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[o]" accesskey="o" class=""><span>로그인</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="더 많은 옵션" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="개인 도구" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">개인 도구</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="사용자 메뉴" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ko.wikipedia.org&uselang=ko"><span>기부</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%ED%81%B4%EB%9D%BC%EC%9D%B8-%EA%B3%A0%EB%93%A0+%EB%B0%A9%EC%A0%95%EC%8B%9D" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>계정 만들기</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%ED%81%B4%EB%9D%BC%EC%9D%B8-%EA%B3%A0%EB%93%A0+%EB%B0%A9%EC%A0%95%EC%8B%9D" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>로그인</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"> 로그아웃한 편집자를 위한 문서 <a href="/wiki/%EB%8F%84%EC%9B%80%EB%A7%90:%EC%86%8C%EA%B0%9C" aria-label="편집에 관해 더 알아보기"><span>더 알아보기</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/%ED%8A%B9%EC%88%98:%EB%82%B4%EA%B8%B0%EC%97%AC" title="이 IP 주소의 편집 목록 [y]" accesskey="y"><span>기여</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EC%82%AC%EC%9A%A9%EC%9E%90%ED%86%A0%EB%A1%A0" title="현재 사용하는 IP 주소에 대한 토론 문서 [n]" accesskey="n"><span>토론</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="사이트"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="목차" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">목차</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">숨기기</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">처음 위치</div> </a> </li> <li id="toc-정의" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#정의"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>정의</span> </div> </a> <button aria-controls="toc-정의-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>정의 하위섹션 토글하기</span> </button> <ul id="toc-정의-sublist" class="vector-toc-list"> <li id="toc-라그랑지언" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#라그랑지언"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>라그랑지언</span> </div> </a> <ul id="toc-라그랑지언-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-유도" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#유도"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>유도</span> </div> </a> <ul id="toc-유도-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-역사" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#역사"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>역사</span> </div> </a> <ul id="toc-역사-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-각주" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#각주"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>각주</span> </div> </a> <ul id="toc-각주-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-외부_링크" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#외부_링크"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>외부 링크</span> </div> </a> <ul id="toc-외부_링크-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-같이_보기" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#같이_보기"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>같이 보기</span> </div> </a> <ul id="toc-같이_보기-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="목차" 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="목차 토글" > <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">목차 토글</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">클라인-고든 방정식</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="다른 언어로 문서를 방문합니다. 31개 언어로 읽을 수 있습니다" > <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-31" 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">31개 언어</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A9_%D9%83%D9%84%D8%A7%D9%8A%D9%86-%D8%BA%D9%88%D8%B1%D8%AF%D9%88%D9%86" title="معادلة كلاين-غوردون – 아랍어" lang="ar" hreflang="ar" data-title="معادلة كلاين-غوردون" data-language-autonym="العربية" data-language-local-name="아랍어" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Equaci%C3%B3_de_Klein-Gordon" title="Equació de Klein-Gordon – 카탈로니아어" lang="ca" hreflang="ca" data-title="Equació de Klein-Gordon" data-language-autonym="Català" data-language-local-name="카탈로니아어" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kleinova%E2%80%93Gordonova_rovnice" title="Kleinova–Gordonova rovnice – 체코어" lang="cs" hreflang="cs" data-title="Kleinova–Gordonova rovnice" data-language-autonym="Čeština" data-language-local-name="체코어" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Klein-Gordon-ligningen" title="Klein-Gordon-ligningen – 덴마크어" lang="da" hreflang="da" data-title="Klein-Gordon-ligningen" data-language-autonym="Dansk" data-language-local-name="덴마크어" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Klein-Gordon-Gleichung" title="Klein-Gordon-Gleichung – 독일어" lang="de" hreflang="de" data-title="Klein-Gordon-Gleichung" data-language-autonym="Deutsch" data-language-local-name="독일어" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CE%BE%CE%AF%CF%83%CF%89%CF%83%CE%B7_%CE%9A%CE%BB%CE%AC%CE%B9%CE%BD-%CE%93%CE%BA%CF%8C%CF%81%CE%BD%CF%84%CE%BF%CE%BD" title="Εξίσωση Κλάιν-Γκόρντον – 그리스어" lang="el" hreflang="el" data-title="Εξίσωση Κλάιν-Γκόρντον" data-language-autonym="Ελληνικά" data-language-local-name="그리스어" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation – 영어" lang="en" hreflang="en" data-title="Klein–Gordon equation" data-language-autonym="English" data-language-local-name="영어" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ecuaci%C3%B3n_de_Klein-Gordon" title="Ecuación de Klein-Gordon – 스페인어" lang="es" hreflang="es" data-title="Ecuación de Klein-Gordon" data-language-autonym="Español" data-language-local-name="스페인어" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D9%87_%DA%A9%D9%84%D8%A7%DB%8C%D9%86%E2%80%93%DA%AF%D9%88%D8%B1%D8%AF%D9%88%D9%86" title="معادله کلاین–گوردون – 페르시아어" lang="fa" hreflang="fa" data-title="معادله کلاین–گوردون" data-language-autonym="فارسی" data-language-local-name="페르시아어" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89quation_de_Klein-Gordon" title="Équation de Klein-Gordon – 프랑스어" lang="fr" hreflang="fr" data-title="Équation de Klein-Gordon" data-language-autonym="Français" data-language-local-name="프랑스어" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%AA_%D7%A7%D7%9C%D7%99%D7%99%D7%9F-%D7%92%D7%95%D7%A8%D7%93%D7%95%D7%9F" title="משוואת קליין-גורדון – 히브리어" lang="he" hreflang="he" data-title="משוואת קליין-גורדון" data-language-autonym="עברית" data-language-local-name="히브리어" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Klein-Gordonova_jednad%C5%BEba" title="Klein-Gordonova