自旋-軌道作用 - 维基百科，自由的百科全书

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title="建议你登录，尽管并非必须。[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>登录</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 未登录编辑者的页面 <a href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div 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data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</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">序言</div> </a> </li> <li id="toc-電子的自旋-軌道作用" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#電子的自旋-軌道作用"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>電子的自旋-軌道作用</span> </div> </a> <button aria-controls="toc-電子的自旋-軌道作用-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>开关電子的自旋-軌道作用子章节</span> </button> <ul id="toc-電子的自旋-軌道作用-sublist" class="vector-toc-list"> <li id="toc-磁場" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#磁場"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>磁場</span> </div> </a> <ul id="toc-磁場-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-磁矩" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#磁矩"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>磁矩</span> </div> </a> <ul id="toc-磁矩-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-哈密頓量微擾項目" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#哈密頓量微擾項目"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>哈密頓量微擾項目</span> </div> </a> <ul id="toc-哈密頓量微擾項目-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-能級位移" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#能級位移"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>能級位移</span> </div> </a> <ul id="toc-能級位移-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-參閱" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#參閱"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>參閱</span> </div> </a> <ul id="toc-參閱-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-參考文獻" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#參考文獻"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>參考文獻</span> </div> </a> <ul id="toc-參考文獻-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部連結" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部連結"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>外部連結</span> </div> </a> <ul id="toc-外部連結-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="目录" 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="开关目录" > <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">开关目录</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">自旋-軌道作用</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="前往另一种语言写成的文章。18种语言可用" > <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-18" 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">18种语言</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%AA%D8%A2%D8%AB%D8%B1_%D9%85%D8%BA%D8%B2%D9%84%D9%8A_%D9%85%D8%AF%D8%A7%D8%B1%D9%8A" title="تآثر مغزلي مداري – 阿拉伯语" lang="ar" hreflang="ar" data-title="تآثر مغزلي مداري" data-language-autonym="العربية" data-language-local-name="阿拉伯语" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B8%D0%BD-%D0%BE%D1%80%D0%B1%D0%B8%D1%82%D0%B0%D0%BB%D0%BD%D0%BE_%D0%B2%D0%B7%D0%B0%D0%B8%D0%BC%D0%BE%D0%B4%D0%B5%D0%B9%D1%81%D1%82%D0%B2%D0%B8%D0%B5" title="Спин-орбитално взаимодействие – 保加利亚语" lang="bg" hreflang="bg" data-title="Спин-орбитално взаимодействие" data-language-autonym="Български" data-language-local-name="保加利亚语" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Interacci%C3%B3_esp%C3%ADn-%C3%B2rbita" title="Interacció espín-òrbita – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Interacció espín-òrbita" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Spin-Bahn-Kopplung" title="Spin-Bahn-Kopplung – 德语" lang="de" hreflang="de" data-title="Spin-Bahn-Kopplung" data-language-autonym="Deutsch" data-language-local-name="德语" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Spin%E2%80%93orbit_interaction" title="Spin–orbit interaction – 英语" lang="en" hreflang="en" data-title="Spin–orbit interaction" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Interaction_spin-orbite" title="Interaction spin-orbite – 法语" lang="fr" hreflang="fr" data-title="Interaction spin-orbite" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Interaksi_spin%E2%80%93orbit" title="Interaksi spin–orbit – 印度尼西亚语" lang="id" hreflang="id" data-title="Interaksi spin–orbit" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" 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/Interazione_spin-orbita" title="Interazione spin-orbita – 意大利语" lang="it" hreflang="it" data-title="Interazione spin-orbita" data-language-autonym="Italiano" data-language-local-name="意大利语" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%B9%E3%83%94%E3%83%B3%E8%BB%8C%E9%81%93%E7%9B%B8%E4%BA%92%E4%BD%9C%E7%94%A8" title="スピン軌道相互作用 – 日语" lang="ja" hreflang="ja" data-title="スピン軌道相互作用" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8A%A4%ED%95%80-%EA%B6%A4%EB%8F%84_%EC%83%81%ED%98%B8%EC%9E%91%EC%9A%A9" title="스핀-궤도 상호작용 – 韩语" lang="ko" hreflang="ko" data-title="스핀-궤도 상호작용" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Tindak_balas_spin-petala" title="Tindak balas spin-petala – 马来语" lang="ms" hreflang="ms" data-title="Tindak balas spin-petala" data-language-autonym="Bahasa Melayu" data-language-local-name="马来语" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Sprz%C4%99%C5%BCenie_spinowo-orbitalne" title="Sprzężenie spinowo-orbitalne – 波兰语" lang="pl" hreflang="pl" data-title="Sprzężenie spinowo-orbitalne" data-language-autonym="Polski" data-language-local-name="波兰语" 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/Intera%C3%A7%C3%A3o_spin-%C3%B3rbita" title="Interação spin-órbita – 葡萄牙语" lang="pt" hreflang="pt" data-title="Interação spin-órbita" data-language-autonym="Português" data-language-local-name="葡萄牙语" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B8%D0%BD-%D0%BE%D1%80%D0%B1%D0%B8%D1%82%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B5_%D0%B2%D0%B7%D0%B0%D0%B8%D0%BC%D0%BE%D0%B4%D0%B5%D0%B9%D1%81%D1%82%D0%B2%D0%B8%D0%B5" title="Спин-орбитальное взаимодействие – 俄语" lang="ru" hreflang="ru" data-title="Спин-орбитальное взаимодействие" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AD%E0%B8%B1%E0%B8%99%E0%B8%95%E0%B8%A3%E0%B8%81%E0%B8%B4%E0%B8%A3%E0%B8%B4%E0%B8%A2%E0%B8%B2%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%AA%E0%B8%9B%E0%B8%B4%E0%B8%99%E0%B8%81%E0%B8%B1%E0%B8%9A%E0%B8%AD%E0%B8%AD%E0%B8%A3%E0%B9%8C%E0%B8%9A%E0%B8%B4%E0%B8%97" title="อันตรกิริยาของสปินกับออร์บิท – 泰语" lang="th" hreflang="th" data-title="อันตรกิริยาของสปินกับออร์บิท" data-language-autonym="ไทย" data-language-local-name="泰语" 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/Spin-y%C3%B6r%C3%BCnge_etkile%C5%9Fimi" title="Spin-yörünge etkileşimi – 土耳其语" lang="tr" hreflang="tr" data-title="Spin-yörünge etkileşimi" data-language-autonym="Türkçe" data-language-local-name="土耳其语" 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%A1%D0%BF%D1%96%D0%BD-%D0%BE%D1%80%D0%B1%D1%96%D1%82%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0_%D0%B2%D0%B7%D0%B0%D1%94%D0%BC%D0%BE%D0%B4%D1%96%D1%8F" title="Спін-орбітальна взаємодія – 乌克兰语" lang="uk" hreflang="uk" data-title="Спін-орбітальна взаємодія" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Spin-orbital_ta%CA%BCsir" title="Spin-orbital taʼsir – 乌兹别克语" lang="uz" hreflang="uz" data-title="Spin-orbital taʼsir" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="乌兹别克语" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1570979#sitelinks-wikipedia" title="编辑跨语言链接" class="wbc-editpage">编辑链接</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="命名空间"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a 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.sidebar-below{background:inherit!important;color:inherit!important;border-color:#54595d!