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data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Beginning</div> </a> </li> <li id="toc-Atoms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Atoms"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Atoms</span> </div> </a> <button aria-controls="toc-Atoms-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 Atoms subsection</span> </button> <ul id="toc-Atoms-sublist" class="vector-toc-list"> <li id="toc-Intrinsic_energy_levels" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Intrinsic_energy_levels"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Intrinsic energy levels</span> </div> </a> <ul id="toc-Intrinsic_energy_levels-sublist" class="vector-toc-list"> <li id="toc-Orbital_state_energy_level" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Orbital_state_energy_level"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1</span> <span>Orbital state energy level</span> </div> </a> <ul id="toc-Orbital_state_energy_level-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fine_structure_splitting" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Fine_structure_splitting"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.2</span> <span>Fine structure splitting</span> </div> </a> <ul id="toc-Fine_structure_splitting-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hyperfine_structure" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Hyperfine_structure"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.3</span> <span>Hyperfine structure</span> </div> </a> <ul id="toc-Hyperfine_structure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electrostatic_interaction_of_an_electron_with_other_electrons" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Electrostatic_interaction_of_an_electron_with_other_electrons"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.4</span> <span>Electrostatic interaction of an electron with other electrons</span> </div> </a> <ul id="toc-Electrostatic_interaction_of_an_electron_with_other_electrons-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Energy_levels_due_to_external_fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Energy_levels_due_to_external_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Energy levels due to external fields</span> </div> </a> <ul id="toc-Energy_levels_due_to_external_fields-sublist" class="vector-toc-list"> <li id="toc-Zeeman_effect" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Zeeman_effect"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.1</span> <span>Zeeman effect</span> </div> </a> <ul id="toc-Zeeman_effect-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zeeman_effect_taking_spin_into_account" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Zeeman_effect_taking_spin_into_account"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.2</span> <span>Zeeman effect taking spin into account</span> </div> </a> <ul id="toc-Zeeman_effect_taking_spin_into_account-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stark_effect" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Stark_effect"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2.3</span> <span>Stark effect</span> </div> </a> <ul id="toc-Stark_effect-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Molecules" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Molecules"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Molecules</span> </div> </a> <ul id="toc-Molecules-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Crystalline_materials" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Crystalline_materials"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Crystalline materials</span> </div> </a> <ul id="toc-Crystalline_materials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_pages" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_pages"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Related pages</span> </div> </a> <ul id="toc-Related_pages-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" 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Available in 42 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-42" 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">42 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B3%D8%AA%D9%88%D9%89_%D8%B7%D8%A7%D9%82%D8%A9" title="مستوى طاقة – Arabic" lang="ar" hreflang="ar" data-title="مستوى طاقة" 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/Nivel_enerx%C3%A9ticu" title="Nivel enerxéticu – Asturian" lang="ast" hreflang="ast" data-title="Nivel enerxéticu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%AD%D0%BD%D0%B5%D1%80%D0%B3%D0%B5%D1%82%D1%8B%D1%87%D0%BD%D1%8B_%D1%9E%D0%B7%D1%80%D0%BE%D0%B2%D0%B5%D0%BD%D1%8C" title="Энергетычны ўзровень – Belarusian" lang="be" hreflang="be" data-title="Энергетычны ўзровень" 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%95%D0%BD%D0%B5%D1%80%D0%B3%D0%B5%D1%82%D0%B8%D1%87%D0%BD%D0%BE_%D0%BD%D0%B8%D0%B2%D0%BE" title="Енергетично ниво – Bulgarian" lang="bg" hreflang="bg" data-title="Енергетично ниво" 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/Energetski_nivo" title="Energetski nivo – Bosnian" lang="bs" hreflang="bs" data-title="Energetski nivo" 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/Nivell_d%27energia" title="Nivell