jednadžba – 크로아티아어" lang="hr" hreflang="hr" data-title="Klein-Gordonova jednadžba" data-language-autonym="Hrvatski" data-language-local-name="크로아티아어" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%AC%D5%A1%D5%B5%D5%B6-%D4%B3%D5%B8%D6%80%D5%A4%D5%B8%D5%B6%D5%AB_%D5%B0%D5%A1%D5%BE%D5%A1%D5%BD%D5%A1%D6%80%D5%B8%D6%82%D5%B4" title="Կլայն-Գորդոնի հավասարում – 아르메니아어" lang="hy" hreflang="hy" data-title="Կլայն-Գորդոնի հավասարում" data-language-autonym="Հայերեն" data-language-local-name="아르메니아어" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Persamaan_Klein-Gordon" title="Persamaan Klein-Gordon – 인도네시아어" lang="id" hreflang="id" data-title="Persamaan Klein-Gordon" data-language-autonym="Bahasa Indonesia" data-language-local-name="인도네시아어" 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/Equazione_di_Klein-Gordon" title="Equazione di Klein-Gordon – 이탈리아어" lang="it" hreflang="it" data-title="Equazione di Klein-Gordon" data-language-autonym="Italiano" data-language-local-name="이탈리아어" 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%AF%E3%83%A9%E3%82%A4%E3%83%B3-%E3%82%B4%E3%83%AB%E3%83%89%E3%83%B3%E6%96%B9%E7%A8%8B%E5%BC%8F" title="クライン-ゴルドン方程式 – 일본어" lang="ja" hreflang="ja" data-title="クライン-ゴルドン方程式" data-language-autonym="日本語" data-language-local-name="일본어" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Klein-Gordonvergelijking" title="Klein-Gordonvergelijking – 네덜란드어" lang="nl" hreflang="nl" data-title="Klein-Gordonvergelijking" data-language-autonym="Nederlands" data-language-local-name="네덜란드어" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Klein-Gordon-ligning" title="Klein-Gordon-ligning – 노르웨이어(보크말)" lang="nb" hreflang="nb" data-title="Klein-Gordon-ligning" data-language-autonym="Norsk bokmål" data-language-local-name="노르웨이어(보크말)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A8%B2%E0%A9%87%E0%A8%87%E0%A8%A8-%E0%A8%97%E0%A9%8C%E0%A8%B0%E0%A8%A1%E0%A8%A8_%E0%A8%87%E0%A8%95%E0%A9%81%E0%A8%8F%E0%A8%B8%E0%A8%BC%E0%A8%A8" title="ਕਲੇਇਨ-ਗੌਰਡਨ ਇਕੁਏਸ਼ਨ – 펀잡어" lang="pa" hreflang="pa" data-title="ਕਲੇਇਨ-ਗੌਰਡਨ ਇਕੁਏਸ਼ਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="펀잡어" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/R%C3%B3wnanie_Kleina-Gordona" title="Równanie Kleina-Gordona – 폴란드어" lang="pl" hreflang="pl" data-title="Równanie Kleina-Gordona" data-language-autonym="Polski" data-language-local-name="폴란드어" 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/Equa%C3%A7%C3%A3o_de_Klein%E2%80%93Gordon" title="Equação de Klein–Gordon – 포르투갈어" lang="pt" hreflang="pt" data-title="Equação de Klein–Gordon" data-language-autonym="Português" data-language-local-name="포르투갈어" 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%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%9A%D0%BB%D0%B5%D0%B9%D0%BD%D0%B0_%E2%80%94_%D0%93%D0%BE%D1%80%D0%B4%D0%BE%D0%BD%D0%B0" title="Уравнение Клейна — Гордона – 러시아어" lang="ru" hreflang="ru" data-title="Уравнение Клейна — Гордона" data-language-autonym="Русский" data-language-local-name="러시아어" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Klein-Gordonova_jednad%C5%BEba" title="Klein-Gordonova jednadžba – 세르비아-크로아티아어" lang="sh" hreflang="sh" data-title="Klein-Gordonova jednadžba" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="세르비아-크로아티아어" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Klein-Gordonova_ena%C4%8Dba" title="Klein-Gordonova enačba – 슬로베니아어" lang="sl" hreflang="sl" data-title="Klein-Gordonova enačba" data-language-autonym="Slovenščina" data-language-local-name="슬로베니아어" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Ekuacioni_Klein-Gordon" title="Ekuacioni Klein-Gordon – 알바니아어" lang="sq" hreflang="sq" data-title="Ekuacioni Klein-Gordon" data-language-autonym="Shqip" data-language-local-name="알바니아어" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Klein%E2%80%93Gordon-ekvationen" title="Klein–Gordon-ekvationen – 스웨덴어" lang="sv" hreflang="sv" data-title="Klein–Gordon-ekvationen" data-language-autonym="Svenska" 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href="mw-data:TemplateStyles:r36480591"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36480591"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist nowraplinks" style="width:19.0em;"><tbody><tr><td class="sidebar-pretitle">관련 기사 시리즈의 일부</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/%EC%96%91%EC%9E%90%EC%97%AD%ED%95%99" title="양자역학">양자역학</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/%EC%8A%88%EB%A2%B0%EB%94%A9%EA%B1%B0_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="슈뢰딩거 방정식">슈뢰딩거 방정식</a></div></td></tr><tr><td class="sidebar-above hlist nowrap" style="display:block;margin-bottom:0.4em;"> <ul><li><a href="/wiki/%EC%96%91%EC%9E%90%EC%97%AD%ED%95%99_%EA%B0%9C%EB%A1%A0" title="양자역학 개론">개론</a></li> <li><a href="/wiki/%EC%96%91%EC%9E%90%EC%97%AD%ED%95%99%EC%9D%98_%EC%88%98%ED%95%99%EC%A0%81_%EA%B3%B5%EC%8B%9D%ED%99%94" class="mw-redirect" title="양자역학의 수학적 공식화">수학적 공식화</a></li> <li><a href="/wiki/%EC%96%91%EC%9E%90%EC%97%AD%ED%95%99%EC%9D%98_%EC%97%AD%EC%82%AC" title="양자역학의 역사">역사</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;">배경</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/%EA%B3%A0%EC%A0%84%EC%97%AD%ED%95%99" title="고전역학">고전역학</a></li> <li><a href="/wiki/%EC%B4%88%EA%B8%B0_%EC%96%91%EC%9E%90%EB%A1%A0" title="초기 양자론">초기 양자론</a></li> <li><a href="/wiki/%EB%B8%8C%EB%9D%BC-%EC%BC%93_%ED%91%9C%EA%B8%B0%EB%B2%95" title="브라-켓 표기법">브라-켓 표기법</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/%ED%95%B4%EB%B0%80%ED%86%A0%EB%8B%88%EC%96%B8_(%EC%96%91%EC%9E%90%EC%97%AD%ED%95%99)" title="해밀토니언 (양자역학)">해밀토니언</a></li> <li><a href="/wiki/%EA%B0%84%EC%84%AD_(%ED%8C%8C%EB%8F%99_%EC%A0%84%ED%8C%8C)" title="간섭 (파동 전파)">간섭</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;">기본</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/%EC%83%81%EB%B3%B4%EC%84%B1_(%EB%AC%BC%EB%A6%AC%ED%95%99)" title="상보성 (물리학)">상보성</a></li> <li><a href="/wiki/%EC%96%91%EC%9E%90_%EA%B2%B0%EC%96%B4%EA%B8%8B%EB%82%A8" title="양자 결어긋남">결어긋남</a></li> <li><a href="/wiki/%EC%96%91%EC%9E%90_%EC%96%BD%ED%9E%98" title="양자 얽힘">얽힘</a></li> <li><a href="/wiki/%EC%97%90%EB%84%88%EC%A7%80_%EC%A4%80%EC%9C%84" title="에너지 준위">에너지 준위</a></li> <li><a href="https://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics" class="extiw" title="w:Measurement in quantum mechanics">측정(measurement)</a></li> <li><a href="https://en.wikipedia.org/wiki/Quantum_nonlocality" class="extiw" title="w:Quantum nonlocality">비국소성(nonlocality)</a></li> <li><a