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar a:not(.new):not(.mw-selflink):link{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:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist nowraplinks" style="width:19.0em;"><tbody><tr><td class="sidebar-pretitle">系列条目</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%A6" title="量子力学">量子力学</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 {\partial }{\partial t}}|\psi (t)\rangle ={\hat {H}}|\psi (t)\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <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>ψ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle ={\hat {H}}|\psi (t)\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0de8741a7d26ae98689c7b3339e97dfafea9fd26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.692ex; height:5.509ex;" alt="{\displaystyle i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle ={\hat {H}}|\psi (t)\rangle }"></span><div class="sidebar-caption" style="font-size:90%;padding-top:0.4em;font-style:italic;"><a href="/wiki/%E8%96%9B%E5%AE%9A%E8%B0%94%E6%96%B9%E7%A8%8B" title="薛定谔方程">薛定谔方程</a></div></td></tr><tr><td class="sidebar-above hlist nowrap" style="display:block;margin-bottom:0.4em;"> <ul><li><a href="/wiki/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8%E5%85%A5%E9%96%80" title="量子力學入門">入门</a></li> <li><span class="ilh-all" data-orig-title="初级量子力学词汇" data-lang-code="en" data-lang-name="英语" data-foreign-title="Glossary of elementary quantum mechanics"><span class="ilh-page"><a href="/w/index.php?title=%E5%88%9D%E7%BA%A7%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%A6%E8%AF%8D%E6%B1%87&action=edit&redlink=1" class="new" title="初级量子力学词汇（页面不存在）">术语</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Glossary_of_elementary_quantum_mechanics" class="extiw" title="en:Glossary of elementary quantum mechanics"><span lang="en" dir="auto">Glossary of elementary quantum mechanics</span></a></span>）</span></span></li> <li><a href="/wiki/%E7%89%A9%E7%90%86%E5%AD%A6%E5%8F%B2#量子理论" title="物理学史">历史</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;">背景</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/%E7%BB%8F%E5%85%B8%E5%8A%9B%E5%AD%A6" title="经典力学">经典力学</a></li> <li><a href="/wiki/%E8%88%8A%E9%87%8F%E5%AD%90%E8%AB%96" title="舊量子論">舊量子論</a></li> <li><a href="/wiki/%E7%8B%84%E6%8B%89%E5%85%8B%E7%AC%A6%E5%8F%B7" title="狄拉克符号">狄拉克符号</a></li></ul> <div class="hlist" style="margin-left: 0em;"> <ul><li><a href="/wiki/%E5%93%88%E5%AF%86%E9%A1%BF%E7%AE%97%E7%AC%A6" title="哈密顿算符">哈密顿算符</a></li> <li><a href="/wiki/%E5%B9%B2%E6%B6%89_(%E7%89%A9%E7%90%86%E5%AD%A6)" title="干涉 (物理学)">干涉</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;">基本原理</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist" style="margin-left: 0em;"> <ul><li><a href="/wiki/%E4%BA%92%E8%A1%A5%E5%8E%9F%E7%90%86" title="互补原理">互补</a></li> <li><a href="/wiki/%E9%87%8F%E5%AD%90%E9%80%80%E7%9B%B8%E5%B9%B2" title="量子退相干">退相干</a></li> <li><a href="/wiki/%E9%87%8F%E5%AD%90%E7%BA%8F%E7%B5%90" title="量子纏結">纠缠</a></li> <li><a href="/wiki/%E8%83%BD%E7%BA%A7" title="能级">能级</a></li> <li><a href="/wiki/%E9%87%8F%E5%AD%90%E6%B8%AC%E9%87%8F" title="量子測量">测量</a></li> <li><span class="ilh-all" data-orig-title="量子非局域性" data-lang-code="en" data-lang-name="英语" data-foreign-title="Quantum nonlocality"><span class="ilh-page"><a href="/w/index.php?title=%E9%87%8F%E5%AD%90%E9%9D%9E%E5%B1%80%E5%9F%9F%E6%80%A7&action=edit&redlink=1" class="new" title="量子非局域性（页面不存在）">非局域性</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Quantum_nonlocality" class="extiw" title="en:Quantum nonlocality"><span lang="en" dir="auto">Quantum nonlocality</span></a></span>）</span></span></li> <li><a href="/wiki/%E9%87%8F%E5%AD%90%E6%95%B0" title="量子数">量子数</a></li> <li><a href="/wiki/%E9%87%8F%E5%AD%90%E6%85%8B" title="量子態">量子態</a></li> <li><a href="/wiki/%E6%80%81%E5%8F%A0%E5%8A%A0%E5%8E%9F%E7%90%86" title="态叠加原理">态叠加原理</a></li> <li><span class="ilh-all" data-orig-title="量子力学中的对称性" data-lang-code="en" data-lang-name="英语" data-foreign-title="Symmetry in quantum mechanics"><span class="ilh-page"><a href="/w/index.php?title=%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%A6%E4%B8%AD%E7%9A%84%E5%AF%B9%E7%A7%B0%E6%80%A7&action=edit&redlink=1" class="new" title="量子力学中的对称性（页面不存在）">对称性</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Symmetry_in_quantum_mechanics" class="extiw" title="en:Symmetry in quantum mechanics"><span lang="en" dir="auto">Symmetry in quantum mechanics</span></a></span>）</span></span></li> <li><a href="/wiki/%E9%87%8F%E5%AD%90%E7%A9%BF%E9%9A%A7%E6%95%88%E6%87%89" title="量子穿隧效應">量子穿隧效應</a></li> <li><a href="/wiki/%E4%B8%8D%E7%A1%AE%E5%AE%9A%E6%80%A7%E5%8E%9F%E7%90%86" title="不确定性原理">不确定性</a></li> <li><a href="/wiki/%E6%B3%A2%E5%87%BD%E6%95%B0" title="波函数">波函数</a> <ul><li><a href="/wiki/%E6%B3%A2%E5%87%BD%E6%95%B0%E5%9D%8D%E7%BC%A9" title="波函数坍缩">波函数坍缩</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;">实验</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist" style="margin-left: 0em;"> <ul><li><a href="/wiki/%E8%B2%9D%E7%88%BE%E5%AE%9A%E7%90%86%E7%9A%84%E5%AF%A6%E9%A9%97%E9%A9%97%E8%AD%89" title="貝爾定理的實驗驗證">貝爾定理的實驗驗證</a></li> <li><a href="/wiki/%E6%88%B4%E7%B6%AD%E6%A3%AE-%E9%9D%A9%E6%9C%AB%E5%AF%A6%E9%A9%97" title="戴維森-革末實驗">戴維森-革末實驗</a></li> <li><a href="/wiki/%E9%9B%99%E7%B8%AB%E5%AF%A6%E9%A9%97" title="雙縫實驗">雙縫實驗</a></li> <li><a href="/wiki/%E4%BC%8A%E5%88%A9%E6%BE%A4-%E5%A8%81%E5%BE%B7%E6%9B%BC%E7%82%B8%E5%BD%88%E6%B8%AC%E8%A9%A6%E5%95%8F%E9%A1%8C" title="伊利澤-威德曼炸彈測試問題">伊利澤-威德曼炸彈測試問題</a></li> <li><a href="/wiki/%E6%B3%95%E8%98%AD%E5%85%8B-%E8%B5%AB%E8%8C%B2%E5%AF%A6%E9%A9%97" title="法蘭克-赫茲實驗">法蘭克-赫茲實驗</a></li> <li><span class="ilh-all" data-orig-title="Leggett-Garg不平等现象" data-lang-code="en" data-lang-name="英语" data-foreign-title="Leggett–Garg inequality"><span class="ilh-page"><a href="/w/index.php?title=Leggett-Garg%E4%B8%8D%E5%B9%B3%E7%AD%89%E7%8E%B0%E8%B1%A1&action=edit&redlink=1" class="new" title="Leggett-Garg不平等现象（页面不存在）">Leggett-Garg不平等现象</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Leggett%E2%80%93Garg_inequality" class="extiw" title="en:Leggett–Garg inequality"><span lang="en" dir="auto">Leggett–Garg inequality</span></a></span>）</span></span></li> <li><a href="/wiki/%E9%A6%AC%E8%B5%AB-%E6%9B%BE%E5%BE%B7%E7%88%BE%E5%B9%B2%E6%B6%89%E5%84%80" title="馬赫-曾德爾干涉儀">馬赫-曾德爾干涉儀</a></li> <li><a href="/wiki/%E6%B3%A2%E6%99%AE%E5%B0%94%E5%AE%9E%E9%AA%8C" title="波普尔实验">波普尔实验</a></li></ul> </div> <ul><li><a href="/wiki/%E9%87%8F%E5%AD%90%E6%93%A6%E9%99%A4%E5%AF%A6%E9%A9%97" title="量子擦除實驗">量子擦除實驗</a> <ul><li><a href="/wiki/%E5%BB%B6%E9%81%B2%E9%81%B8%E6%93%87%E9%87%8F%E5%AD%90%E6%93%A6%E9%99%A4%E5%AF%A6%E9%A9%97" title="延遲選擇量子擦除實驗">延遲選擇</a></li></ul></li></ul> <div class="hlist" style="margin-left: 0em;"> <ul><li><a href="/wiki/%E8%96%9B%E5%AE%9A%E8%B0%94%E7%8C%AB" title="薛定谔猫">薛定谔猫</a></li> <li><a href="/wiki/%E6%96%BD%E7%89%B9%E6%81%A9-%E6%A0%BC%E6%8B%89%E8%B5%AB%E5%AE%9E%E9%AA%8C" title="施特恩-格拉赫实验">施特恩-格拉赫实验</a></li> <li><a href="/wiki/%E6%83%A0%E5%8B%92%E5%BB%B6%E8%BF%9F%E9%80%89%E6%8B%A9%E5%AE%9E%E9%AA%8C" title="惠勒延迟选择实验">惠勒延迟选择实验</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;">表述</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/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8%E7%9A%84%E6%95%B8%E5%AD%B8%E8%A1%A8%E8%BF%B0" title="量子力學的數學表述">概览</a></li></ul> <div class="hlist" style="margin-left: 0em;"> <ul><li><a href="/wiki/%E6%B5%B7%E6%A3%AE%E5%A0%A1%E7%B9%AA%E6%99%AF" title="海森堡繪景">海森堡繪景</a></li> <li><a href="/wiki/%E7%9B%B8%E4%BA%92%E4%BD%9C%E7%94%A8%E7%B9%AA%E6%99%AF" title="相互作用繪景">相互作用繪景</a></li> <li><a href="/wiki/%E7%9F%A9%E9%99%A3%E5%8A%9B%E5%AD%B8" title="矩陣力學">矩陣力學</a></li> <li><a href="/wiki/%E7%9B%B8%E7%A9%BA%E9%97%B4%E8%A1%A8%E8%BF%B0" title="相空间表述">相空间表述</a></li> <li><a href="/wiki/%E8%96%9B%E4%B8%81%E6%A0%BC%E7%B9%AA%E6%99%AF" title="薛丁格繪景">薛丁格繪景</a></li> <li><a href="/wiki/%E8%B7%AF%E5%BE%91%E7%A9%8D%E5%88%86%E8%A1%A8%E8%BF%B0" title="路徑積分表述">路徑積分表述</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;">方程</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist" style="margin-left: 0em;"> <ul><li><a href="/wiki/%E7%8B%84%E6%8B%89%E5%85%8B%E6%96%B9%E7%A8%8B%E5%BC%8F" title="狄拉克方程式">狄拉克方程式</a></li> <li><a href="/wiki/%E5%85%8B%E8%8E%B1%E5%9B%A0-%E6%88%88%E5%B0%94%E7%99%BB%E6%96%B9%E7%A8%8B" title="克莱因-戈尔登方程">克莱因-戈尔登方程</a></li> <li><a href="/wiki/%E5%8C%85%E7%AB%8B%E6%96%B9%E7%A8%8B%E5%BC%8F" title="包立方程式">包立方程式</a></li> <li><a href="/wiki/%E9%87%8C%E5%BE%B7%E4%BC%AF%E5%85%AC%E5%BC%8F" title="里德伯公式">里德伯公式</a></li> <li><a href="/wiki/%E8%96%9B%E5%AE%9A%E8%B0%94%E6%96%B9%E7%A8%8B" title="薛定谔方程">薛定谔方程</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;">诠释</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/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8%E8%A9%AE%E9%87%8B" title="量子力學詮釋">概览</a></li></ul> <div class="hlist" style="margin-left: 0em;"> <ul><li><span class="ilh-all" data-orig-title="量子贝叶斯主义" data-lang-code="en" data-lang-name="英语" data-foreign-title="Quantum Bayesianism"><span class="ilh-page"><a href="/w/index.php?title=%E9%87%8F%E5%AD%90%E8%B4%9D%E5%8F%B6%E6%96%AF%E4%B8%BB%E4%B9%89&action=edit&redlink=1" class="new" title="量子贝叶斯主义（页面不存在）">量子贝叶斯主义</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Quantum_Bayesianism" class="extiw" title="en:Quantum Bayesianism"><span lang="en" dir="auto">Quantum Bayesianism</span></a></span>）</span></span></li> <li><a href="/wiki/%E4%B8%80%E8%87%B4%E6%80%A7%E5%8E%86%E5%8F%B2" title="一致性历史">一致性历史</a></li> <li><a href="/wiki/%E5%93%A5%E6%9C%AC%E5%93%88%E6%A0%B9%E8%A9%AE%E9%87%8B" title="哥本哈根詮釋">哥本哈根詮釋</a></li> <li><a href="/wiki/%E5%BE%B7%E5%B8%83%E7%BD%97%E6%84%8F-%E7%8E%BB%E5%A7%86%E7%90%86%E8%AE%BA" title="德布罗意-玻姆理论">德布罗意-玻姆理论</a></li> <li><a href="/wiki/%E7%B3%BB%E7%B6%9C%E8%A9%AE%E9%87%8B" title="系綜詮釋">系綜詮釋</a></li> <li><a href="/wiki/%E9%9A%B1%E8%AE%8A%E9%87%8F%E7%90%86%E8%AB%96" title="隱變量理論">隱變量理論</a> <ul><li><span class="ilh-all" data-orig-title="局部隐变量理论" data-lang-code="en" data-lang-name="英语" data-foreign-title="Local hidden-variable theory"><span class="ilh-page"><a href="/w/index.php?title=%E5%B1%80%E9%83%A8%E9%9A%90%E5%8F%98%E9%87%8F%E7%90%86%E8%AE%BA&action=edit&redlink=1" class="new" title="局部隐变量理论（页面不存在）">局部隐变量</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Local_hidden-variable_theory" class="extiw" title="en:Local hidden-variable theory"><span lang="en" dir="auto">Local