d'energia – Catalan" lang="ca" hreflang="ca" data-title="Nivell d'energia" 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%AD%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D0%BB%D0%BB%D0%B5_%D1%88%D0%B0%D0%B9" title="Энергилле шай – Chuvash" lang="cv" hreflang="cv" data-title="Энергилле шай" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Energiniveau" title="Energiniveau – Danish" lang="da" hreflang="da" data-title="Energiniveau" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Energieniveau" title="Energieniveau – German" lang="de" hreflang="de" data-title="Energieniveau" 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/Energiatase_(kvantf%C3%BC%C3%BCsika)" title="Energiatase (kvantfüüsika) – Estonian" lang="et" hreflang="et" data-title="Energiatase (kvantfüüsika)" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Energy_level" title="Energy level – English" lang="en" hreflang="en" data-title="Energy level" data-language-autonym="English" data-language-local-name="English" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Nivel_energ%C3%A9tico" title="Nivel energético – Spanish" lang="es" hreflang="es" data-title="Nivel energético" 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/Energinivelo" title="Energinivelo – Esperanto" lang="eo" hreflang="eo" data-title="Energinivelo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%B1%D8%A7%D8%B2_%D8%A7%D9%86%D8%B1%DA%98%DB%8C" title="تراز انرژی – Persian" lang="fa" hreflang="fa" data-title="تراز انرژی" 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/Niveau_d%27%C3%A9nergie" title="Niveau d'énergie – French" lang="fr" hreflang="fr" data-title="Niveau d'énergie" 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/Leibh%C3%A9al_fuinnimh" title="Leibhéal fuinnimh – Irish" lang="ga" hreflang="ga" data-title="Leibhéal fuinnimh" 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%97%90%EB%84%88%EC%A7%80_%EC%A4%80%EC%9C%84" title="에너지 준위 – Korean" lang="ko" hreflang="ko" data-title="에너지 준위" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Energijske_razine" title="Energijske razine – Croatian" lang="hr" hreflang="hr" data-title="Energijske razine" 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/Tingkat_energi" title="Tingkat energi – Indonesian" lang="id" hreflang="id" data-title="Tingkat energi" 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/Livello_energetico" title="Livello energetico – Italian" lang="it" hreflang="it" data-title="Livello energetico" 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%A8%D7%9E%D7%AA_%D7%90%D7%A0%D7%A8%D7%92%D7%99%D7%94" title="רמת אנרגיה – Hebrew" lang="he" hreflang="he" data-title="רמת אנרגיה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%AD%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%8F_%D0%B4%D0%B5%D2%A3%D0%B3%D0%B5%D0%B9%D1%96" title="Энергия деңгейі – Kazakh" lang="kk" hreflang="kk" data-title="Энергия деңгейі" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Nivo_en%C3%A8jetik" title="Nivo enèjetik – Haitian Creole" lang="ht" hreflang="ht" data-title="Nivo enèjetik" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%95%D0%BD%D0%B5%D1%80%D0%B3%D0%B5%D1%82%D1%81%D0%BA%D0%BE_%D0%BD%D0%B8%D0%B2%D0%BE" title="Енергетско ниво – Macedonian" lang="mk" hreflang="mk" data-title="Енергетско ниво" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Energieniveau" title="Energieniveau – Dutch" lang="nl" hreflang="nl" data-title="Energieniveau" 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/%E3%82%A8%E3%83%8D%E3%83%AB%E3%82%AE%E3%83%BC%E6%BA%96%E4%BD%8D" title="エネルギー準位 – Japanese" lang="ja" hreflang="ja" data-title="エネルギー準位" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Energiniv%C3%A5" title="Energinivå – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Energinivå" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%8A%E0%A8%B0%E0%A8%9C%E0%A8%BE_%E0%A8%B2%E0%A9%88%E0%A8%B5%E0%A8%B2" title="ਊਰਜਾ ਲੈਵਲ – Punjabi" lang="pa" hreflang="pa" data-title="ਊਰਜਾ ਲੈਵਲ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Poziom_energetyczny" title="Poziom energetyczny – Polish" lang="pl" hreflang="pl" data-title="Poziom energetyczny" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%ADvel_de_energia" title="Nível de energia – Portuguese" lang="pt" hreflang="pt" data-title="Nível de energia" 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/Nivel_de_energie" title="Nivel de energie – Romanian" lang="ro" hreflang="ro" data-title="Nivel de energie" 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%AD%D0%BD%D0%B5%D1%80%D0%B3%D0%B5%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D1%83%D1%80%D0%BE%D0%B2%D0%B5%D0%BD%D1%8C" title="Энергетический уровень – Russian" lang="ru" hreflang="ru" data-title="Энергетический уровень" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Nivelet_energjetike" title="Nivelet energjetike – Albanian" lang="sq" hreflang="sq" data-title="Nivelet energjetike" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Energijski_nivo" title="Energijski nivo – Slovenian" lang="sl" hreflang="sl" data-title="Energijski nivo" 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%95%D0%BD%D0%B5%D1%80%D0%B3%D0%B5%D1%82%D1%81%D0%BA%D0%B8_%D0%BD%D0%B8%D0%B2%D0%BE%D0%B8" title="Енергетски нивои – Serbian" lang="sr" hreflang="sr" data-title="Енергетски нивои" 