href="/wiki/%EC%96%91%EC%9E%90%EC%88%98" title="양자수">양자수</a></li> <li><a href="https://en.wikipedia.org/wiki/Quantum_state" class="extiw" title="w:Quantum state">상태(state)</a></li> <li><a href="/wiki/%EC%96%91%EC%9E%90_%EC%A4%91%EC%B2%A9" title="양자 중첩">중첩</a></li> <li><a href="https://en.wikipedia.org/wiki/Symmetry_in_quantum_mechanics" class="extiw" title="w:Symmetry in quantum mechanics">Symmetry</a></li> <li><a href="/wiki/%ED%84%B0%EB%84%90_%ED%9A%A8%EA%B3%BC" title="터널 효과">터널링</a></li> <li><a href="/wiki/%EB%B6%88%ED%99%95%EC%A0%95%EC%84%B1_%EC%9B%90%EB%A6%AC" title="불확정성 원리">불확정성</a></li> <li><a href="/wiki/%ED%8C%8C%EB%8F%99_%ED%95%A8%EC%88%98" title="파동 함수">파동 함수</a> <ul><li><a href="/wiki/%ED%8C%8C%EB%8F%99_%ED%95%A8%EC%88%98_%EB%B6%95%EA%B4%B4" title="파동 함수 붕괴">붕괴</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;">실험</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/%EB%B2%A8_%EB%B6%80%EB%93%B1%EC%8B%9D_%EC%8B%A4%ED%97%98" title="벨 부등식 실험">벨 부등식</a></li> <li><a href="/wiki/%EB%8D%B0%EC%9D%B4%EB%B9%84%EC%8A%A8-%EA%B1%B0%EB%A8%B8_%EC%8B%A4%ED%97%98" title="데이비슨-거머 실험">데이비슨-거머</a></li> <li><a href="/wiki/%EC%9D%B4%EC%A4%91%EC%8A%AC%EB%A6%BF_%EC%8B%A4%ED%97%98" title="이중슬릿 실험">이중슬릿</a></li> <li><a href="https://en.wikipedia.org/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" class="extiw" title="w:Elitzur–Vaidman bomb tester">엘리추르–바이드만(Elitzur–Vaidman)</a></li> <li><a href="/wiki/%ED%94%84%EB%9E%91%ED%81%AC-%ED%97%A4%EB%A5%B4%EC%B8%A0_%EC%8B%A4%ED%97%98" title="프랑크-헤르츠 실험">프랑크-헤르츠</a></li> <li><a href="https://en.wikipedia.org/wiki/Leggett%E2%80%93Garg_inequality" class="extiw" title="w:Leggett–Garg inequality">레게트-가르그 부등식(Leggett–Garg inequality)</a></li> <li><a href="https://en.wikipedia.org/wiki/Mach%E2%80%93Zehnder_interferometer" class="extiw" title="w:Mach–Zehnder interferometer">마하-젠더(Mach–Zehnder)</a></li> <li><a href="https://en.wikipedia.org/wiki/Popper%27s_experiment" class="extiw" title="w:Popper's experiment">포퍼(Popper)</a></li></ul> </div> <ul><li><a href="https://en.wikipedia.org/wiki/Quantum_eraser_experiment" class="extiw" title="w:Quantum eraser experiment">양자 지우개(quantum eraser)</a> <ul><li><a href="/wiki/%EC%A7%80%EC%97%B0%EC%84%A0%ED%83%9D_%EC%96%91%EC%9E%90_%EC%A7%80%EC%9A%B0%EA%B0%9C" title="지연선택 양자 지우개">지연선택</a></li></ul></li></ul> <div class="hlist"> <ul><li><a href="/wiki/%EC%8A%88%EB%A2%B0%EB%94%A9%EA%B1%B0%EC%9D%98_%EA%B3%A0%EC%96%91%EC%9D%B4" title="슈뢰딩거의 고양이">슈뢰딩거의 고양이</a></li> <li><a href="/wiki/%EC%8A%88%ED%85%8C%EB%A5%B8-%EA%B2%8C%EB%A5%BC%EB%9D%BC%ED%9D%90_%EC%8B%A4%ED%97%98" title="슈테른-게를라흐 실험">슈테른-게를라흐</a></li> <li><a href="https://en.wikipedia.org/wiki/Wheeler%27s_delayed-choice_experiment" class="extiw" title="w:Wheeler's delayed-choice experiment">휠러의 지연된 선택(Wheeler's delayed-choice)</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;">공식화</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/%EC%96%91%EC%9E%90%EC%97%AD%ED%95%99%EC%9D%98_%EC%88%98%ED%95%99_%EA%B3%B5%EC%8B%9D%ED%99%94" title="양자역학의 수학 공식화">개요</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/%ED%95%98%EC%9D%B4%EC%A0%A0%EB%B2%A0%EB%A5%B4%ED%81%AC_%EB%AC%98%EC%82%AC" title="하이젠베르크 묘사">하이젠베르크</a></li> <li><a href="/wiki/%EC%83%81%ED%98%B8%EC%9E%91%EC%9A%A9_%EB%AC%98%EC%82%AC" title="상호작용 묘사">상호작용 묘사</a></li> <li><a href="/wiki/%ED%96%89%EB%A0%AC_%EC%97%AD%ED%95%99" title="행렬 역학">행렬</a></li> <li><a href="https://en.wikipedia.org/wiki/Phase_space_formulation" class="extiw" title="w:Phase space formulation">Phase-space</a></li> <li><a href="/wiki/%EC%8A%88%EB%A2%B0%EB%94%A9%EA%B1%B0_%EB%AC%98%EC%82%AC" title="슈뢰딩거 묘사">슈뢰딩거</a></li> <li><a href="/wiki/%EA%B2%BD%EB%A1%9C_%EC%A0%81%EB%B6%84_%EA%B3%B5%EC%8B%9D%ED%99%94" title="경로 적분 공식화">합계 기록(경로 적분)</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;">방정식</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/%EB%94%94%EB%9E%99_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="디랙 방정식">디랙</a></li> <li><a class="mw-selflink selflink">클라인-고든</a></li> <li><a href="https://en.wikipedia.org/wiki/Pauli_equation" class="extiw" title="w:Pauli equation">파울리(Pauli)</a></li> <li><a href="/wiki/%EB%A4%BC%EB%93%9C%EB%B2%A0%EB%A6%AC_%EA%B3%B5%EC%8B%9D" title="뤼드베리 공식">뤼드베리</a></li> <li><a href="/wiki/%EC%8A%88%EB%A2%B0%EB%94%A9%EA%B1%B0_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="슈뢰딩거 방정식">슈뢰딩거</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;"><a href="/wiki/%EC%96%91%EC%9E%90%EC%97%AD%ED%95%99%EC%9D%98_%ED%95%B4%EC%84%9D" title="양자역학의 해석">해석</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/%EC%96%91%EC%9E%90_%EB%B2%A0%EC%9D%B4%EC%A6%88%EC%A3%BC%EC%9D%98" title="양자 베이즈주의">베이지안</a></li> <li><a href="/wiki/%EC%A0%95%ED%95%A9%EC%A0%81_%EC%97%AD%EC%82%AC" title="정합적 역사">정합적 역사</a></li> <li><a href="/wiki/%EC%BD%94%ED%8E%9C%ED%95%98%EA%B2%90_%ED%95%B4%EC%84%9D" title="코펜하겐 해석">코펜하겐</a></li> <li><a href="/wiki/%EB%93%9C_%EB%B8%8C%EB%A1%9C%EC%9D%B4-%EB%B4%84_%EC%9D%B4%EB%A1%A0" title="드 브로이-봄 이론">드 브로이-봄</a></li> <li><a href="/wiki/%EC%95%99%EC%83%81%EB%B8%94_%ED%95%B4%EC%84%9D" title="앙상블 해석">앙상블</a></li> <li><a href="/wiki/%EC%88%A8%EC%9D%80_%EB%B3%80%EC%88%98_%EC%9D%B4%EB%A1%A0" title="숨은 변수 이론">숨은 변수</a> <ul><li><a href="/wiki/%EA%B5%AD%EC%86%8C%EC%A0%81_%EC%88%A8%EC%9D%80_%EB%B3%80%EC%88%98_%EC%9D%B4%EB%A1%A0" title="국소적 숨은 변수 이론">국소적</a></li></ul></li> <li><a href="/wiki/%EB%8B%A4%EC%84%B8%EA%B3%84_%ED%95%B4%EC%84%9D" title="다세계 해석">다세계</a></li> <li><a href="/wiki/%EA%B0%9D%EA%B4%80%EC%A0%81_%EB%B6%95%EA%B4%B4_%EC%9D%B4%EB%A1%A0" title="객관적 붕괴 이론">객관적 붕괴</a></li> <li><a href="/wiki/%EC%96%91%EC%9E%90_%EB%85%BC%EB%A6%AC" title="양자 논리">양자 논리</a></li> <li><a href="/wiki/%EA%B4%80%EA%B3%84%EC%A0%81_%EC%96%91%EC%9E%90_%EC%97%AD%ED%95%99" title="관계적 양자 역학">관계적</a></li> <li><a href="https://en.wikipedia.org/wiki/Transactional_interpretation" class="extiw" title="w:Transactional interpretation">트랜잭션(transactional)</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;">고급 주제</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/%EC%83%81%EB%8C%80%EB%A1%A0%EC%A0%81_%EC%96%91%EC%9E%90%EC%97%AD%ED%95%99" title="상대론적 양자역학">상대론적 양자역학</a></li> <li><a href="/wiki/%EC%96%91%EC%9E%90%EC%9E%A5%EB%A1%A0" title="양자장론">양자장론</a></li> <li><a href="/wiki/%EC%96%91%EC%9E%90_%EC%A0%95%EB%B3%B4_%EA%B3%BC%ED%95%99" class="mw-redirect" title="양자 정보 과학">양자 정보 과학</a></li> <li><a href="https://en.wikipedia.org/wiki/Quantum_computing" class="extiw" title="w:Quantum computing">양자 컴퓨팅(quantum computing)</a></li> <li><a