hidden-variable theory</span></a></span>）</span></span></li></ul></li> <li><a href="/wiki/%E5%A4%9A%E4%B8%96%E7%95%8C%E8%AF%A0%E9%87%8A" title="多世界诠释">多世界诠释</a></li> <li><a href="/wiki/%E5%AE%A2%E8%A7%80%E5%9D%8D%E7%B8%AE%E7%90%86%E8%AB%96" title="客觀坍縮理論">客觀坍縮理論</a></li> <li><span class="ilh-all" data-orig-title="量子逻辑" data-lang-code="en" data-lang-name="英语" data-foreign-title="Quantum logic"><span class="ilh-page"><a href="/w/index.php?title=%E9%87%8F%E5%AD%90%E9%80%BB%E8%BE%91&action=edit&redlink=1" class="new" title="量子逻辑（页面不存在）">量子逻辑</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Quantum_logic" class="extiw" title="en:Quantum logic"><span lang="en" dir="auto">Quantum logic</span></a></span>）</span></span></li> <li><a href="/wiki/%E9%97%9C%E4%BF%82%E6%80%A7%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8" title="關係性量子力學">關係性量子力學</a></li> <li><a href="/wiki/%E4%BA%A4%E6%98%93%E8%A9%AE%E9%87%8B" title="交易詮釋">交易詮釋</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;">高阶</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><span class="ilh-all" data-orig-title="相对论量子力学" data-lang-code="en" data-lang-name="英语" data-foreign-title="Relativistic quantum mechanics"><span class="ilh-page"><a href="/wiki/%E7%9B%B8%E5%B0%8D%E8%AB%96%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8" class="mw-redirect" title="相對論量子力學">相对论量子力学</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Relativistic_quantum_mechanics" class="extiw" title="en:Relativistic quantum mechanics"><span lang="en" dir="auto">Relativistic quantum mechanics</span></a></span>）</span></span></li> <li><a href="/wiki/%E9%87%8F%E5%AD%90%E5%9C%BA%E8%AE%BA" title="量子场论">量子场论</a></li> <li><a href="/wiki/%E9%87%8F%E5%AD%90%E4%BF%A1%E6%81%AF%E7%A7%91%E5%AD%A6" title="量子信息科学">量子信息科学</a></li> <li><a href="/wiki/%E9%87%8F%E5%AD%90%E8%AE%A1%E7%AE%97%E6%9C%BA" title="量子计算机">量子计算</a></li> <li><span class="ilh-all" data-orig-title="量子混沌" data-lang-code="en" data-lang-name="英语" data-foreign-title="Quantum chaos"><span class="ilh-page"><a href="/w/index.php?title=%E9%87%8F%E5%AD%90%E6%B7%B7%E6%B2%8C&action=edit&redlink=1" class="new" title="量子混沌（页面不存在）">量子混沌</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Quantum_chaos" class="extiw" title="en:Quantum chaos"><span lang="en" dir="auto">Quantum chaos</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%AF%86%E5%BA%A6%E7%9F%A9%E9%99%A3" title="密度矩陣">密度矩陣</a></li> <li><span class="ilh-all" data-orig-title="散射理论" data-lang-code="en" data-lang-name="英语" data-foreign-title="Scattering theory"><span class="ilh-page"><a href="/w/index.php?title=%E6%95%A3%E5%B0%84%E7%90%86%E8%AE%BA&action=edit&redlink=1" class="new" title="散射理论（页面不存在）">散射理论</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Scattering_theory" class="extiw" title="en:Scattering theory"><span lang="en" dir="auto">Scattering theory</span></a></span>）</span></span></li> <li><a href="/wiki/%E9%87%8F%E5%AD%90%E7%BB%9F%E8%AE%A1%E5%8A%9B%E5%AD%A6" title="量子统计力学">量子统计力学</a></li> <li><a href="/wiki/%E9%87%8F%E5%AD%90%E6%A9%9F%E5%99%A8%E5%AD%B8%E7%BF%92" title="量子機器學習">量子機器學習</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;">科学家</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist" style="margin-left: 0em;"> <ul><li><a href="/wiki/%E4%BA%9A%E5%9F%BA%E5%B0%94%C2%B7%E9%98%BF%E5%93%88%E7%BD%97%E8%AF%BA%E5%A4%AB" title="亚基尔·阿哈罗诺夫">阿哈罗诺夫</a></li> <li><a href="/wiki/%E7%B4%84%E7%BF%B0%C2%B7%E8%B2%9D%E7%88%BE" class="mw-disambig" title="約翰·貝爾">貝爾</a></li> <li><a href="/wiki/%E6%B1%89%E6%96%AF%C2%B7%E8%B4%9D%E7%89%B9" title="汉斯·贝特">贝特</a></li> <li><a href="/wiki/%E5%B8%95%E7%89%B9%E9%87%8C%E5%85%8B%C2%B7%E5%B8%83%E8%8E%B1%E5%85%8B%E7%89%B9" title="帕特里克·布莱克特">布莱克特</a></li> <li><a href="/wiki/%E8%B4%B9%E5%88%A9%E5%85%8B%E6%96%AF%C2%B7%E5%B8%83%E6%B4%9B%E8%B5%AB" title="费利克斯·布洛赫">布洛赫</a></li> <li><a href="/wiki/%E6%88%B4%E7%BB%B4%C2%B7%E7%8E%BB%E5%A7%86" title="戴维·玻姆">玻姆</a></li> <li><a href="/wiki/%E5%B0%BC%E5%B0%94%E6%96%AF%C2%B7%E7%8E%BB%E5%B0%94" title="尼尔斯·玻尔">玻尔</a></li> <li><a href="/wiki/%E9%A9%AC%E5%85%8B%E6%96%AF%C2%B7%E7%8E%BB%E6%81%A9" title="马克斯·玻恩">玻恩</a></li> <li><a href="/wiki/%E8%96%A9%E7%89%B9%E5%BB%B6%E5%BE%B7%E6%8B%89%C2%B7%E7%B4%8D%E7%89%B9%C2%B7%E7%8E%BB%E8%89%B2" title="薩特延德拉·納特·玻色">玻色</a></li> <li><a href="/wiki/%E8%B7%AF%E6%98%93%C2%B7%E5%BE%B7%E5%B8%83%E7%BD%97%E6%84%8F" title="路易·德布罗意">德布罗意</a></li> <li><a href="/wiki/%E9%98%BF%E7%91%9F%C2%B7%E5%BA%B7%E6%99%AE%E9%A1%BF" title="阿瑟·康普顿">康普顿</a></li> <li><a href="/wiki/%E4%BF%9D%E7%BD%97%C2%B7%E7%8B%84%E6%8B%89%E5%85%8B" title="保罗·狄拉克">狄拉克</a></li> <li><a href="/wiki/%E5%85%8B%E6%9E%97%E9%A1%BF%C2%B7%E6%88%B4%E7%BB%B4%E5%AD%99" title="克林顿·戴维孙">戴维孙</a></li> <li><a href="/wiki/%E5%BD%BC%E5%BE%97%C2%B7%E5%BE%B7%E6%8B%9C" title="彼得·德拜">德拜</a></li> <li><a href="/wiki/%E4%BF%9D%E7%BD%97%C2%B7%E5%9F%83%E4%BC%A6%E8%B4%B9%E6%96%AF%E7%89%B9" title="保罗·埃伦费斯特">埃伦费斯特</a></li> <li><a href="/wiki/%E9%98%BF%E5%B0%94%E4%BC%AF%E7%89%B9%C2%B7%E7%88%B1%E5%9B%A0%E6%96%AF%E5%9D%A6" title="阿尔伯特·爱因斯坦">爱因斯坦</a></li> <li><a href="/wiki/%E4%BC%91%C2%B7%E8%89%BE%E5%BC%97%E9%9B%B7%E7%89%B9%E4%B8%89%E4%B8%96" title="休·艾弗雷特三世">艾弗雷特三世</a></li> <li><a href="/wiki/%E5%BC%97%E6%8B%89%E5%9F%BA%E7%B1%B3%E5%B0%94%C2%B7%E7%A6%8F%E5%85%8B" title="弗拉基米尔·福克">福克</a></li> <li><a href="/wiki/%E6%81%A9%E9%87%8C%E7%A7%91%C2%B7%E8%B4%B9%E7%B1%B3" title="恩里科·费米">费米</a></li> <li><a href="/wiki/%E7%90%86%E6%9F%A5%E5%BE%B7%C2%B7%E8%B2%BB%E6%9B%BC" title="理查德·費曼">費曼</a></li> <li><a href="/wiki/%E7%BD%97%E4%BC%8A%C2%B7%E6%A0%BC%E5%8A%B3%E4%BC%AF" title="罗伊·格劳伯">格劳伯</a></li> <li><a href="/wiki/%E5%8F%A4%E8%8C%A8%E5%A8%81%E5%8B%92" title="古茨威勒">古茨威勒</a></li> <li><a href="/wiki/%E7%BB%B4%E5%B0%94%E7%BA%B3%C2%B7%E6%B5%B7%E6%A3%AE%E5%A0%A1" title="维尔纳·海森堡">海森堡</a></li> <li><a href="/wiki/%E5%A4%A7%E5%8D%AB%C2%B7%E5%B8%8C%E5%B0%94%E4%BC%AF%E7%89%B9" title="大卫·希尔伯特">希尔伯特</a></li> <li><a href="/wiki/%E5%B8%95%E6%96%AF%E5%A4%B8%E5%B0%94%C2%B7%E7%BA%A6%E5%B0%94%E6%97%A6" class="mw-redirect" title="帕斯夸尔·约尔旦">约尔旦</a></li> <li><a href="/wiki/%E6%B1%89%E6%96%AF%C2%B7%E5%85%8B%E5%96%87%E6%9C%AB" title="汉斯·克喇末">克喇末</a></li> <li><a href="/wiki/%E6%B2%83%E5%B0%94%E5%A4%AB%E5%86%88%C2%B7%E6%B3%A1%E5%88%A9" title="沃尔夫冈·泡利">泡利</a></li> <li><a href="/wiki/%E5%A8%81%E5%88%A9%E6%96%AF%C2%B7%E5%85%B0%E5%A7%86" title="威利斯·兰姆">兰姆</a></li> <li><a href="/wiki/%E5%88%97%E5%A4%AB%C2%B7%E6%9C%97%E9%81%93" title="列夫·朗道">朗道</a></li> <li><a href="/wiki/%E9%A9%AC%E5%85%8B%E6%96%AF%C2%B7%E5%86%AF%C2%B7%E5%8A%B3%E5%8E%84" title="马克斯·冯·劳厄">劳厄</a></li> <li><a href="/wiki/%E4%BA%A8%E5%88%A9%C2%B7%E8%8E%AB%E5%A1%9E%E8%8E%B1" title="亨利·莫塞莱">莫塞莱</a></li> <li><a href="/wiki/%E7%BD%97%E4%BC%AF%E7%89%B9%C2%B7%E5%AF%86%E7%AB%8B%E6%A0%B9" title="罗伯特·密立根">密立根</a></li> <li><a href="/wiki/%E6%B5%B7%E5%85%8B%C2%B7%E5%8D%A1%E6%9C%AB%E6%9E%97%C2%B7%E6%98%82%E5%85%A7%E6%96%AF" title="海克·卡末林·昂內斯">昂內斯</a></li> <li><a href="/wiki/%E9%A9%AC%E5%85%8B%E6%96%AF%C2%B7%E6%99%AE%E6%9C%97%E5%85%8B" title="马克斯·普朗克">普朗克</a></li> <li><a href="/wiki/%E4%BC%8A%E8%A5%BF%E5%A4%9A%C2%B7%E6%8B%89%E6%AF%94" title="伊西多·拉比">拉比</a></li> <li><a href="/wiki/%E9%92%B1%E5%BE%B7%E6%8B%89%E5%A1%9E%E5%8D%A1%E6%8B%89%C2%B7%E6%8B%89%E6%9B%BC" title="钱德拉塞卡拉·拉曼">拉曼</a></li> <li><a href="/wiki/%E7%BA%A6%E7%BF%B0%E5%86%85%E6%96%AF%C2%B7%E9%87%8C%E5%BE%B7%E4%BC%AF" title="约翰内斯·里德伯">里德伯</a></li> <li><a href="/wiki/%E5%9F%83%E5%B0%94%E6%B8%A9%C2%B7%E8%96%9B%E5%AE%9A%E8%B0%94" title="埃尔温·薛定谔">薛定谔</a></li> <li><span class="ilh-all" data-orig-title="米歇尔·西蒙斯" data-lang-code="en" data-lang-name="英语" data-foreign-title="Michelle Simmons"><span class="ilh-page"><a href="/w/index.php?title=%E7%B1%B3%E6%AD%87%E5%B0%94%C2%B7%E8%A5%BF%E8%92%99%E6%96%AF&action=edit&redlink=1" class="new" title="米歇尔·西蒙斯（页面不存在）">米歇尔·西蒙斯</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Michelle_Simmons" class="extiw" title="en:Michelle Simmons"><span lang="en" dir="auto">Michelle Simmons</span></a></span>）</span></span></li> <li><a href="/wiki/%E9%98%BF%E8%AB%BE%C2%B7%E7%B4%A2%E6%9C%AB%E8%8F%B2" title="阿諾·索末菲">索末菲</a></li> <li><a href="/wiki/%E7%BA%A6%E7%BF%B0%C2%B7%E5%86%AF%C2%B7%E8%AF%BA%E4%BC%8A%E6%9B%BC" class="mw-redirect" title="约翰·冯·诺伊曼">冯·诺伊曼</a></li> <li><a href="/wiki/%E8%B5%AB%E5%B0%94%E6%9B%BC%C2%B7%E5%A4%96%E5%B0%94" title="赫尔曼·外尔">外尔</a></li> <li><a href="/wiki/%E5%A8%81%E5%BB%89%C2%B7%E7%BB%B4%E6%81%A9" title="威廉·维恩">维恩</a></li> <li><a href="/wiki/%E5%B0%A4%E9%87%91%C2%B7%E7%BB%B4%E6%A0%BC%E7%BA%B3" title="尤金·维格纳">维格纳</a></li> <li><a href="/wiki/%E5%BD%BC%E5%BE%97%C2%B7%E5%A1%9E%E6%9B%BC" title="彼得·塞曼">塞曼</a></li> <li><a href="/wiki/%E5%AE%89%E4%B8%9C%C2%B7%E8%94%A1%E6%9E%97%E6%A0%BC" title="安东·蔡林格">蔡林格</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar" style="line-height:1.6;border-top:1px solid #aaa;padding-top:0.1em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><style data-mw-deduplicate="TemplateStyles:r84244141">.mw-parser-output .navbar{display:inline;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:110%;margin:0 8em}.mw-parser-output .navbar-ct-mini{font-size:110%;margin:0 5em}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:%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8" title="Template:量子力學"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8" title="Template talk:量子力學"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8" title="Special:编辑页面/Template:量子力學"><abbr title="编辑该模板">编</abbr></a></li></ul></div></td></tr></tbody></table> <p>在<a href="/wiki/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8" class="mw-redirect" title="量子力學">量子力學</a>裏，一個粒子因為<a href="/wiki/%E8%87%AA%E6%97%8B" title="自旋">自旋</a>與<a href="/wiki/%E8%BB%8C%E9%81%93%E9%81%8B%E5%8B%95" title="軌道運動">軌道運動</a>而產生的作用，稱為<b>自旋-軌道作用</b>（英語：<span lang="en"><b>Spin–orbit interaction</b></span>），也稱作<b>自旋-軌道效應</b>或<b>自旋-軌道耦合</b>。