1">Peleg, Y.; Pnini, R.; Zaarur, E.; Hecht, E. (2010). <i>Quantum Mechanics</i>. Schuam's Outlines (2nd ed.). McGraw Hill (USA). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-162358-2" title="Special:BookSources/978-0-07-162358-2"><bdi>978-0-07-162358-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics&rft.series=Schuam%27s+Outlines&rft.edition=2nd&rft.pub=McGraw+Hill+%28USA%29&rft.date=2010&rft.isbn=978-0-07-162358-2&rft.aulast=Peleg&rft.aufirst=Y.&rft.au=Pnini%2C+R.&rft.au=Zaarur%2C+E.&rft.au=Hecht%2C+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (February 2019)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-Jolly-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-Jolly_16-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJolly1984" class="citation book cs1">Jolly, William L. (1984). <a rel="nofollow" class="external text" href="https://archive.org/details/trent_0116300649799/page/10"><i>Modern Inorganic Chemistry</i></a> (1st ed.). McGraw-Hill. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/trent_0116300649799/page/10">10–12</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-032760-2" title="Special:BookSources/0-07-032760-2"><bdi>0-07-032760-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Inorganic+Chemistry&rft.pages=10-12&rft.edition=1st&rft.pub=McGraw-Hill&rft.date=1984&rft.isbn=0-07-032760-2&rft.aulast=Jolly&rft.aufirst=William+L.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftrent_0116300649799%2Fpage%2F10&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLevine1983" class="citation book cs1">Levine, Ira N. (1983). <i>Physical chemistry</i> (2 ed.). New York: McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-037421-8" title="Special:BookSources/978-0-07-037421-8"><bdi>978-0-07-037421-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+chemistry&rft.place=New+York&rft.edition=2&rft.pub=McGraw-Hill&rft.date=1983&rft.isbn=978-0-07-037421-8&rft.aulast=Levine&rft.aufirst=Ira+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAtkins1977" class="citation book cs1">Atkins, P. W. (1977). <i>Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry</i>. Vol. 1. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-855129-0" title="Special:BookSources/0-19-855129-0"><bdi>0-19-855129-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Molecular+Quantum+Mechanics+Parts+I+and+II%3A+An+Introduction+to+Quantum+Chemistry&rft.pub=Oxford+University+Press&rft.date=1977&rft.isbn=0-19-855129-0&rft.aulast=Atkins&rft.aufirst=P.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (February 2019)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-Atkins_1977-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-Atkins_1977_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Atkins_1977_19-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAtkins1977" class="citation book cs1">Atkins, P. W. (1977). <i>Molecular Quantum Mechanics Part III: An Introduction to Quantum Chemistry</i>. Vol. 2. Oxford University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Molecular+Quantum+Mechanics+Part+III%3A+An+Introduction+to+Quantum+Chemistry&rft.pub=Oxford+University+Press&rft.date=1977&rft.aulast=Atkins&rft.aufirst=P.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="Please supply an ISBN for this book.">ISBN missing</span></a></i>]</sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (February 2019)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-Krane_1988-20"><span class="mw-cite-backlink">^ <a href="#cite_ref-Krane_1988_20-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Krane_1988_20-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrane1988" class="citation book cs1">Krane, K. S. (1988). <i>Introductory Nuclear Physics</i>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-80553-3" title="Special:BookSources/978-0-471-80553-3"><bdi>978-0-471-80553-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introductory+Nuclear+Physics&rft.pub=John+Wiley+%26+Sons&rft.date=1988&rft.isbn=978-0-471-80553-3&rft.aulast=Krane&rft.aufirst=K.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (February 2019)">page needed</span></a></i>]</sup></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_number&action=edit&section=18" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDirac,_Paul_A._M.1982" class="citation book cs1"><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac, Paul A. M.</a> (1982). <i>Principles of Quantum Mechanics</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-852011-5" title="Special:BookSources/0-19-852011-5"><bdi>0-19-852011-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Quantum+Mechanics&rft.pub=Oxford+University+Press&rft.date=1982&rft.isbn=0-19-852011-5&rft.au=Dirac%2C+Paul+A.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffiths,_David_J.2004" class="citation book cs1"><a href="/wiki/David_J._Griffiths" title="David J. Griffiths">Griffiths, David J.</a> (2004). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoel00grif_0"><i>Introduction to Quantum Mechanics</i></a></span> (2nd ed.). Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-805326-X" title="Special:BookSources/0-13-805326-X"><bdi>0-13-805326-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=Introduction+to+Quantum+Mechanics&rft.edition=2nd&rft.pub=Prentice+Hall&rft.date=2004&rft.isbn=0-13-805326-X&rft.au=Griffiths%2C+David+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoel00grif_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalzen,_FrancisMartin,_Alan_D.1984" class="citation book cs1"><a href="/wiki/Francis_Halzen" title="Francis Halzen">Halzen, Francis</a> & <a href="/wiki/Alan_Martin_(physicist)" title="Alan Martin (physicist)">Martin, Alan D.</a> (1984). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/quarksleptonsint0000halz"><i>Quarks and Leptons: An Introductory Course in Modern Particle Physics</i></a></span>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-88741-2" title="Special:BookSources/0-471-88741-2"><bdi>0-471-88741-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quarks+and+Leptons%3A+An+Introductory+Course+in+Modern+Particle+Physics&rft.pub=John+Wiley+%26+Sons&rft.date=1984&rft.isbn=0-471-88741-2&rft.au=Halzen%2C+Francis&rft.au=Martin%2C+Alan+D.