href="https://en.wikipedia.org/wiki/Quantum_chaos" class="extiw" title="w:Quantum chaos">양자 카오스(quantum chaos)</a></li> <li><a href="/wiki/EPR_%EC%97%AD%EC%84%A4" title="EPR 역설">EPR 역설</a></li> <li><a href="/wiki/%EB%B0%80%EB%8F%84_%ED%96%89%EB%A0%AC" title="밀도 행렬">밀도 행렬</a></li> <li><a href="/wiki/%EC%82%B0%EB%9E%80_%EC%9D%B4%EB%A1%A0" title="산란 이론">산란 이론</a></li> <li><a href="/wiki/%EC%96%91%EC%9E%90%ED%86%B5%EA%B3%84%EC%97%AD%ED%95%99" title="양자통계역학">양자통계역학</a></li> <li><a href="https://en.wikipedia.org/wiki/Quantum_machine_learning" class="extiw" title="w:Quantum machine learning">양자 기계 학습(quantum machine learning)</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;">과학자</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="https://en.wikipedia.org/wiki/Yakir_Aharonov" class="extiw" title="w:Yakir Aharonov">아하로노프<sub>Aharonov</sub></a></li> <li><a href="/wiki/%EC%A1%B4_%EC%8A%A4%ED%8A%9C%EC%96%B4%ED%8A%B8_%EB%B2%A8" title="존 스튜어트 벨">벨</a></li> <li><a href="/wiki/%ED%95%9C%EC%8A%A4_%EB%B2%A0%ED%85%8C" title="한스 베테">베테</a></li> <li><a href="/wiki/%ED%8C%A8%ED%8A%B8%EB%A6%AD_%EB%B8%94%EB%9E%98%ED%82%B7" title="패트릭 블래킷">블래킷</a></li> <li><a href="/wiki/%ED%8E%A0%EB%A6%AD%EC%8A%A4_%EB%B8%94%EB%A1%9C%ED%9D%90" title="펠릭스 블로흐">블로흐</a></li> <li><a href="/wiki/%EB%8D%B0%EC%9D%B4%EB%B9%84%EB%93%9C_%EB%B4%84" title="데이비드 봄">봄</a></li> <li><a href="/wiki/%EB%8B%90%EC%8A%A4_%EB%B3%B4%EC%96%B4" title="닐스 보어">보어</a></li> <li><a href="/wiki/%EB%A7%89%EC%8A%A4_%EB%B3%B4%EB%A5%B8" title="막스 보른">보른</a></li> <li><a href="/wiki/%EC%82%AC%ED%8B%B0%EC%97%94%EB%93%9C%EB%9D%BC_%EB%82%98%ED%8A%B8_%EB%B3%B4%EC%8A%A4" title="사티엔드라 나트 보스">보스</a></li> <li><a href="/wiki/%EB%A3%A8%EC%9D%B4_%EB%93%9C_%EB%B8%8C%EB%A1%9C%EC%9D%B4" title="루이 드 브로이">드브로이</a></li> <li><a href="/wiki/%EC%95%84%EC%84%9C_%EC%BD%A4%ED%94%84%ED%84%B4" title="아서 콤프턴">콤프턴</a></li> <li><a href="/wiki/%ED%8F%B4_%EB%94%94%EB%9E%99" title="폴 디랙">디랙</a></li> <li><a href="/wiki/%ED%81%B4%EB%A6%B0%ED%84%B4_%EC%A1%B0%EC%A7%80%ED%94%84_%EB%8D%B0%EC%9D%B4%EB%B9%84%EC%8A%A8" title="클린턴 조지프 데이비슨">데이비슨</a></li> <li><a href="/wiki/%ED%94%BC%ED%84%B0_%EB%94%94%EB%B0%94%EC%9D%B4" title="피터 디바이">디바이</a></li> <li><a href="/wiki/%ED%8C%8C%EC%9A%B8_%EC%97%90%EB%A0%8C%ED%8E%98%EC%8A%A4%ED%8A%B8" title="파울 에렌페스트">에렌페스트</a></li> <li><a href="/wiki/%EC%95%8C%EB%B2%A0%EB%A5%B4%ED%8A%B8_%EC%95%84%EC%9D%B8%EC%8A%88%ED%83%80%EC%9D%B8" title="알베르트 아인슈타인">아인슈타인</a></li> <li><a href="/wiki/%ED%9C%B4_%EC%97%90%EB%B2%84%EB%A0%9B_3%EC%84%B8" title="휴 에버렛 3세">에버렛</a></li> <li><a href="/wiki/%EB%B8%94%EB%9D%BC%EB%94%94%EB%AF%B8%EB%A5%B4_%ED%8F%AC%ED%81%AC" title="블라디미르 포크">포크</a></li> <li><a href="/wiki/%EC%97%94%EB%A6%AC%EC%BD%94_%ED%8E%98%EB%A5%B4%EB%AF%B8" title="엔리코 페르미">페르미</a></li> <li><a href="/wiki/%EB%A6%AC%EC%B2%98%EB%93%9C_%ED%8C%8C%EC%9D%B8%EB%A8%BC" class="mw-redirect" title="리처드 파인먼">파인먼</a></li> <li><a href="/wiki/%EB%A1%9C%EC%9D%B4_J._%EA%B8%80%EB%9D%BC%EC%9A%B0%EB%B2%84" title="로이 J. 글라우버">글라우버</a></li> <li><a href="https://en.wikipedia.org/wiki/Martin_Gutzwiller" class="extiw" title="w:Martin Gutzwiller">구츠윌러<sub>Gutzwiller</sub></a></li> <li><a href="/wiki/%EB%B2%A0%EB%A5%B4%EB%84%88_%ED%95%98%EC%9D%B4%EC%A0%A0%EB%B2%A0%EB%A5%B4%ED%81%AC" title="베르너 하이젠베르크">하이젠베르크</a></li> <li><a href="/wiki/%EB%8B%A4%EB%B9%84%ED%8A%B8_%ED%9E%90%EB%B2%A0%EB%A5%B4%ED%8A%B8" title="다비트 힐베르트">힐베르트</a></li> <li><a href="/wiki/%ED%8C%8C%EC%8A%A4%EC%BF%A0%EC%95%8C_%EC%9A%94%EB%A5%B4%EB%8B%A8" title="파스쿠알 요르단">요르단</a></li> <li><a href="/wiki/%ED%97%A8%EB%93%9C%EB%A6%AD_%EC%95%88%ED%86%A0%EB%8B%88_%ED%81%AC%EB%9D%BC%EB%A8%B8%EB%A5%B4%EC%8A%A4" title="헨드릭 안토니 크라머르스">크라머르스</a></li> <li><a href="/wiki/%EB%B3%BC%ED%94%84%EA%B0%95_%ED%8C%8C%EC%9A%B8%EB%A6%AC" title="볼프강 파울리">파울리</a></li> <li><a href="/wiki/%EC%9C%8C%EB%A6%AC%EC%8A%A4_%EC%9C%A0%EC%A7%84_%EB%9E%A8" title="윌리스 유진 램">램</a></li> <li><a href="/wiki/%EB%A0%88%ED%94%84_%EB%9E%80%EB%8B%A4%EC%9A%B0" title="레프 란다우">란다우</a></li> <li><a href="/wiki/%EB%A7%89%EC%8A%A4_%ED%8F%B0_%EB%9D%BC%EC%9A%B0%EC%97%90" title="막스 폰 라우에">라우에</a></li> <li><a href="/wiki/%ED%97%A8%EB%A6%AC_%EB%AA%A8%EC%A6%90%EB%A6%AC_(%EB%AC%BC%EB%A6%AC%ED%95%99%EC%9E%90)" title="헨리 모즐리 (물리학자)">모즐리</a></li> <li><a href="/wiki/%EB%A1%9C%EB%B2%84%ED%8A%B8_%EC%95%A4%EB%93%9C%EB%A3%A8%EC%8A%A4_%EB%B0%80%EB%A6%AC%EC%BB%A8" title="로버트 앤드루스 밀리컨">밀리컨</a></li> <li><a href="/wiki/%ED%97%A4%EC%9D%B4%EC%BB%A4_%EC%B9%B4%EB%A9%94%EB%A5%BC%EB%A7%81_%EC%98%A4%EB%84%88%EC%8A%A4" title="헤이커 카메를링 오너스">오너스</a></li> <li><a href="/wiki/%EB%A7%89%EC%8A%A4_%ED%94%8C%EB%9E%91%ED%81%AC" title="막스 플랑크">플랑크</a></li> <li><a href="/wiki/%EC%9D%B4%EC%A7%80%EB%8F%84%EC%96%B4_%EC%95%84%EC%9D%B4%EC%9E%91_%EB%9D%BC%EB%B9%84" title="이지도어 아이작 라비">라비</a></li> <li><a href="/wiki/%EC%B0%AC%EB%93%9C%EB%9D%BC%EC%84%B8%EC%B9%B4%EB%9D%BC_%EB%B2%B5%EC%B9%B4%ED%83%80_%EB%9D%BC%EB%A7%8C" title="찬드라세카라 벵카타 라만">라만</a></li> <li><a href="/wiki/%EC%9A%94%ED%95%9C%EB%84%A4%EC%8A%A4_%EB%A4%BC%EB%93%9C%EB%B2%A0%EB%A6%AC" title="요한네스 뤼드베리">뤼드베리</a></li> <li><a href="/wiki/%EC%97%90%EB%A5%B4%EB%B9%88_%EC%8A%88%EB%A2%B0%EB%94%A9%EA%B1%B0" title="에르빈 슈뢰딩거">슈뢰딩거</a></li> <li><a href="https://en.wikipedia.org/wiki/Michelle_Simmons" class="extiw" title="w:Michelle Simmons">시몬스<sub>Simmons</sub></a></li> <li><a href="/wiki/%EC%95%84%EB%A5%B4%EB%86%80%ED%8A%B8_%EC%A1%B0%EB%A8%B8%ED%8E%A0%ED%8A%B8" title="아르놀트 조머펠트">조머펠트</a></li> <li><a href="/wiki/%EC%A1%B4_%ED%8F%B0_%EB%85%B8%EC%9D%B4%EB%A7%8C" title="존 폰 노이만">폰 노이만</a></li> <li><a href="/wiki/%ED%97%A4%EB%A5%B4%EB%A7%8C_%EB%B0%94%EC%9D%BC" title="헤르만 바일">바일</a></li> <li><a href="/wiki/%EB%B9%8C%ED%97%AC%EB%A6%84_%EB%B9%88" title="빌헬름 빈">빈</a></li> <li><a href="/wiki/%EC%9C%A0%EC%A7%84_%EC%9C%84%EA%B7%B8%EB%84%88" title="유진 위그너">위그너</a></li> <li><a href="/wiki/%ED%94%BC%ED%84%B0%EB%A5%B4_%EC%A0%9C%EC%9D%B4%EB%A7%8C" title="피터르 제이만">제이만</a></li> <li><a href="/wiki/%EC%95%88%ED%86%A4_%EC%B0%A8%EC%9D%BC%EB%A7%81%EA%B1%B0" title="안톤 차일링거">차일링거</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar" style="border-top:1px solid #aaa;padding-top:0.1em; [[분류:역학 틀]] [[분류:물리학 사이드바 틀]]"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36480591"><style data-mw-deduplicate="TemplateStyles:r34311309">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-보기"><a href="/wiki/%ED%8B%80:%EC%96%91%EC%9E%90%EC%97%AD%ED%95%99" title="틀:양자역학"><abbr title="이 틀을 보기">v</abbr></a></li><li class="nv-토론"><a href="/wiki/%ED%8B%80%ED%86%A0%EB%A1%A0:%EC%96%91%EC%9E%90%EC%97%AD%ED%95%99" title="틀토론:양자역학"><abbr title="이 틀에 관해 토론하기">t</abbr></a></li><li