最著名的例子是<a href="/wiki/%E9%9B%BB%E5%AD%90" class="mw-redirect" title="電子">電子</a><a href="/wiki/%E8%83%BD%E7%B4%9A" class="mw-redirect" title="能級">能級</a>的位移。電子移動經過<a href="/wiki/%E5%8E%9F%E5%AD%90%E6%A0%B8" title="原子核">原子核</a>的<a href="/wiki/%E9%9B%BB%E5%A0%B4" title="電場">電場</a>時，會產生<a href="/wiki/%E9%9B%BB%E7%A3%81%E4%BD%9C%E7%94%A8" class="mw-redirect" title="電磁作用">電磁作用</a>．電子的自旋與這電磁作用的耦合，形成了自旋-軌道作用。<a href="/wiki/%E8%AD%9C%E7%B7%9A" title="譜線">譜線</a>分裂實驗明顯地偵測到電子能級的位移，證實了自旋-軌道作用理論的正確性。另外一個類似的例子是<a href="/wiki/%E5%8E%9F%E5%AD%90%E6%A0%B8" title="原子核">原子核</a><a href="/wiki/%E6%A0%B8%E6%AE%BC%E5%B1%A4%E6%A8%A1%E5%9E%8B" title="核殼層模型">殼層模型</a><a href="/wiki/%E8%83%BD%E7%B4%9A" class="mw-redirect" title="能級">能級</a>的位移。 </p><p><a href="/wiki/%E5%8D%8A%E5%B0%8E%E9%AB%94" class="mw-redirect" title="半導體">半導體</a>或其它新穎材料常常會涉及電子的自旋-軌道效應。<a href="/wiki/%E8%87%AA%E6%97%8B%E9%9B%BB%E5%AD%90%E5%AD%B8" title="自旋電子學">自旋電子學</a>專門研究與應用這方面的問題。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="電子的自旋-軌道作用"><span id=".E9.9B.BB.E5.AD.90.E7.9A.84.E8.87.AA.E6.97.8B-.E8.BB.8C.E9.81.93.E4.BD.9C.E7.94.A8"></span>電子的自旋-軌道作用</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E8%87%AA%E6%97%8B-%E8%BB%8C%E9%81%93%E4%BD%9C%E7%94%A8&action=edit&section=1" title="编辑章节：電子的自旋-軌道作用"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>在這篇文章裏，會以相當簡單與公式化的方式，詳細地講解一個束縛於<a href="/wiki/%E5%8E%9F%E5%AD%90" title="原子">原子</a>內的電子的自旋-軌道作用理論。這會用到<a href="/wiki/%E9%9B%BB%E7%A3%81%E5%AD%B8" class="mw-redirect" title="電磁學">電磁學</a>、<a href="/wiki/%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8" class="mw-redirect" title="量子力學">非相對論性量子力學</a>、<a href="/wiki/%E5%BE%AE%E6%93%BE%E7%90%86%E8%AB%96_(%E9%87%8F%E5%AD%90%E5%8A%9B%E5%AD%B8)" title="微擾理論 (量子力學)">一階微擾理論</a>。這自旋-軌道作用理論給出的答案，雖然與實驗結果並不完全相同，但相當的符合。更嚴謹的導引應該從<a href="/wiki/%E7%8B%84%E6%8B%89%E5%85%8B%E6%96%B9%E7%A8%8B%E5%BC%8F" title="狄拉克方程式">狄拉克方程式</a>開始，也會求得相同的答案。若想得到更準確的答案，則必須用<a href="/wiki/%E9%87%8F%E5%AD%90%E9%9B%BB%E5%8B%95%E5%8A%9B%E5%AD%B8" title="量子電動力學">量子電動力學</a>來計算微小的修正。這兩種方法都在本條目範圍之外。 </p> <div class="mw-heading mw-heading3"><h3 id="磁場"><span id=".E7.A3.81.E5.A0.B4"></span>磁場</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E8%87%AA%E6%97%8B-%E8%BB%8C%E9%81%93%E4%BD%9C%E7%94%A8&action=edit&section=2" title="编辑章节：磁場"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>雖然在原子核的<a href="/wiki/%E9%9D%9C%E6%AD%A2%E5%8F%83%E8%80%83%E7%B3%BB" title="靜止參考系">靜止參考系</a> (<span lang="en">rest frame</span>) ，並沒有作用在電子上的磁場；在電子的靜止參考系，有作用在電子上的磁場存在。暫時假設電子的靜止參考系為<a href="/wiki/%E6%83%AF%E6%80%A7%E5%8F%82%E8%80%83%E7%B3%BB" title="惯性参考系">慣性參考系</a>，則根據<a href="/wiki/%E7%8B%B9%E7%BE%A9%E7%9B%B8%E5%B0%8D%E8%AB%96" class="mw-redirect" title="狹義相對論">狹義相對論</a><sup id="cite_ref-French1968_1-0" class="reference"><a href="#cite_note-French1968-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>，磁場 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c90d341e1b108a9aae7335c5ed1a1e849c491833" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.288ex; height:2.176ex;" alt="{\displaystyle \mathbf {B} \,\!}"></span> 是 </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 \mathbf {B} =-\,{\frac {\mathbf {v} \times \mathbf {E} }{c^{2}}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mo>−</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =-\,{\frac {\mathbf {v} \times \mathbf {E} }{c^{2}}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738833e0bad31083fdc7bd5ff49f749fd3167008" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; margin-right: -0.387ex; width:14.426ex; height:5.509ex;" alt="{\displaystyle \mathbf {B} =-\,{\frac {\mathbf {v} \times \mathbf {E} }{c^{2}}}\,\!}"></span> ；<span style="position:absolute;right:15%">(1)</span></dd></dl> <p>其中，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df6e8492015c331122565c580264dcc31144461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.798ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} \,\!}"></span> 是電子的速度，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/155fe529b7afbd9f951fcebbfeb220be27bdad0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.144ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} \,\!}"></span> 是電子運動經過的電場，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a26f20ddaaf267143f0e92c1648a49fbe5687e6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.394ex; height:1.676ex;" alt="{\displaystyle c\,\!}"></span> 是<a href="/wiki/%E5%85%89%E9%80%9F" title="光速">光速</a>。 </p><p>以質子的位置為<a href="/wiki/%E5%8E%9F%E9%BB%9E" title="原點">原點</a>，則從<a href="/wiki/%E8%B3%AA%E5%AD%90" title="質子">質子</a>產生的電場是 </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 \mathbf {E} ={\frac {Ze}{4\pi \epsilon _{0}r^{2}}}{\hat {\mathbf {r} }}={\frac {Ze}{4\pi \epsilon _{0}r^{3}}}\mathbf {r} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>e</mi> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>e</mi> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ={\frac {Ze}{4\pi \epsilon _{0}r^{2}}}{\hat {\mathbf {r} }}={\frac {Ze}{4\pi \epsilon _{0}r^{3}}}\mathbf {r} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3407262ff35265eda585e48bb0faf0b6f9716c78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-right: -0.387ex; width:25.469ex; height:5.843ex;" alt="{\displaystyle \mathbf {E} ={\frac {Ze}{4\pi \epsilon _{0}r^{2}}}{\hat {\mathbf {r} }}={\frac {Ze}{4\pi \epsilon _{0}r^{3}}}\mathbf {r} \,\!}"></span> ；</dd></dl> <p>其中，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d77b3560e6c0fc0484ec0622d1b424af08d9ddc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.067ex; height:2.176ex;" alt="{\displaystyle Z\,\!}"></span> 是質子數量（<a href="/wiki/%E5%8E%9F%E5%AD%90%E5%BA%8F%E6%95%B8" class="mw-redirect" title="原子序數">原子序數</a>），<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32308a9beabe5e8a2c683aa285cfd6e8abafb02c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.471ex; height:1.676ex;" alt="{\displaystyle e\,\!}"></span> 是<a href="/wiki/%E5%96%AE%E4%BD%8D%E9%9B%BB%E8%8D%B7%E9%87%8F" class="mw-redirect" title="單位電荷量">單位電荷量</a>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{0}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{0}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/320886200372fe77d6fea272c34564def3d8cca9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:2.385ex; height:2.009ex;" alt="{\displaystyle \epsilon _{0}\,\!}"></span> 是<a href="/wiki/%E7%9C%9F%E7%A9%BA%E7%94%B5%E5%AE%B9%E7%8E%87" title="真空电容率">真空電容率</a>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {r}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {r}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca11f8c534f8c0aa34f6b7f46877d8fdd6c4d0bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.679ex; height:2.176ex;" alt="{\displaystyle {\hat {r}}\,\!}"></span> 是徑向單位向量，<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\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86937851b7cb70e8802197a7b5ddcf39c0b02445" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.436ex; height:1.676ex;" alt="{\displaystyle r\,\!}"></span> 是徑向距離，徑向向量 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a1ffb0f7dee0c7f11111906fe55becd542d4d82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.489ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} \,\!}"></span> 是電子的位置。 </p><p>電子的<a href="/wiki/%E5%8B%95%E9%87%8F" class="mw-redirect" title="動量">動量</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60a9e20e22328367d07629008314cf17f0366638" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:1.872ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} \,\!}"></span> 是 </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 \mathbf {p} =m\mathbf {v} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =m\mathbf {v} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf4871e181e4bd8693eec4d2690d7bac63473ed3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:8.422ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} =m\mathbf {v} \,\!}"></span> ；</dd></dl> <p>其中，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/030f4b5505401a882fc1ae0ad190d41639b62eed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.427ex; height:1.676ex;" alt="{\displaystyle m\,\!}"></span> 是電子的質量。 </p><p>所以，作用於電子的磁場是 </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 \mathbf {B} ={\frac {Ze}{4\pi \epsilon _{0}mc^{2}r^{3}}}\,\mathbf {r} \times \mathbf {p} ={\frac {Ze}{4\pi \epsilon _{0}mc^{2}r^{3}}}\,\mathbf {L} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>e</mi> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>e</mi> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} ={\frac {Ze}{4\pi \epsilon _{0}mc^{2}r^{3}}}\,\mathbf {r} \times \mathbf {p} ={\frac {Ze}{4\pi \epsilon _{0}mc^{2}r^{3}}}\,\mathbf {L} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73d6dd0ae84bd6b2f5967a622659623a1932a882" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-right: -0.387ex; width:39.362ex; height:5.843ex;" alt="{\displaystyle \mathbf {B} ={\frac {Ze}{4\pi \epsilon _{0}mc^{2}r^{3}}}\,\mathbf {r} \times \mathbf {p} ={\frac {Ze}{4\pi \epsilon _{0}mc^{2}r^{3}}}\,\mathbf {L} \,\!}"></span> ；<span style="position:absolute;right:15%">(2)</span></dd></dl> <p>其中，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b6839f8ca2690529910c38ca9c4b1d72f993ecb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.995ex; height:2.176ex;" alt="{\displaystyle \mathbf {L} \,\!}"></span> 是<a href="/wiki/%E8%A7%92%E5%8B%95%E9%87%8F" class="mw-redirect" title="角動量">角動量</a>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1fea6a7f63a90a3b4b52c9745b7bbf980954738" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:10.522ex; height:2.509ex;" alt="{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} \,\!}"></span> 。 </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c90d341e1b108a9aae7335c5ed1a1e849c491833" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.288ex; height:2.176ex;" alt="{\displaystyle \mathbf {B} \,\!}"></span> 是一個正值因子乘以 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b6839f8ca2690529910c38ca9c4b1d72f993ecb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.995ex; height:2.176ex;" alt="{\displaystyle \mathbf {L} \,\!}"></span> ，也就是說，磁場與電子的軌道角動量平行。 </p> <div class="mw-heading mw-heading3"><h3 id="磁矩"><span id=".E7.A3.81.E7.9F.A9"></span>磁矩</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E8%87%AA%E6%97%8B-%E8%BB%8C%E9%81%93%E4%BD%9C%E7%94%A8&action=edit&section=3" title="编辑章节：磁矩"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>電子自旋的<a href="/wiki/%E7%A3%81%E7%9F%A9" title="磁矩">磁矩</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\mu }}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\mu }}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b1f46c7a806fd5b03b79273638325893ec3bb42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:2.033ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {\mu }}\,\!}"></span> 是 </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 {\boldsymbol {\mu }}=\gamma \,\mathbf {S} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> <mo>=</mo> <mi>γ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\mu }}=\gamma \,\mathbf {S} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ecea1ddee0dc0fc2ac55575601186db537f56dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:8.266ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {\mu }}=\gamma \,\mathbf {S} \,\!}"></span> ；</dd></dl> <p>其中，<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 \gamma ={\frac {g_{s}q_{e}}{2m}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\frac {g_{s}q_{e}}{2m}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3399fb4221079f7d7c2ef0eb7323776870d015d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:9.732ex; height:4.843ex;" alt="{\displaystyle \gamma ={\frac {g_{s}q_{e}}{2m}}\,\!}"></span> 是<a href="/wiki/%E6%97%8B%E7%A3%81%E6%AF%94" title="旋磁比">旋磁比</a> (<span lang="en">gyromagnetic ratio</span>) ，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {S} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f664a468992732e87c225fa92d078732756a2ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.872ex; height:2.176ex;" alt="{\displaystyle \mathbf {S} \,\!}"></span> 是自旋角动量，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{s}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{s}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9028626b4b26a89cbeb0e58364e7326e4ce032" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:2.499ex; height:2.009ex;" alt="{\displaystyle g_{s}\,\!}"></span> 是<a href="/wiki/%E6%9C%97%E5%BE%B7g%E5%9B%A0%E5%AD%90" title="朗德g因子">朗德g因子</a>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{e}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{e}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92eaf9ad821a1e6a579191f9bd23379226d02ae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:2.423ex; height:2.009ex;" alt="{\displaystyle q_{e}\,\!}"></span> 是<a href="/wiki/%E7%94%B5%E8%8D%B7%E9%87%8F" class="mw-redirect" title="电荷量">電荷量</a>。 </p><p>電子的<a href="/wiki/%E6%9C%97%E5%BE%B7g%E5%9B%A0%E5%AD%90" title="朗德g因子">朗德g因子</a>（g-factor）是 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4646d6f5a2ff38f91d8eeefddc4a7463ea123ea1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle 2\,\!}"></span> ，電荷量是 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -e\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>e</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -e\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d17457c87ae2fca60bd2176f2be58aeeb5df64c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.387ex; width:3.279ex; height:2.176ex;" alt="{\displaystyle -e\,\!}"></span> 。所以， </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 {\boldsymbol {\mu }}=-{\frac {e}{m}}\mathbf {S} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>e</mi> <mi>m</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\mu }}=-{\frac {e}{m}}\mathbf {S} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4303396b93ab643314328f405aca15473a39d91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:11.301ex; height:4.676ex;" alt="{\displaystyle {\boldsymbol {\mu }}=-{\frac {e}{m}}\mathbf {S} \,\!}"></span> 。<span style="position:absolute;right:15%">(3)</span></dd></dl> <p>電子的磁矩與自旋反平行。 </p> <div class="mw-heading mw-heading3"><h3 id="哈密頓量微擾項目"><span id=".E5.93.88.E5.AF.86.E9.A0.93.E9.87.8F.E5.BE.AE.E6.93.BE.E9.A0.85.E7.9B.AE"></span>哈密頓量微擾項目</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E8%87%AA%E6%97%8B-%E8%BB%8C%E9%81%93%E4%BD%9C%E7%94%A8&action=edit&section=4" title="编辑章节：哈密頓量微擾項目"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>自旋-軌道作用的<a href="/wiki/%E5%93%88%E5%AF%86%E9%A0%93%E9%87%8F" class="mw-redirect" title="哈密頓量">哈密頓量</a>微擾項目是 </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 H'=-{\boldsymbol {\mu }}\cdot \mathbf {B} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mo>′</mo> </msup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H'=-{\boldsymbol {\mu }}\cdot \mathbf {B} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568f78232d13a6e94c31b7cd32b9c247ecdf7c05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:13.308ex; height:2.843ex;" alt="{\displaystyle H'=-{\boldsymbol {\mu }}\cdot \mathbf {B} \,\!}"></span> 。</dd></dl> <p>代入 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\mu }}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\mu }}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b1f46c7a806fd5b03b79273638325893ec3bb42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:2.033ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {\mu }}\,\!}"></span> 的公式 (3) 和 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c90d341e1b108a9aae7335c5ed1a1e849c491833" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.288ex; height:2.176ex;" alt="{\displaystyle \mathbf {B} \,\!}"></span> 的公式(2)，經過一番運算，可以得到 </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 H'={\frac {Ze^{2}}{4\pi \epsilon _{0}m^{2}c^{2}}}\ {\frac {\mathbf {L} \cdot \mathbf {S} }{r^{3}}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H'={\frac {Ze^{2}}{4\pi \epsilon _{0}m^{2}c^{2}}}\ {\frac {\mathbf {L} \cdot \mathbf {S} }{r^{3}}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c014d856b2f47c95a6a443e01b6de5ec5e32a120" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-right: -0.387ex; width:22.948ex; height:6.343ex;" alt="{\displaystyle H'={\frac {Ze^{2}}{4\pi \epsilon _{0}m^{2}c^{2}}}\ {\frac {\mathbf {L} \cdot \mathbf {S} }{r^{3}}}\,\!}"></span></dd></dl> <p>一直到現在，都還沒有考慮到電子靜止坐標乃非慣性坐標。這事實引發的效應稱為<a href="/wiki/%E6%89%98%E9%A9%AC%E6%96%AF%E8%BF%9B%E5%8A%A8" class="mw-redirect" title="托马斯进动">托馬斯進動</a> (<span lang="en">Thomas precession</span>) 。因為這效應，必須添加因子 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/2\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/2\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/238d7181da7b3b1ba2d59ae58e5cfebc29ad1ca1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:3.874ex; height:2.843ex;" alt="{\displaystyle 1/2\,\!}"></span> 在公式裏。所以， </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 H'={\frac {Ze^{2}}{8\pi \epsilon _{0}m^{2}c^{2}}}\ {\frac {\mathbf {L} \cdot \mathbf {S} }{r^{3}}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H'={\frac {Ze^{2}}{8\pi \epsilon _{0}m^{2}c^{2}}}\ {\frac {\mathbf {L} \cdot \mathbf {S} }{r^{3}}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfaa9ca4507e8d83a07e3f97d69fb8dd7081a32c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-right: -0.387ex; width:22.948ex; height:6.343ex;" alt="{\displaystyle H'={\frac {Ze^{2}}{8\pi \epsilon _{0}m^{2}c^{2}}}\ {\frac {\mathbf {L} \cdot \mathbf {S} }{r^{3}}}\,\!