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/Energetski_nivo" title="Energetski nivo – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Energetski nivo" 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-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Energiniv%C3%A5" title="Energinivå – Swedish" lang="sv" hreflang="sv" data-title="Energinivå" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%86%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%B2%E0%AF%8D_%E0%AE%AE%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%AE%E0%AF%8D" title="ஆற்றல் மட்டம் – Tamil" lang="ta" hreflang="ta" data-title="ஆற்றல் மட்டம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B0%E0%B8%94%E0%B8%B1%E0%B8%9A%E0%B8%9E%E0%B8%A5%E0%B8%B1%E0%B8%87%E0%B8%87%E0%B8%B2%E0%B8%99" title="ระดับพลังงาน – Thai" lang="th" hreflang="th" data-title="ระดับพลังงาน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Enerji_seviyesi" title="Enerji seviyesi – Turkish" lang="tr" hreflang="tr" data-title="Enerji seviyesi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%95%D0%BD%D0%B5%D1%80%D0%B3%D0%B5%D1%82%D0%B8%D1%87%D0%BD%D0%B8%D0%B9_%D1%80%D1%96%D0%B2%D0%B5%D0%BD%D1%8C" title="Енергетичний рівень – Ukrainian" lang="uk" hreflang="uk" data-title="Енергетичний рівень" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%83%BD%E7%BA%A7" title="能级 – Chinese" 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For compounds' energy levels, see <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potential</a>.</i></dd></dl> <p><b>Energy level</b> is simply defined as the different states of <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> for <a href="/wiki/Electron" title="Electron">electrons</a> in an <a href="/wiki/Atom" title="Atom">atom</a>. A <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanical</a> system can only be in certain <a href="/w/index.php?title=State_(physics)&action=edit&redlink=1" class="new" title="State (physics) (not yet started)">states</a>, so that only certain <b>energy levels</b> are possible. The term energy level is most commonly used in reference to the <a href="/wiki/Electron_configuration" title="Electron configuration">electron configuration</a> in <a href="/wiki/Atom" title="Atom">atoms</a> or <a href="/wiki/Molecule" title="Molecule">molecules</a>. In other words, the energy spectrum can be <a href="/w/index.php?title=Quantization&action=edit&redlink=1" class="new" title="Quantization (not yet started)">quantized</a>. </p><p>As with classical potentials, the potential energy is usually set to zero at infinity, leading to a negative potential energy for <a href="/w/index.php?title=Bound_states&action=edit&redlink=1" class="new" title="Bound states (not yet started)">bound electron states</a>. </p><p>Energy levels are said to be degenerate, if the same energy level is obtained by more than one quantum mechanical <a href="/wiki/Quantum_state" title="Quantum state">state</a>. They are then called <a href="/w/index.php?title=Degenerate_energy_level&action=edit&redlink=1" class="new" title="Degenerate energy level (not yet started)">degenerate energy levels</a>. </p><p>The following sections of this article gives an overview over the most important factors that determine the energy levels of atoms and molecules. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Atoms">Atoms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Energy_level&veaction=edit&section=1" title="Change section: Atoms" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Energy_level&action=edit&section=1" title="Edit section's source code: Atoms"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Intrinsic_energy_levels">Intrinsic energy levels</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Energy_level&veaction=edit&section=2" title="Change section: Intrinsic energy levels" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Energy_level&action=edit&section=2" title="Edit section's source code: Intrinsic energy levels"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Orbital_state_energy_level">Orbital state energy level</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Energy_level&veaction=edit&section=3" title="Change section: Orbital state energy level" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Energy_level&action=edit&section=3" title="Edit section's source code: Orbital state energy level"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Assume an electron in a given <a href="/wiki/Atomic_orbital" title="Atomic orbital">atomic orbital</a>. The energy of its state is mainly determined by the <a href="/wiki/Electrostatic_force" class="mw-redirect" title="Electrostatic force">electrostatic i</a>nteraction of the (negative) <a href="/wiki/Electron" title="Electron">electron</a> with the (positive) <a href="/wiki/Atomic_nucleus" title="Atomic nucleus">nucleus</a>. The energy levels of an electron around a nucleus are given by : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}=-hcR_{\infty }{\frac {Z^{2}}{n^{2}}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>h</mi> <mi>c</mi> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}=-hcR_{\infty }{\frac {Z^{2}}{n^{2}}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c5ce020b448754fb680317d1a5c57b19f4360ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.005ex; height:6.009ex;" alt="{\displaystyle