class="nv-편집"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%AC%B8%EC%84%9C%ED%8E%B8%EC%A7%91/%ED%8B%80:%EC%96%91%EC%9E%90%EC%97%AD%ED%95%99" title="특수:문서편집/틀:양자역학"><abbr title="이 틀을 편집하기">e</abbr></a></li></ul></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r34311307">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}body.skin-minerva .mw-parser-output .infobox-header,body.skin-minerva .mw-parser-output .infobox-subheader,body.skin-minerva .mw-parser-output .infobox-above,body.skin-minerva .mw-parser-output .infobox-title,body.skin-minerva .mw-parser-output .infobox-image,body.skin-minerva .mw-parser-output .infobox-full-data,body.skin-minerva .mw-parser-output .infobox-below{text-align:center}</style><table class="infobox"><caption class="infobox-title">장방정식</caption><tbody><tr><th scope="row" class="infobox-label" style="background:#ddf;">스핀 0</th><td class="infobox-data"><a class="mw-selflink selflink">클라인-고든 방정식</a></td></tr><tr><th scope="row" class="infobox-label" style="background:#ddf;">스핀 ½</th><td class="infobox-data"><a href="/wiki/%EB%94%94%EB%9E%99_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="디랙 방정식">디랙 방정식</a> · <a href="/wiki/%EB%B0%94%EC%9D%BC_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="바일 방정식">바일 방정식</a> · <a href="/wiki/%EB%A7%88%EC%9A%94%EB%9D%BC%EB%82%98_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="마요라나 방정식">마요라나 방정식</a></td></tr><tr><th scope="row" class="infobox-label" style="background:#ddf;">스핀 1</th><td class="infobox-data"><a href="/wiki/%EB%A7%A5%EC%8A%A4%EC%9B%B0_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="맥스웰 방정식">맥스웰 방정식</a> · <a href="/wiki/%ED%94%84%EB%A1%9C%EC%B9%B4_%EC%9E%91%EC%9A%A9" title="프로카 작용">프로카 방정식</a></td></tr><tr><th scope="row" class="infobox-label" style="background:#ddf;">스핀 1½</th><td class="infobox-data"><a href="/wiki/%EB%9D%BC%EB%A6%AC%ED%83%80-%EC%8A%88%EC%9C%99%EA%B1%B0_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="라리타-슈윙거 방정식">라리타-슈윙거 방정식</a></td></tr><tr><th scope="row" class="infobox-label" style="background:#ddf;">스핀 2</th><td class="infobox-data"><a href="/wiki/%EC%95%84%EC%9D%B8%EC%8A%88%ED%83%80%EC%9D%B8_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="아인슈타인 방정식">아인슈타인 방정식</a></td></tr><tr><td colspan="2" class="infobox-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36480591"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311309"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-보기"><a href="/wiki/%ED%8B%80:%EC%9E%A5%EB%B0%A9%EC%A0%95%EC%8B%9D" title="틀:장방정식"><abbr title="이 틀을 보기">v</abbr></a></li><li class="nv-토론"><a href="/w/index.php?title=%ED%8B%80%ED%86%A0%EB%A1%A0:%EC%9E%A5%EB%B0%A9%EC%A0%95%EC%8B%9D&action=edit&redlink=1" class="new" title="틀토론:장방정식 (없는 문서)"><abbr title="이 틀에 관해 토론하기">t</abbr></a></li><li class="nv-편집"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%AC%B8%EC%84%9C%ED%8E%B8%EC%A7%91/%ED%8B%80:%EC%9E%A5%EB%B0%A9%EC%A0%95%EC%8B%9D" title="특수:문서편집/틀:장방정식"><abbr title="이 틀을 편집하기">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><a href="/wiki/%EC%96%91%EC%9E%90%EC%9E%A5%EB%A1%A0" title="양자장론">양자장론</a>에서 <b>클라인-고든 방정식</b>(Klein-Gordon方程式, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Klein–Gordon equation</span>) 또는 <b>클레인-고르돈 방정식</b>은 (<a href="/w/index.php?title=%EC%9C%A0%EC%82%AC%EC%8A%A4%EC%B9%BC%EB%9D%BC&action=edit&redlink=1" class="new" title="유사스칼라 (없는 문서)">유사</a>) <a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC" class="mw-disambig" title="스칼라">스칼라</a> <a href="/wiki/%EC%9E%A5_(%EB%AC%BC%EB%A6%AC)" class="mw-redirect" title="장 (물리)">장</a>을 다루는 상대론적 <a href="/wiki/%ED%8C%8C%EB%8F%99_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="파동 방정식">파동 방정식</a>이다. 상대론적인 <a href="/wiki/%EC%A7%88%EB%9F%89-%EC%97%90%EB%84%88%EC%A7%80_%EB%8F%99%EB%93%B1%EC%84%B1" class="mw-redirect" title="질량-에너지 동등성">질량-에너지 동등성</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 p^{2}=m^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{2}=m^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7e338af54fca21f40cc6a819c94c9e485c75ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.506ex; height:3.009ex;" alt="{\displaystyle p^{2}=m^{2}}"></span>을 나타낸다. </p><p>클라인-고든 방정식을 따르는 장은 슈뢰딩거 방정식처럼 단입자의 <a href="/wiki/%ED%99%95%EB%A5%A0_%EC%A7%84%ED%8F%AD" title="확률 진폭">확률 진폭</a>으로 해석할 수 없는데, 이는 이 방정식이 시간에 대하여 2차 편미분 방정식이어서 음의 에너지가 존재하고, 또 <a href="/wiki/%ED%99%95%EB%A5%A0_%ED%9D%90%EB%A6%84" title="확률 흐름">확률 흐름</a>을 보존하지 않기 때문이다. (다만, <a href="/w/index.php?title=%ED%8C%8C%EC%9D%B8%EB%A8%BC-%EC%8A%A4%ED%8A%80%EC%BC%88%EB%B2%A0%EB%A5%B4%ED%81%AC_%ED%95%B4%EC%84%9D&action=edit&redlink=1" class="new" title="파인먼-스튀켈베르크 해석 (없는 문서)">파인먼-스튀켈베르크 해석</a>(Feynman-Stückelberg interpretation)에 따라, 시간에 대해 앞뒤로 전파하는 입자에 대한 기술이라고 해석할 수 있다.) 최근에 발견된 것으로 추측되는 <a href="/wiki/%ED%9E%89%EC%8A%A4_%EB%B3%B4%EC%86%90" title="힉스 보손">힉스 보손</a>이나, 다른 스칼라 또는 유사스칼라 <a href="/wiki/%EA%B8%B0%EB%B3%B8_%EC%9E%85%EC%9E%90" title="기본 입자">기본 입자</a>(<a href="/wiki/%EC%B4%88%EB%8C%80%EC%B9%AD" title="초대칭">초대칭</a>에서의 여러 입자 등)나 스핀 0의 복합 입자(스칼라 <a href="/wiki/%EC%A4%91%EA%B0%84%EC%9E%90" title="중간자">중간자</a> 따위)를 다룰 때 유용하다. </p><p>클라인-고든 방정식은 <a href="/wiki/%ED%8A%B9%EC%88%98_%EC%83%81%EB%8C%80%EC%84%B1%EC%9D%B4%EB%A1%A0" title="특수 상대성이론">특수 상대성이론</a>의 <a href="/wiki/%EC%A7%88%EB%9F%89-%EC%97%90%EB%84%88%EC%A7%80_%EB%93%B1%EA%B0%80%EC%84%B1" class="mw-redirect" title="질량-에너지 등가성">질량-에너지 등가성</a>을 <a href="/wiki/%EC%96%91%EC%9E%90%EC%97%AD%ED%95%99" title="양자역학">양자역학</a>적으로 쓴 것이므로, 다른 모든 상대론적 <a href="/wiki/%ED%8C%8C%EB%8F%99_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="파동 방정식">파동 방정식</a>의 기본을 이룬다. 예를 들어, <a href="/wiki/%EC%8A%A4%ED%95%80" class="mw-disambig" title="스핀">스핀</a> 1/2의 <a href="/wiki/%EB%94%94%EB%9E%99_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="디랙 방정식">디랙 방정식</a>이나 <a href="/wiki/%EC%8A%A4%ED%95%80" class="mw-disambig" title="스핀">스핀</a> 1의 <a href="/wiki/%ED%94%84%EB%A1%9C%EC%B9%B4_%EB%B0%A9%EC%A0%95%EC%8B%9D" class="mw-redirect" title="프로카 방정식">프로카 방정식</a>은 클라인-고든 방정식의 특수한 경우다. 즉, 모든 디랙 방정식의 해와 프로카 방정식의 해는 클라인-고든 방정식을 만족한다. (그러나 그 역은 성립하지 않는다.) </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="정의"><span id=".EC.A0.95.EC.9D.98"></span>정의</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%81%B4%EB%9D%BC%EC%9D%B8-%EA%B3%A0%EB%93%A0_%EB%B0%A9%EC%A0%95%EC%8B%9D&action=edit&section=1" title="부분 편집: 정의"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>+−−− <a href="/wiki/%EA%B3%84%EB%9F%89_%EB%B6%80%ED%98%B8%EC%88%98" title="계량 부호수">계량 부호수</a>를 사용하고, <span class="mwe-math-element"><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=\hbar =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=\hbar =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da53d95302cc3d484fa56968fced5ddbb324c96c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.673ex; height:2.176ex;" alt="{\displaystyle c=\hbar =1}"></span>로 놓자. 