}"></span> 。</dd></dl> <div class="mw-heading mw-heading3"><h3 id="能級位移"><span id=".E8.83.BD.E7.B4.9A.E4.BD.8D.E7.A7.BB"></span>能級位移</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E8%87%AA%E6%97%8B-%E8%BB%8C%E9%81%93%E4%BD%9C%E7%94%A8&action=edit&section=5" title="编辑章节：能級位移"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>在準備好了自旋-軌道作用的哈密頓量微擾項目以後，現在可以估算這項目會造成的能量位移。特別地，想要找到 <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_{0}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed0c442b4d740f2d8545c9959fcc55dcbf2800b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:3.373ex; height:2.509ex;" alt="{\displaystyle H_{0}\,\!}"></span> 的<a href="/wiki/%E6%9C%AC%E5%BE%B5%E5%87%BD%E6%95%B8" title="本徵函數">本徵函數</a>形成的<a href="/wiki/%E5%9F%BA%E5%BA%95" class="mw-redirect" title="基底">基底</a>，使 <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"> <msup> <mi>H</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </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/90a1588ceee45686d430487d2bc541eec5e645b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:3.175ex; height:2.509ex;" alt="{\displaystyle H'\,\!}"></span> 能夠<a href="/wiki/%E5%8F%AF%E5%AF%B9%E8%A7%92%E5%8C%96%E7%9F%A9%E9%98%B5" title="可对角化矩阵">對角化</a>。為了找到這基底，先定義總角動量算符 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/171a3da4b70b33db97722a689fa121eb83d7f079" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.768ex; height:2.176ex;" alt="{\displaystyle \mathbf {J} \,\!}"></span> ： </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 \mathbf {J} =\mathbf {L} +\mathbf {S} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} =\mathbf {L} +\mathbf {S} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/518d23b23c173d46b31377eeca987f70eb2d60a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.387ex; width:10.8ex; height:2.343ex;" alt="{\displaystyle \mathbf {J} =\mathbf {L} +\mathbf {S} \,\!}"></span> 。</dd></dl> <p>總角動量算符與自己的內積是 </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 \mathbf {J} ^{2}=\mathbf {L} ^{2}+\mathbf {S} ^{2}+2\mathbf {L} \cdot \mathbf {S} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} ^{2}=\mathbf {L} ^{2}+\mathbf {S} ^{2}+2\mathbf {L} \cdot \mathbf {S} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e1ee4af5030f274bb7a021757bd051ed6a7b279" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.387ex; width:22.739ex; height:2.843ex;" alt="{\displaystyle \mathbf {J} ^{2}=\mathbf {L} ^{2}+\mathbf {S} ^{2}+2\mathbf {L} \cdot \mathbf {S} \,\!}"></span> 。</dd></dl> <p>所以， </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 \mathbf {L} \cdot \mathbf {S} ={1 \over 2}(\mathbf {J} ^{2}-\mathbf {L} ^{2}-\mathbf {S} ^{2})\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} \cdot \mathbf {S} ={1 \over 2}(\mathbf {J} ^{2}-\mathbf {L} ^{2}-\mathbf {S} ^{2})\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e03957e6edc8546598f60fc7e6904cf45635fd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:25.384ex; height:5.176ex;" alt="{\displaystyle \mathbf {L} \cdot \mathbf {S} ={1 \over 2}(\mathbf {J} ^{2}-\mathbf {L} ^{2}-\mathbf {S} ^{2})\,\!}"></span> 。</dd></dl> <p>請注意 <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"> <msup> <mi>H</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </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/90a1588ceee45686d430487d2bc541eec5e645b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:3.175ex; height:2.509ex;" alt="{\displaystyle H'\,\!}"></span> 與 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b6839f8ca2690529910c38ca9c4b1d72f993ecb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.995ex; height:2.176ex;" alt="{\displaystyle \mathbf {L} \,\!}"></span> 互相不<a href="/wiki/%E5%B0%8D%E6%98%93%E7%AE%97%E7%AC%A6" class="mw-redirect" title="對易算符">對易</a>， <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H'\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </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/90a1588ceee45686d430487d2bc541eec5e645b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:3.175ex; height:2.509ex;" alt="{\displaystyle H'\,\!}"></span> 與 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {S} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f664a468992732e87c225fa92d078732756a2ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.872ex; height:2.176ex;" alt="{\displaystyle \mathbf {S} \,\!}"></span> 互相不對易。讀者可以很容易地證明這兩個事實。由於這兩個事實，<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_{0}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed0c442b4d740f2d8545c9959fcc55dcbf2800b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:3.373ex; height:2.509ex;" alt="{\displaystyle H_{0}\,\!}"></span> 與 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b6839f8ca2690529910c38ca9c4b1d72f993ecb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.995ex; height:2.176ex;" alt="{\displaystyle \mathbf {L} \,\!}"></span> 的共同本徵函數不能被當做零微擾<a href="/wiki/%E6%B3%A2%E5%87%BD%E6%95%B8" class="mw-redirect" title="波函數">波函數</a>，用來計算一階能量位移 <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^{(1)}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{(1)}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/926aabe89651909a31c6b3294d2b3457d00f3853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:4.515ex; height:2.843ex;" alt="{\displaystyle E^{(1)}\,\!}"></span> 。<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed0c442b4d740f2d8545c9959fcc55dcbf2800b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:3.373ex; height:2.509ex;" alt="{\displaystyle H_{0}\,\!}"></span> 與 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {S} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f664a468992732e87c225fa92d078732756a2ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.872ex; height:2.176ex;" alt="{\displaystyle \mathbf {S} \,\!}"></span> 的共同本徵函數也不能被當做零微擾波函數，用來計算一階能量位移 <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^{(1)}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{(1)}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/926aabe89651909a31c6b3294d2b3457d00f3853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:4.515ex; height:2.843ex;" alt="{\displaystyle E^{(1)}\,\!}"></span> 。可是， <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"> <msup> <mi>H</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </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/90a1588ceee45686d430487d2bc541eec5e645b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:3.175ex; height:2.509ex;" alt="{\displaystyle H'\,\!}"></span> 、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{2}\,\!}"> <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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J^{2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abf017e5d94bf79cbea10f30307f458e2d57f3c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.967ex; height:2.676ex;" alt="{\displaystyle J^{2}\,\!}"></span> 、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}\,\!}"> <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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e24e2e5cae805e283d9279de6457d807918459c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:3.024ex; height:2.676ex;" alt="{\displaystyle L^{2}\,\!}"></span> 、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f8f4b8894e9616d347420e9d6d5e5db3eac8061" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.963ex; height:2.676ex;" alt="{\displaystyle S^{2}\,\!}"></span> ，這四個算符都互相對易。<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_{0}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed0c442b4d740f2d8545c9959fcc55dcbf2800b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:3.373ex; height:2.509ex;" alt="{\displaystyle H_{0}\,\!}"></span> 、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{2}\,\!}"> <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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J^{2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abf017e5d94bf79cbea10f30307f458e2d57f3c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.967ex; height:2.676ex;" alt="{\displaystyle J^{2}\,\!}"></span> 、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}\,\!}"> <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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e24e2e5cae805e283d9279de6457d807918459c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:3.024ex; height:2.676ex;" alt="{\displaystyle L^{2}\,\!}"></span> 、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f8f4b8894e9616d347420e9d6d5e5db3eac8061" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.963ex; height:2.676ex;" alt="{\displaystyle S^{2}\,\!}"></span> ，這四個算符也都互相對易。所以，<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_{0}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed0c442b4d740f2d8545c9959fcc55dcbf2800b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:3.373ex; height:2.509ex;" alt="{\displaystyle H_{0}\,\!}"></span> 、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{2}\,\!