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquarksleptonsint0000halz&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEisbergResnick1985" class="citation book cs1">Eisberg, Robert Martin; <a href="/wiki/Robert_Resnick" title="Robert Resnick">Resnick, Robert</a> (1985). <a rel="nofollow" class="external text" href="https://archive.org/details/quantumphysicsof00eisb"><i>Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles</i></a> (2nd ed.). John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-87373-0" title="Special:BookSources/978-0-471-87373-0"><bdi>978-0-471-87373-0</bdi></a> – via <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</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=978-0-471-87373-0&rft.aulast=Eisberg&rft.aufirst=Robert+Martin&rft.au=Resnick%2C+Robert&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumphysicsof00eisb&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+number" class="Z3988"></span></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style 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selflink">Quantum numbers</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/Principal_quantum_number" title="Principal quantum number">Principal quantum number (<span class="texhtml mvar" style="font-style:italic;">n</span>)</a></li> <li><a href="/wiki/Azimuthal_quantum_number" title="Azimuthal quantum number">Azimuthal quantum number (<span class="texhtml mvar" style="font-style:italic;">ℓ</span>)</a></li> <li><a href="/wiki/Magnetic_quantum_number" title="Magnetic quantum number">Magnetic quantum number (<span class="texhtml mvar" style="font-style:italic;">m</span>)</a></li> <li><a href="/wiki/Spin_quantum_number" title="Spin quantum number">Spin quantum number (<span class="texhtml mvar" style="font-style:italic;">s</span>)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Ground-state configurations</th><td class="navbox-list-with-group 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</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Electron pairing</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/Electron_pair" title="Electron pair">Electron pair</a></li> <li><a href="/wiki/Unpaired_electron" title="Unpaired electron">Unpaired electron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Bonding participation</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/Valence_electron" title="Valence electron">Valence electron</a></li> <li><a href="/wiki/Core_electron" title="Core electron">Core electron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Electron_counting" title="Electron counting">Electron counting</a> rules</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/Octet_rule" title="Octet rule">Octet rule</a></li> <li><a href="/wiki/18-electron_rule" title="18-electron rule">18-electron rule</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Quantum_mechanics" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li 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mechanics">Introduction</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a> <ul><li><a href="/wiki/Timeline_of_quantum_mechanics" title="Timeline of quantum mechanics">Timeline</a></li></ul></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamentals</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Born_rule" title="Born rule">Born rule</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li> <li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)"> Complementarity</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a> <ul><li><a href="/wiki/Ground_state" title="Ground state">Ground state</a></li> <li><a href="/wiki/Excited_state" title="Excited state">Excited state</a></li> <li><a href="/wiki/Degenerate_energy_levels" title="Degenerate energy levels">Degenerate levels</a></li> <li><a href="/wiki/Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li></ul></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Formulations</a></li> <li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Equations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Weyl_equation" title="Weyl equation">Weyl</a></li> <li><a href="/wiki/Majorana_equation" title="Majorana equation">Majorana</a></li> <li><a href="/wiki/Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell test</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li> <li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed choice</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_nanoscience" class="mw-redirect" title="Quantum nanoscience">Science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_biology" title="Quantum biology">Quantum biology</a></li> <li><a href="/wiki/Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li> <li><a href="/wiki/Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li> <li><a href="/wiki/Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li> <li><a href="/wiki/Quantum_mind" title="Quantum mind">Quantum mind</a></li> <li><a href="/wiki/Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li> <li><a href="/wiki/Quantum_spacetime" title="Quantum spacetime">Quantum spacetime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li> <li><a href="/wiki/Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">Quantum bus</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a> <ul><li><a href="/wiki/Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li> <li><a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">Quantum electronics</a></li> <li><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li> <li><a href="/wiki/Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li> <li><a href="/wiki/Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li> <li><a href="/wiki/Quantum_machine" title="Quantum machine">Quantum machine</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li> <li><a href="/wiki/Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li> <li><a href="/wiki/Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">Quantum network</a></li> <li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li> <li><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Extensions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li> <li><a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a> <ul><li><a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></li></ul></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat_in_popular_culture" title="Schrödinger's cat in popular culture">in popular culture</a></li></ul></li> <li><a href="/wiki/Wigner%27s_friend" title="Wigner's friend">Wigner's friend</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Quantum_mysticism" title="Quantum mysticism">Quantum mysticism</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" 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