실수 (전하를 가지지 않는) 스칼라 마당의 클라인-고든 방정식은 다음과 같다. </p> <dl><dd><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 (\partial ^{2}+m^{2})\psi =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\partial ^{2}+m^{2})\psi =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2cd6c781f3b760586c467c3ad8107dd6fd7d531" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.563ex; height:3.176ex;" alt="{\displaystyle (\partial ^{2}+m^{2})\psi =0.}"></span></dd></dl></dd></dl> <p>여기서 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>은 장의 <a href="/wiki/%EC%96%91%EC%9E%90" class="mw-disambig" title="양자">양자</a>의 <a href="/wiki/%EC%A7%88%EB%9F%89" title="질량">질량</a>이다. </p><p>이 꼴은 방정식 </p> <dl><dd><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 \psi =e^{-ip\cdot x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>p</mi> <mo>⋅</mo> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =e^{-ip\cdot x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/925345d96168afb01198cf24bd8bb03afb82a554" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.998ex; height:3.009ex;" alt="{\displaystyle \psi =e^{-ip\cdot x}}"></span></dd></dl></dd></dl> <p>의 <a href="/wiki/%ED%8F%89%EB%A9%B4%ED%8C%8C" title="평면파">평면파</a> 해에 <a href="/wiki/%ED%8A%B9%EC%88%98_%EC%83%81%EB%8C%80%EC%84%B1%EC%9D%B4%EB%A1%A0" title="특수 상대성이론">특수 상대성이론</a>의 <a href="/wiki/%EC%A7%88%EB%9F%89-%EC%97%90%EB%84%88%EC%A7%80_%EB%8F%99%EB%93%B1%EC%84%B1" class="mw-redirect" title="질량-에너지 동등성">질량-에너지 동등성</a>, 즉 </p> <dl><dd><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 p^{2}=m^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{2}=m^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7e338af54fca21f40cc6a819c94c9e485c75ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.506ex; height:3.009ex;" alt="{\displaystyle p^{2}=m^{2}}"></span></dd></dl></dd></dl> <p>을 가하여 얻어진다. <a href="/wiki/%EC%8A%88%EB%A2%B0%EB%94%A9%EA%B1%B0_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="슈뢰딩거 방정식">슈뢰딩거 방정식</a>과는 달리, 어떤 주어진 3차원 운동량 <span class="mwe-math-element"><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> </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/dd73e3862cb92b016721b8c492eadb4e8a577527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }"></span>에 대해 가능한 에너지 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b969ada68a88e2aeba9a2d2096abaf1fd53c21d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{0}}"></span>값이 양과 음 두가지다. </p> <div class="mw-heading mw-heading3"><h3 id="라그랑지언"><span id=".EB.9D.BC.EA.B7.B8.EB.9E.91.EC.A7.80.EC.96.B8"></span>라그랑지언</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%81%B4%EB%9D%BC%EC%9D%B8-%EA%B3%A0%EB%93%A0_%EB%B0%A9%EC%A0%95%EC%8B%9D&action=edit&section=2" title="부분 편집: 라그랑지언"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>클라인-고든 방정식은 다음 <a href="/wiki/%EB%9D%BC%EA%B7%B8%EB%9E%91%EC%A7%80%EC%96%B8" title="라그랑지언">라그랑지언</a>의 <a href="/wiki/%EC%98%A4%EC%9D%BC%EB%9F%AC-%EB%9D%BC%EA%B7%B8%EB%9E%91%EC%A3%BC_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="오일러-라그랑주 방정식">오일러-라그랑주 방정식</a>이다. </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 {\mathcal {L}}={\frac {1}{2}}(\partial \psi )^{2}-{\frac {1}{2}}m^{2}\psi ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∂</mi> <mi>ψ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial \psi )^{2}-{\frac {1}{2}}m^{2}\psi ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba26afd86b05d05186c16d7528f51cfbdf576c97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.897ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}={\frac {1}{2}}(\partial \psi )^{2}-{\frac {1}{2}}m^{2}\psi ^{2}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="유도"><span id=".EC.9C.A0.EB.8F.84"></span>유도</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%81%B4%EB%9D%BC%EC%9D%B8-%EA%B3%A0%EB%93%A0_%EB%B0%A9%EC%A0%95%EC%8B%9D&action=edit&section=3" title="부분 편집: 유도"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/w/index.php?title=%EC%9E%90%EC%9C%A0_%EC%9E%85%EC%9E%90&action=edit&redlink=1" class="new" title="자유 입자 (없는 문서)">자유 입자</a>의 비상대론적 에너지는 다음과 같다. </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 {\frac {\mathbf {p} ^{2}}{2m}}=E.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>E</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathbf {p} ^{2}}{2m}}=E.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ceee69f37ded361b2dac87b337532ea346bc467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.56ex; height:5.676ex;" alt="{\displaystyle {\frac {\mathbf {p} ^{2}}{2m}}=E.}"></span></dd></dl> <p>이를 양자화하면, <a href="/w/index.php?title=%EC%9E%90%EC%9C%A0_%EC%9E%85%EC%9E%90&action=edit&redlink=1" class="new" title="자유 입자 (없는 문서)">자유 입자</a>의 <a href="/wiki/%EC%8A%88%EB%A2%B0%EB%94%A9%EA%B1%B0_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="슈뢰딩거 방정식">슈뢰딩거 방정식</a>이 된다. </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 {\frac {\mathbf {p} ^{2}}{2m}}\psi =i\hbar {\frac {\partial }{\partial t}}\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo>=</mo> <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> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathbf {p} ^{2}}{2m}}\psi =i\hbar {\frac {\partial }{\partial t}}\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22f70ea50fd8d590f2ffb4e501e95b9cea443305" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.267ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathbf {p} ^{2}}{2m}}\psi =i\hbar {\frac {\partial }{\partial t}}\psi }"></span></dd></dl> <p>여기서 <span class="mwe-math-element"><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> </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/dd73e3862cb92b016721b8c492eadb4e8a577527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }"></span>는 운동량 <a href="/wiki/%EC%97%B0%EC%82%B0%EC%9E%90" title="연산자">연산자</a>이다. </p><p>이를 상대화하기 위하여, <a href="/wiki/%ED%8A%B9%EC%88%98_%EC%83%81%EB%8C%80%EC%84%B1%EC%9D%B4%EB%A1%A0" title="특수 상대성이론">특수 상대성이론</a>의 에너지 공식을 사용하자. </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 {\sqrt {\mathbf {p} ^{2}c^{2}+m^{2}c^{4}}}=E.