}"> <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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J^{2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abf017e5d94bf79cbea10f30307f458e2d57f3c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.967ex; height:2.676ex;" alt="{\displaystyle J^{2}\,\!}"></span> 、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}\,\!}"> <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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e24e2e5cae805e283d9279de6457d807918459c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:3.024ex; height:2.676ex;" alt="{\displaystyle L^{2}\,\!}"></span> 、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f8f4b8894e9616d347420e9d6d5e5db3eac8061" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.963ex; height:2.676ex;" alt="{\displaystyle S^{2}\,\!}"></span> ，這四個算符的共同本徵函數 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |n,j,l,s\rangle \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <mi>s</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |n,j,l,s\rangle \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15c6dddbca65f855f42f42fb2114dc6951cbed4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:9.177ex; height:2.843ex;" alt="{\displaystyle |n,j,l,s\rangle \,\!}"></span> 可以被當做零微擾波函數，用來計算一階能量位移 <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}^{(1)}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}^{(1)}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2535256b4328b0a0115686b933026de17190543" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:4.515ex; height:3.343ex;" alt="{\displaystyle E_{n}^{(1)}\,\!}"></span> ；其中， <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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </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/ddf4a520ae4c1cdb7467e78a29509510ea61a08f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.782ex; height:1.676ex;" alt="{\displaystyle n\,\!}"></span> 是<a href="/wiki/%E4%B8%BB%E9%87%8F%E5%AD%90%E6%95%B8" title="主量子數">主量子數</a>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e466a79da556198f633f4eeb8360add30e40f96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; margin-right: -0.387ex; width:1.372ex; height:2.509ex;" alt="{\displaystyle j\,\!}"></span> 是總角量子數，<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 l\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b29435a849761c988f6c160c9e8bd9dda302c319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.08ex; height:2.176ex;" alt="{\displaystyle l\,\!}"></span> 是<a href="/wiki/%E8%A7%92%E9%87%8F%E5%AD%90%E6%95%B8" class="mw-redirect" title="角量子數">角量子數</a>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd19aeed2e55b7abe70cb770e360639ae133a0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.478ex; height:1.676ex;" alt="{\displaystyle s\,\!}"></span> 是自旋量子數。這一組本徵函數所形成的基底，就是想要尋找的基底。這共同本徵函數 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |n,j,l,s\rangle \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <mi>s</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |n,j,l,s\rangle \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15c6dddbca65f855f42f42fb2114dc6951cbed4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:9.177ex; height:2.843ex;" alt="{\displaystyle |n,j,l,s\rangle \,\!}"></span> 的 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} \cdot \mathbf {S} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} \cdot \mathbf {S} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3780485f1c67c3638475b5730138b7b9aa74b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:5.16ex; height:2.176ex;" alt="{\displaystyle \mathbf {L} \cdot \mathbf {S} \,\!}"></span> 的期望值是 </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 {\begin{aligned}\langle n,j,l,s\,|\,\mathbf {L} \cdot \mathbf {S} \,|\,n,j,l,s\rangle &={1 \over 2}(\langle \mathbf {J} ^{2}\rangle -\langle \mathbf {L} ^{2}\rangle -\langle \mathbf {S} ^{2}\rangle )\\&={\hbar ^{2} \over 2}[j(j+1)-l(l+1)-s(s+1)]\\&={\hbar ^{2} \over 2}[j(j+1)-l(l+1)-3/4]\\\end{aligned}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>n</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <mi>s</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mi>n</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <mi>s</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">⟨</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>j</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>l</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>j</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>l</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo stretchy="false">]</mo> </mtd> </mtr> </mtable> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\langle n,j,l,s\,|\,\mathbf {L} \cdot \mathbf {S} \,|\,n,j,l,s\rangle &={1 \over 2}(\langle \mathbf {J} ^{2}\rangle -\langle \mathbf {L} ^{2}\rangle -\langle \mathbf {S} ^{2}\rangle )\\&={\hbar ^{2} \over 2}[j(j+1)-l(l+1)-s(s+1)]\\&={\hbar ^{2} \over 2}[j(j+1)-l(l+1)-3/4]\\\end{aligned}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fb3bb4d70e0d9b154addbd14293b1f0a119116b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.581ex; margin-right: -0.387ex; margin-bottom: -0.257ex; width:61.245ex; height:16.843ex;" alt="{\displaystyle {\begin{aligned}\langle n,j,l,s\,|\,\mathbf {L} \cdot \mathbf {S} \,|\,n,j,l,s\rangle &={1 \over 2}(\langle \mathbf {J} ^{2}\rangle -\langle \mathbf {L} ^{2}\rangle -\langle \mathbf {S} ^{2}\rangle )\\&={\hbar ^{2} \over 2}[j(j+1)-l(l+1)-s(s+1)]\\&={\hbar ^{2} \over 2}[j(j+1)-l(l+1)-3/4]\\\end{aligned}}\,\!}"></span><span style="vertical-align:bottom">；</span></dd></dl> <p>其中，電子的自旋 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=1/2\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=1/2\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/897f2f0a3a4cbdf27e2b473723f3ebf3c094c91e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:8.063ex; height:2.843ex;" alt="{\displaystyle s=1/2\,\!}"></span> 。 </p><p>經過一番繁瑣的運算<sup id="cite_ref-Griffiths2004_2-0" class="reference"><a href="#cite_note-Griffiths2004-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>，可以得到 <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^{-3}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r^{-3}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5319a99e5cdc79987abbe9bef76d9eece6be23bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:3.769ex; height:2.676ex;" alt="{\displaystyle r^{-3}\,\!}"></span> 的期望值 </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 \langle n,j,l,s\,|\,r^{-3}\,|\,n,j,l,s\rangle ={\frac {2Z^{3}}{a_{0}^{3}n^{3}l(l+1)(2l+1)}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>n</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <mi>s</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mi>n</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <mi>s</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>l</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle n,j,l,s\,|\,r^{-3}\,|\,n,j,l,s\rangle ={\frac {2Z^{3}}{a_{0}^{3}n^{3}l(l+1)(2l+1)}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c460c671f5d25bdba865afa0627ed349b9550260" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-right: -0.387ex; width:46.431ex; height:6.843ex;" alt="{\displaystyle \langle n,j,l,s\,|\,r^{-3}\,|\,n,j,l,s\rangle ={\frac {2Z^{3}}{a_{0}^{3}n^{3}l(l+1)(2l+1)}}\,\!}"></span> ；</dd></dl> <p>其中，<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_{0}={\frac {4\pi \epsilon _{0}\hbar ^{2}}{me^{2}}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>m</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}={\frac {4\pi \epsilon _{0}\hbar ^{2}}{me^{2}}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d09afdffbfbc1d5766ef9db622a02d8bc45e4a50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; margin-right: -0.387ex; width:13.475ex; height:6.009ex;" alt="{\displaystyle a_{0}={\frac {4\pi \epsilon _{0}\hbar ^{2}}{me^{2}}}\,\!}"></span> 是<a href="/wiki/%E6%B3%A2%E8%80%B3%E5%8D%8A%E5%BE%91" class="mw-redirect" title="波耳半徑">波耳半徑</a>。 </p><p>將這兩個期望值的公式代入，能級位移是 </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}^{(1)}={\frac {Z^{4}e^{2}\hbar ^{2}}{8\pi \epsilon _{0}m^{2}c^{2}a_{0}^{3}}}\ {\frac {[j(j+1)-l(l+1)-3/4]}{n^{3}\,l(l+1)(2l+1)}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mi>j</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>l</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo stretchy="false">]</mo> </mrow> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>l</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}^{(1)}={\frac {Z^{4}e^{2}\hbar ^{2}}{8\pi \epsilon _{0}m^{2}c^{2}a_{0}^{3}}}\ {\frac {[j(j+1)-l(l+1)-3/4]}{n^{3}\,l(l+1)(2l+1)}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6420ba21c90b55712ef158d20b72a244f5160f3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-right: -0.387ex; width:47.187ex; height:6.843ex;" alt="{\displaystyle E_{n}^{(1)}={\frac {Z^{4}e^{2}\hbar ^{2}}{8\pi \epsilon _{0}m^{2}c^{2}a_{0}^{3}}}\ {\frac {[j(j+1)-l(l+1)-3/4]}{n^{3}\,l(l+1)(2l+1)}}\,\!}"></span> 。