E_{n}=-hcR_{\infty }{\frac {Z^{2}}{n^{2}}}\ }"></span>,</dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\infty }\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\infty }\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cbc598098f53720b4acd528ab7fe8159c062336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.22ex; height:2.509ex;" alt="{\displaystyle R_{\infty }\ }"></span> is the <a href="/wiki/Rydberg_constant" title="Rydberg constant">Rydberg constant</a> (typically between 1 <a href="/wiki/Electronvolt" title="Electronvolt">eV</a> and 10<sup>3</sup> eV), <i>Z</i> is the charge of the atom's nucleus, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81a2f2e76ab2d87f5ed37aeb7018991a8d82b04b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.975ex; height:1.676ex;" alt="{\displaystyle n\ }"></span> is the <a href="/wiki/Principal_quantum_number" class="mw-redirect" title="Principal quantum number">principal quantum number</a>, <i>e</i> is the charge of the electron, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> is <a href="/wiki/Planck%27s_constant" class="mw-redirect" title="Planck's constant">Planck's constant</a>, and <i>c</i> is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>. We The Rydberg levels depend only on the principal quantum number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81a2f2e76ab2d87f5ed37aeb7018991a8d82b04b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.975ex; height:1.676ex;" alt="{\displaystyle n\ }"></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Fine_structure_splitting">Fine structure splitting</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Energy_level&veaction=edit&section=4" title="Change section: Fine structure splitting" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Energy_level&action=edit&section=4" title="Edit section's source code: Fine structure splitting"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/w/index.php?title=Fine_structure&action=edit&redlink=1" class="new" title="Fine structure (not yet started)">Fine structure</a> arises from relativistic <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> corrections, <a href="/w/index.php?title=Spin-orbit_coupling&action=edit&redlink=1" class="new" title="Spin-orbit coupling (not yet started)">spin-orbit coupling</a> (an electrodynamic interaction between the electron's <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> and motion and the nucleus's electric field) and the <a href="/w/index.php?title=Darwin_term&action=edit&redlink=1" class="new" title="Darwin term (not yet started)">Darwin term</a> (contact term interaction of s-shell electrons inside the nucleus). Typical magnitude <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{-3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c78cf9ec2db87a7a95990ef3a2e59ac4320c49e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.658ex; height:2.676ex;" alt="{\displaystyle 10^{-3}}"></span> eV. </p> <div class="mw-heading mw-heading4"><h4 id="Hyperfine_structure">Hyperfine structure</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Energy_level&veaction=edit&section=5" title="Change section: Hyperfine structure" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Energy_level&action=edit&section=5" title="Edit section's source code: Hyperfine structure"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Spin-nuclear-spin coupling (<i>see <a href="/w/index.php?title=Hyperfine_structure&action=edit&redlink=1" class="new" title="Hyperfine structure (not yet started)">hyperfine structure</a></i>). Typical magnitude <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{-4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{-4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9588d1bcf3d738e9ed5c044770795e06f7e4eb28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.658ex; height:2.676ex;" alt="{\displaystyle 10^{-4}}"></span> eV. </p> <div class="mw-heading mw-heading4"><h4 id="Electrostatic_interaction_of_an_electron_with_other_electrons">Electrostatic interaction of an electron with other electrons</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Energy_level&veaction=edit&section=6" title="Change section: Electrostatic interaction of an electron with other electrons" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Energy_level&action=edit&section=6" title="Edit section's source code: Electrostatic interaction of an electron with other electrons"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If there is more than one electron around the atom, electron-electron-interactions raise the energy level. These interactions are often neglected if the spatial overlap of the electron <a href="/wiki/Wave_function" title="Wave function">wavefunctions</a> is low. </p> <div class="mw-heading mw-heading3"><h3 id="Energy_levels_due_to_external_fields">Energy levels due to external fields</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Energy_level&veaction=edit&section=7" title="Change section: Energy levels due to external fields" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Energy_level&action=edit&section=7" title="Edit section's source code: Energy levels due to external fields"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Zeeman_effect">Zeeman