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mi>E</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\mathbf {p} ^{2}c^{2}+m^{2}c^{4}}}=E.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bbeb7b2f0bf127303831428316a563a28e4b84e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:20.441ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\mathbf {p} ^{2}c^{2}+m^{2}c^{4}}}=E.}"></span></dd></dl> <p>마찬가지로 <a href="/wiki/%EC%96%91%EC%9E%90%ED%99%94_(%EB%AC%BC%EB%A6%AC%ED%95%99)" title="양자화 (물리학)">양자화</a>하면 다음과 같다. </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 {\sqrt {(-i\hbar \mathbf {\nabla } )^{2}c^{2}+m^{2}c^{4}}}\psi =i\hbar {\frac {\partial }{\partial t}}\psi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </msqrt> </mrow> <mi>ψ</mi> <mo>=</mo> <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> <mi>ψ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {(-i\hbar \mathbf {\nabla } )^{2}c^{2}+m^{2}c^{4}}}\psi =i\hbar {\frac {\partial }{\partial t}}\psi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e707b5b7dd47662223cc313907d076016e0fe38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.972ex; height:5.509ex;" alt="{\displaystyle {\sqrt {(-i\hbar \mathbf {\nabla } )^{2}c^{2}+m^{2}c^{4}}}\psi =i\hbar {\frac {\partial }{\partial t}}\psi .}"></span></dd></dl> <p>그러나 이 공식은 제곱근이 들어가 있기 때문에 다루기 힘들며, 또 <a href="/w/index.php?title=%EA%B5%AD%EC%86%8C%EC%84%B1&action=edit&redlink=1" class="new" title="국소성 (없는 문서)">비국소적</a>이다. 대신, 에너지 공식의 양변을 제곱하자. </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} ^{2}c^{2}+m^{2}c^{4}=E^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} ^{2}c^{2}+m^{2}c^{4}=E^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3560b378aa5179a14af405c2e92b1002de34783e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.543ex; height:3.009ex;" alt="{\displaystyle \mathbf {p} ^{2}c^{2}+m^{2}c^{4}=E^{2}}"></span></dd></dl> <p>이를 양자화하면 다음과 같다. </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 ((-i\hbar \mathbf {\nabla } )^{2}c^{2}+m^{2}c^{4})\psi =(i\hbar {\frac {\partial }{\partial t}})^{2}\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <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> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((-i\hbar \mathbf {\nabla } )^{2}c^{2}+m^{2}c^{4})\psi =(i\hbar {\frac {\partial }{\partial t}})^{2}\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1e632a9e5125720c49287ed6e9c57ccc3ccb208" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.674ex; height:5.509ex;" alt="{\displaystyle ((-i\hbar \mathbf {\nabla } )^{2}c^{2}+m^{2}c^{4})\psi =(i\hbar {\frac {\partial }{\partial t}})^{2}\psi }"></span></dd></dl> <p>고쳐 쓰면, </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 -\hbar ^{2}c^{2}\mathbf {\nabla } ^{2}\psi +m^{2}c^{4}\psi =-\hbar ^{2}{\frac {\partial ^{2}}{(\partial t)^{2}}}\psi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msup> <mi class="MJX-variant">ℏ</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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> <mo>+</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>ψ</mi> <mo>=</mo> <mo>−</mo> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∂</mi> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\hbar ^{2}c^{2}\mathbf {\nabla } ^{2}\psi +m^{2}c^{4}\psi =-\hbar ^{2}{\frac {\partial ^{2}}{(\partial t)^{2}}}\psi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be14e97d3b2c264dfc0897484a2d0ca570995a0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.558ex; height:6.509ex;" alt="{\displaystyle -\hbar ^{2}c^{2}\mathbf {\nabla } ^{2}\psi +m^{2}c^{4}\psi =-\hbar ^{2}{\frac {\partial ^{2}}{(\partial t)^{2}}}\psi .}"></span></dd></dl> <p>항을 옮기면 다음을 얻는다. </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 {\frac {1}{c^{2}}}{\frac {\partial ^{2}}{(\partial t)^{2}}}\psi -\mathbf {\nabla } ^{2}\psi +{\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∂</mi> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>ψ</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{c^{2}}}{\frac {\partial ^{2}}{(\partial t)^{2}}}\psi -\mathbf {\nabla } ^{2}\psi +{\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f799405b9934a26e2fb8bc45b1ef1908211f948" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.865ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{c^{2}}}{\frac {\partial ^{2}}{(\partial t)^{2}}}\psi -\mathbf {\nabla } ^{2}\psi +{\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi =0.}"></span></dd></dl> <p>모든 복소수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>가 사라졌으므로, 이 방정식은 복소 마당뿐만 아니라 실수값을 가지는 마당에도 적용할 수 있다. </p><p>상대론적 표기법으로 쓰면, 다음과 같이 된다. </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 (\Box +\mu ^{2})\psi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>◻</mi> <mo>+</mo> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Box +\mu ^{2})\psi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3502634a4834c0725f0d0c5021f79e764e9fadb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.688ex; height:3.176ex;" alt="{\displaystyle (\Box +\mu ^{2})\psi =0}"></span></dd></dl> <p>여기서 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></span>는 <a href="/wiki/%EB%8B%AC%EB%9E%91%EB%B2%A0%EB%A5%B4_%EC%97%B0%EC%82%B0%EC%9E%90" title="달랑베르 연산자">달랑베르 연산자</a>이다. </p> <div class="mw-heading mw-heading2"><h2 id="역사"><span id=".EC.97.AD.EC.82.AC"></span>역사</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%81%B4%EB%9D%BC%EC%9D%B8-%EA%B3%A0%EB%93%A0_%EB%B0%A9%EC%A0%95%EC%8B%9D&action=edit&section=4" title="부분 편집: 역사"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%EC%97%90%EB%A5%B4%EB%B9%88_%EC%8A%88%EB%A2%B0%EB%94%A9%EA%B1%B0" title="에르빈 슈뢰딩거">에르빈 슈뢰딩거</a>는 <a href="/wiki/%EC%A0%84%EC%9E%90" title="전자">전자</a>의 <a href="/wiki/%EB%AC%BC%EC%A7%88%ED%8C%8C" title="물질파">물질파</a>를 기술하는 방정식을 찾다가 1925년경에 오늘날 클라인-고든 방정식이라 불리는 방정식을 고안하였다. 이 방정식은 전자의 스핀을 무시하기 때문에, <a href="/wiki/%EC%88%98%EC%86%8C" title="수소">수소</a> 원자의 <a href="/w/index.php?title=%EC%A0%84%EC%9E%90_%EA%B5%AC%EC%A1%B0&action=edit&redlink=1" class="new" title="전자 구조 (없는 문서)">전자 구조</a>를 올바르게 예측하지 못한다. 그러나 슈뢰딩거는 이 방정식의 비상대적 <a href="/wiki/%EA%B7%B9%ED%95%9C" title="극한">극한</a>이 유용하다는 것을 깨닫고, 이를 1926년 1월에 출판하였다. 이 방정식은 이후 '<a href="/wiki/%EC%8A%88%EB%A2%B0%EB%94%A9%EA%B1%B0_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="슈뢰딩거 방정식">슈뢰딩거 방정식</a>'이라고 알려진다. 