</dd></dl> <p>經過一番運算，可以得到 </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}^{(1)}={\frac {(E_{n}^{(0)})^{2}}{mc^{2}}}\ {\frac {2n[j(j+1)-l(l+1)-3/4]}{l(l+1)(2l+1)}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mo stretchy="false">[</mo> <mi>j</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>l</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo stretchy="false">]</mo> </mrow> <mrow> <mi>l</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}^{(1)}={\frac {(E_{n}^{(0)})^{2}}{mc^{2}}}\ {\frac {2n[j(j+1)-l(l+1)-3/4]}{l(l+1)(2l+1)}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8012622497b6f875b639d0a32437fecfc25c658" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-right: -0.387ex; width:44.803ex; height:7.176ex;" alt="{\displaystyle E_{n}^{(1)}={\frac {(E_{n}^{(0)})^{2}}{mc^{2}}}\ {\frac {2n[j(j+1)-l(l+1)-3/4]}{l(l+1)(2l+1)}}\,\!}"></span> ；</dd></dl> <p>其中，<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}^{(0)}={\frac {Z^{2}\hbar ^{2}}{2ma_{0}^{2}n^{2}}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>m</mi> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}^{(0)}={\frac {Z^{2}\hbar ^{2}}{2ma_{0}^{2}n^{2}}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9767a8576a9a94b1003ff11a3d4b56b523de5ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-right: -0.387ex; width:16.385ex; height:6.843ex;" alt="{\displaystyle E_{n}^{(0)}={\frac {Z^{2}\hbar ^{2}}{2ma_{0}^{2}n^{2}}}\,\!}"></span> 是主量子數為 <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> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </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/ddf4a520ae4c1cdb7467e78a29509510ea61a08f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.782ex; height:1.676ex;" alt="{\displaystyle n\,\!}"></span> 的零微擾能級。 </p><p>特別注意，當 <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 l=0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l=0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c61880fa703287e15fdfdcc14ecb7be6c555580e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:5.341ex; height:2.176ex;" alt="{\displaystyle l=0\,\!}"></span> 時，這方程式會遇到<a href="/wiki/%E9%99%A4%E4%BB%A5%E9%9B%B6" title="除以零">除以零</a>的不可定義運算；雖然<a href="/wiki/%E5%88%86%E6%95%B8" title="分數">分子</a>項目 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j(j+1)-l(l+1)-3/4=0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>l</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j(j+1)-l(l+1)-3/4=0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e126d79c249734498ab020a0cee5f481def9fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; margin-right: -0.387ex; width:28.77ex; height:2.843ex;" alt="{\displaystyle j(j+1)-l(l+1)-3/4=0\,\!}"></span> 也等於零。零除以零，仍舊無法計算這方程式的值。很幸運地，在<a href="/wiki/%E7%B2%BE%E7%B4%B0%E7%B5%90%E6%A7%8B" title="精細結構">精細結構</a>能量微擾的計算裏，這不可定義問題自動地會消失。事實上，當 <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 l=0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l=0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c61880fa703287e15fdfdcc14ecb7be6c555580e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:5.341ex; height:2.176ex;" alt="{\displaystyle l=0\,\!}"></span> 時，電子的軌道運動是<a href="/wiki/%E7%90%83%E5%B0%8D%E7%A8%B1%E4%BD%8D%E5%8B%A2" title="球對稱位勢">球對稱</a>的。這可以從電子的波函數的角部分觀察出來，<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 l=0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l=0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c61880fa703287e15fdfdcc14ecb7be6c555580e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:5.341ex; height:2.176ex;" alt="{\displaystyle l=0\,\!}"></span> <a href="/wiki/%E7%90%83%E8%B0%90%E5%87%BD%E6%95%B0" title="球谐函数">球諧函數</a>是 </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 Y_{0}^{0}={\frac {1}{\sqrt {4\pi }}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>4</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{0}^{0}={\frac {1}{\sqrt {4\pi }}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/992372db0ba8fccf79299a1deec7c2de6c06fed7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; margin-right: -0.387ex; width:11.706ex; height:6.176ex;" alt="{\displaystyle Y_{0}^{0}={\frac {1}{\sqrt {4\pi }}}\,\!}"></span> ，</dd></dl> <p>由於完全跟角度無關，角動量也是零，電子並不會感覺到任何磁場，所以，電子的 <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 l=0\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l=0\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c61880fa703287e15fdfdcc14ecb7be6c555580e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:5.341ex; height:2.176ex;" alt="{\displaystyle l=0\,\!}"></span> 軌道沒有自旋-軌道作用。 </p> <div class="mw-heading mw-heading2"><h2 id="參閱"><span id=".E5.8F.83.E9.96.B1"></span>參閱</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E8%87%AA%E6%97%8B-%E8%BB%8C%E9%81%93%E4%BD%9C%E7%94%A8&action=edit&section=6" title="编辑章节：參閱"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E6%96%AF%E5%A1%94%E5%85%8B%E6%95%88%E5%BA%94" title="斯塔克效应">斯塔克效應</a></li> <li><a href="/wiki/%E5%A1%9E%E6%9B%BC%E6%95%88%E6%87%89" class="mw-redirect" title="塞曼效應">塞曼效應</a></li> <li><a href="/wiki/%E8%B6%85%E7%B2%BE%E7%BB%86%E7%BB%93%E6%9E%84" title="超精细结构">超精細結構</a></li> <li><a href="/wiki/%E8%98%AD%E5%A7%86%E4%BD%8D%E7%A7%BB" title="蘭姆位移">蘭姆位移</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="參考文獻"><span id=".E5.8F.83.E8.80.83.E6.96.87.E7.8D.BB"></span>參考文獻</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E8%87%AA%E6%97%8B-%E8%BB%8C%E9%81%93%E4%BD%9C%E7%94%A8&action=edit&section=7" title="编辑章节：參考文獻"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-French1968-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-French1968_1-0">^</a></b></span> <span class="reference-text"><cite class="citation book">French, A. P. Special Relativity (The M.I.T Introductory Physics Series). W. W. Norton & Company, Inc. 1968: pp. 237–250. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0748764224" title="Special:网络书源/0748764224"><span title="国际标准书号">ISBN</span> 0748764224</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E8%87%AA%E6%97%8B-%E8%BB%8C%E9%81%93%E4%BD%9C%E7%94%A8&rft.au=French%2C+A.+P.&rft.btitle=Special+Relativity+%28The+M.I.T+Introductory+Physics+Series%29&rft.date=1968&rft.genre=book&rft.isbn=0748764224&rft.pages=pp.+237-250&rft.pub=W.+W.+Norton+%26+Company%2C+Inc.&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span><span class="citation-comment" style="display:none; color:#33aa33"> 引文格式1维护：冗余文本 (<a href="/wiki/Category:%E5%BC%95%E6%96%87%E6%A0%BC%E5%BC%8F1%E7%BB%B4%E6%8A%A4%EF%BC%9A%E5%86%97%E4%BD%99%E6%96%87%E6%9C%AC" title="Category:引文格式1维护：冗余文本">link</a>)</span></span> </li> <li id="cite_note-Griffiths2004-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Griffiths2004_2-0">^</a></b></span> <span class="reference-text"><cite class="citation book">Griffiths, David J. Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. 2004: pp. 266–276. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-13-111892-7" title="Special:网络书源/0-13-111892-7"><span title="国际标准书号">ISBN</span> 0-13-111892-7</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E8%87%AA%E6%97%8B-%E8%BB%8C%E9%81%93%E4%BD%9C%E7%94%A8&rft.au=Griffiths%2C+David+J.&rft.btitle=Introduction+to+Quantum+Mechanics+%282nd+ed.%29&rft.date=2004&rft.genre=book&rft.isbn=0-13-111892-7&rft.pages=pp.+266-276&rft.pub=Prentice+Hall&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span><span class="citation-comment" style="display:none; color:#33aa33"> 引文格式1维护：冗余文本 (<a href="/wiki/Category:%E5%BC%95%E6%96%87%E6%A0%BC%E5%BC%8F1%E7%BB%B4%E6%8A%A4%EF%BC%9A%E5%86%97%E4%BD%99%E6%96%87%E6%9C%AC" title="Category:引文格式1维护：冗余文本">link</a>)</span></span> </li> </ol></div> <ul><li><cite class="citation book">E. U. Condon and G. H. Shortley. <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.212979">The Theory of Atomic Spectra</a>. Cambridge University Press. 1935. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0521092094" class="internal mw-magiclink-isbn">ISBN 0-521-09209-4</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E8%87%AA%E6%97%8B-%E8%BB%8C%E9%81%93%E4%BD%9C%E7%94%A8&rft.au=E.+U.+Condon+and+G.+H.+Shortley&rft.btitle=The+Theory+of+Atomic+Spectra&rft.date=1935&rft.genre=book&rft.pub=Cambridge+University+Press&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fin.ernet.dli.2015.212979&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="外部連結"><span id=".E5.A4.96.E9.83.A8.E9.80.A3.E7.B5.90"></span>外部連結</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E8%87%AA%E6%97%8B-%E8%BB%8C%E9%81%93%E4%BD%9C%E7%94%A8&action=edit&section=8" title="编辑章节：外部連結"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E8%81%96%E5%9C%B0%E7%89%99%E5%93%A5%E5%8A%A0%E5%B7%9E%E5%A4%A7%E5%AD%B8" class="mw-redirect" title="聖地牙哥加州大學">圣地牙哥加州大学</a>物理系量子力学視聽教學： <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100624231316/http://physicsstream.ucsd.edu/courses/fall2003/physics130b/movies/2003-10-24_full.mov">自旋-軌道作用</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="">检索自“<a dir="ltr" href="https://zh.wikipedia.org/w/index.php?title=自旋-軌道作用&oldid=81805231">https://zh.wikipedia.org/w/index.php?title=自旋-軌道作用&oldid=81805231</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:%E9%A1%B5%E9%9D%A2%E5%88%86%E7%B1%BB" 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