effect</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Energy_level&veaction=edit&section=8" title="Change section: Zeeman effect" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Energy_level&action=edit&section=8" title="Edit section's source code: Zeeman effect"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The interaction energy is: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=-\mu B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mo>−</mo> <mi>μ</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=-\mu B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be1e35ab8444b2f1673862d313f730232ab93ec6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.855ex; height:2.676ex;" alt="{\displaystyle U=-\mu B}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =qL/2m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>q</mi> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =qL/2m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28d02ef4e4977d75817e25af880eb2aff6311085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.518ex; height:2.843ex;" alt="{\displaystyle \mu =qL/2m}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Zeeman_effect_taking_spin_into_account">Zeeman effect taking spin into account</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Energy_level&veaction=edit&section=9" title="Change section: Zeeman effect taking spin into account" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Energy_level&action=edit&section=9" title="Edit section's source code: Zeeman effect taking spin into account"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This takes both the magnetic <a href="/wiki/Dipole" title="Dipole">dipole</a> moment due to the orbital <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> and the <a href="/w/index.php?title=Magnetic_momentum&action=edit&redlink=1" class="new" title="Magnetic momentum (not yet started)">magnetic momentum</a> arising from the electron spin into account. </p><p>Due to relativistic effects (<a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a>), the magnetic moment arising from the electron spin is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =-\mu _{B}gs}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>g</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =-\mu _{B}gs}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/149c8de86011818850e6e9c7712bd8bdfd782148" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.396ex; height:2.509ex;" alt="{\displaystyle \mu =-\mu _{B}gs}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> the <a href="/w/index.php?title=Gyro-magnetic&action=edit&redlink=1" class="new" title="Gyro-magnetic (not yet started)">gyro-magnetic</a> factor (about 2). <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 \mu =\mu _{l}+g\mu _{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>+</mo> <mi>g</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\mu _{l}+g\mu _{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3eed79f8c253f2ebc116b82616115d22f05a3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.986ex; height:2.509ex;" alt="{\displaystyle \mu =\mu _{l}+g\mu _{s}}"></span> The interaction energy therefore gets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{B}=-\mu B=\mu _{B}B(m_{l}+gm_{s})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>μ</mi> <mi>B</mi> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>B</mi> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>+</mo> <mi>g</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{B}=-\mu B=\mu _{B}B(m_{l}+gm_{s})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90b9d568da95393f6cc3bf45ed637a32630d94c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.455ex; height:2.843ex;" alt="{\displaystyle U_{B}=-\mu B=\mu _{B}B(m_{l}+gm_{s})}"></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Stark_effect">Stark effect</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Energy_level&veaction=edit&section=10" title="Change section: Stark effect" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Energy_level&action=edit&section=10" title="Edit section's source code: Stark effect"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Interaction with an external electric field (<i>see <a href="/w/index.php?title=Stark_effect&action=edit&redlink=1" class="new" title="Stark effect (not yet started)">Stark effect</a></i>). </p> <div class="mw-heading mw-heading2"><h2 id="Molecules">Molecules</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Energy_level&veaction=edit&section=11" title="Change section: Molecules" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Energy_level&action=edit&section=11" title="Edit section's source code: Molecules"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Roughly speaking, a <b>molecular energy state</b>, i.e. an <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenstate</a> of the <a href="/w/index.php?title=Molecular_Hamiltonian&action=edit&redlink=1" class="new" title="Molecular Hamiltonian (not yet started)">molecular Hamiltonian</a>, is the sum of an electronic, vibrational, rotational, nuclear and translational component, such that: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=E_{\mathrm {electronic} }+E_{\mathrm {vibrational} }+E_{\mathrm {rotational} }+E_{\mathrm {nuclear} }+E_{\mathrm {translational} }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=E_{\mathrm {electronic} }+E_{\mathrm {vibrational} }+E_{\mathrm {rotational} }+E_{\mathrm {nuclear} }+E_{\mathrm {translational} }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a4812d64c02a0798ebc8d7f0fe844ef7401de6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:62.346ex; height:2.509ex;" alt="{\displaystyle E=E_{\mathrm {electronic} }+E_{\mathrm {vibrational} }+E_{\mathrm {rotational} }+E_{\mathrm {nuclear} }+E_{\mathrm {translational} }\,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {electronic} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {electronic} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b3b7c86bbf0577360395fd5eb607b54062fdf22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.803ex; height:2.509ex;" alt="{\displaystyle E_{\mathrm {electronic} }}"></span> is an <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalue</a> of the <a href="/w/index.php?title=Electronic_molecular_Hamiltonian&action=edit&redlink=1" class="new" title="Electronic molecular Hamiltonian (not yet started)">electronic molecular Hamiltonian</a> (the value of the <a href="/w/index.php?title=Potential_energy_surface&action=edit&redlink=1" class="new" title="Potential energy surface (not yet started)">potential energy surface</a>) at the <a href="/wiki/Molecular_geometry" title="Molecular geometry">equilibrium geometry of the molecule</a>. </p><p>The molecular energy levels are labelled by the <a href="/w/index.php?title=Molecular_term_symbol&action=edit&redlink=1" class="new" title="Molecular term symbol (not yet started)">molecular term symbols</a>. </p><p>The specific energies of these components vary with the specific energy state and the substance. </p><p>In <a href="/w/index.php?title=Molecular_physics&action=edit&redlink=1" class="new" title="Molecular physics (not yet started)">molecular physics</a> and <a href="/w/index.php?title=Quantum_chemistry&action=edit&redlink=1" class="new" title="Quantum chemistry (not yet started)">quantum chemistry</a>, an <b>energy level</b> is a quantized energy of a <a href="/w/index.php?title=Bound_state&action=edit&redlink=1" class="new" title="Bound state (not yet started)">bound</a> <a href="/wiki/Quantum_state" title="Quantum state">quantum mechanical state</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Crystalline_materials">Crystalline materials</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Energy_level&veaction=edit&section=12" title="Change section: Crystalline materials" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Energy_level&action=edit&section=12" title="Edit section's source code: Crystalline materials"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Crystalline" class="mw-redirect" title="Crystalline">Crystalline</a> materials are often characterized by a number of important energy levels. The most important ones are the top of the <a href="/wiki/Valence_band" class="mw-redirect" title="Valence band">valence band</a>, the bottom of the <a href="/w/index.php?title=Conduction_band&action=edit&redlink=1" class="new" title="Conduction band (not yet started)">conduction band</a>, the <a href="/w/index.php?title=Fermi_energy&action=edit&redlink=1" class="new" title="Fermi energy (not yet started)">Fermi energy</a>, the <a href="/w/index.php?title=Vacuum_level&action=edit&redlink=1" class="new" title="Vacuum level (not yet started)">vacuum level</a>, and the energy levels of any <a href="/w/index.php?title=Defect_states&action=edit&redlink=1" class="new" title="Defect states (not yet started)">defect states</a> in the crystals. </p> <div class="mw-heading mw-heading2"><h2 id="Related_pages">Related pages</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Energy_level&veaction=edit&section=13" title="Change section: Related pages" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Energy_level&action=edit&section=13" title="Edit section's source code: Related pages"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Electron_configuration" title="Electron configuration">Electron configuration</a></li> <li><a href="/wiki/Computational_chemistry" title="Computational chemistry">Computational chemistry</a></li> <li><a href="/wiki/Spectroscopy" title="Spectroscopy">Spectroscopy</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://simple.wikipedia.org/w/index.php?title=Energy_level&oldid=9665064">https://simple.wikipedia.org/w/index.php?title=Energy_level&oldid=9665064</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:Categories" title="Special:Categories">Categories</a>: <ul><li><a href="/wiki/Category:Mechanics" title="Category:Mechanics">Mechanics</a></li><li><a href="/wiki/Category:Spectroscopy" title="Category:Spectroscopy">Spectroscopy</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last changed on 24 July 2024, at 19:26.</li> <li id="footer-info-copyright">Text is available under the <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution-ShareAlike License</a> and the <a href="/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License" title="Wikipedia:Text of the GNU Free Documentation License">GFDL</a>; additional terms may apply. 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