같은 해 <a href="/wiki/%EC%86%8C%EB%A0%A8" title="소련">소련</a>의 <a href="/wiki/%EB%B8%94%EB%9D%BC%EB%94%94%EB%AF%B8%EB%A5%B4_%ED%8F%AC%ED%81%AC" title="블라디미르 포크">블라디미르 포크</a>는 슈뢰딩거 방정식을 <a href="/wiki/%EC%9E%90%EA%B8%B0%EC%9E%A5" title="자기장">자기장</a>이 있을 경우로 일반화하여, 클라인-고든 방정식을 유도하였으나, 이는 서방 학계에 잘 알려지지 않았다. </p><p>곧 <a href="/wiki/%EC%98%A4%EC%8A%A4%EC%B9%B4%EB%A5%B4_%ED%81%B4%EB%A0%88%EC%9D%B8" title="오스카르 클레인">오스카르 클레인</a><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>과 <a href="/wiki/%EB%B0%9C%ED%84%B0_%EA%B3%A0%EB%A5%B4%EB%8F%88" title="발터 고르돈">발터 고르돈</a><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>이 상대론적 <a href="/wiki/%EC%A0%84%EC%9E%90" title="전자">전자</a>를 기술하기 위해 이 방정식을 제시하였고, 이를 따 "클라인-고든 방정식"으로 알려지게 되었다. </p> <div class="mw-heading mw-heading2"><h2 id="각주"><span id=".EA.B0.81.EC.A3.BC"></span>각주</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%81%B4%EB%9D%BC%EC%9D%B8-%EA%B3%A0%EB%93%A0_%EB%B0%A9%EC%A0%95%EC%8B%9D&action=edit&section=5" title="부분 편집: 각주"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r35556958">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><cite class="citation journal">Klein, O. (1926년 12월). “Quantentheorie und fünfdimensionale Relativitätstheorie”. 《Zeitschrift für Physik》 (독일어) <b>37</b>: 895–906. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007%2FBF01397481">10.1007/BF01397481</a>. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EC%9D%BC%EB%A0%A8_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 일련 번호">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0044-3328">0044-3328</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zeitschrift+f%C3%BCr+Physik&rft.atitle=Quantentheorie+und+f%C3%BCnfdimensionale+Relativit%C3%A4tstheorie&rft.volume=37&rft.pages=895-906&rft.date=1926-12&rft_id=info%3Adoi%2F10.1007%2FBF01397481&rft.issn=0044-3328&rft.aulast=Klein&rft.aufirst=O.&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%81%B4%EB%9D%BC%EC%9D%B8-%EA%B3%A0%EB%93%A0+%EB%B0%A9%EC%A0%95%EC%8B%9D" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><cite class="citation journal">Gordon, W. (1927년 1월). “Der Comptoneffekt nach der Schrödingerschen Theorie”. 《Zeitschrift für Physik》 (독일어) <b>40</b> (1–2): 117–133. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007%2FBF01390840">10.1007/BF01390840</a>. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EC%9D%BC%EB%A0%A8_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 일련 번호">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0044-3328">0044-3328</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zeitschrift+f%C3%BCr+Physik&rft.atitle=Der+Comptoneffekt+nach+der+Schr%C3%B6dingerschen+Theorie&rft.volume=40&rft.issue=1%E2%80%932&rft.pages=117-133&rft.date=1927-01&rft_id=info%3Adoi%2F10.1007%2FBF01390840&rft.issn=0044-3328&rft.aulast=Gordon&rft.aufirst=W.&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%81%B4%EB%9D%BC%EC%9D%B8-%EA%B3%A0%EB%93%A0+%EB%B0%A9%EC%A0%95%EC%8B%9D" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div></div> <ul><li><cite class="citation book">Peskin, Michael E.; Daniel V. Schroeder (1995년 10월). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140902045539/http://physics.weber.edu/schroeder/qftbook.html">《An introduction to quantum field theory》</a> (영어). Westview Press. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/0-201-50397-2" title="특수:책찾기/0-201-50397-2"><bdi>0-201-50397-2</bdi></a>. 2014년 9월 2일에 <a rel="nofollow" class="external text" href="http://physics.weber.edu/schroeder/qftbook.html">원본 문서</a>에서 보존된 문서<span class="reference-accessdate">. 2014년 10월 1일에 확인함</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+quantum+field+theory&rft.pub=Westview+Press&rft.date=1995-10&rft.isbn=0-201-50397-2&rft.aulast=Peskin&rft.aufirst=Michael+E.&rft_id=http%3A%2F%2Fphysics.weber.edu%2Fschroeder%2Fqftbook.html&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%81%B4%EB%9D%BC%EC%9D%B8-%EA%B3%A0%EB%93%A0+%EB%B0%A9%EC%A0%95%EC%8B%9D" class="Z3988"><span style="display:none;"> </span></span> <span style="display:none;font-size:100%" class="error citation-comment">더 이상 지원되지 않는 변수를 사용함 (<a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EC%9D%B8%EC%9A%A9_%EC%98%A4%EB%A5%98_%EB%8F%84%EC%9B%80%EB%A7%90#deprecated_params" title="위키백과:인용 오류 도움말">도움말</a>)</span></li></ul> <div class="mw-heading mw-heading2"><h2 id="외부_링크"><span id=".EC.99.B8.EB.B6.80_.EB.A7.81.ED.81.AC"></span>외부 링크</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%81%B4%EB%9D%BC%EC%9D%B8-%EA%B3%A0%EB%93%A0_%EB%B0%A9%EC%A0%95%EC%8B%9D&action=edit&section=6" title="부분 편집: 외부 링크"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation web"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Klein-Gordon_equation">“Klein-Gordon equation”</a>. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. 2001. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-1-55608-010-4" title="특수:책찾기/978-1-55608-010-4"><bdi>978-1-55608-010-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Klein-Gordon+equation&rft.date=2001&rft.isbn=978-1-55608-010-4&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FKlein-Gordon_equation&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%81%B4%EB%9D%BC%EC%9D%B8-%EA%B3%A0%EB%93%A0+%EB%B0%A9%EC%A0%95%EC%8B%9D" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Massless_Klein-Gordon_equation">“Massless Klein-Gordon equation”</a>. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. 2001. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-1-55608-010-4" title="특수:책찾기/978-1-55608-010-4"><bdi>978-1-55608-010-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Massless+Klein-Gordon+equation&rft.date=2001&rft.isbn=978-1-55608-010-4&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FMassless_Klein-Gordon_equation&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%81%B4%EB%9D%BC%EC%9D%B8-%EA%B3%A0%EB%93%A0+%EB%B0%A9%EC%A0%95%EC%8B%9D" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web">Weisstein, Eric Wolfgang. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Klein-GordonEquation.html">“Klein-Gordon Equation”</a>. 《<a href="/wiki/%EB%A7%A4%EC%8A%A4%EC%9B%94%EB%93%9C" title="매스월드">Wolfram MathWorld</a>》 (영어). Wolfram Research.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=Klein-Gordon+Equation&rft.aulast=Weisstein&rft.aufirst=Eric+Wolfgang&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FKlein-GordonEquation.html&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%81%B4%EB%9D%BC%EC%9D%B8-%EA%B3%A0%EB%93%A0+%EB%B0%A9%EC%A0%95%EC%8B%9D" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="같이_보기"><span id=".EA.B0.99.EC.9D.B4_.EB.B3.B4.EA.B8.B0"></span>같이 보기</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%81%B4%EB%9D%BC%EC%9D%B8-%EA%B3%A0%EB%93%A0_%EB%B0%A9%EC%A0%95%EC%8B%9D&action=edit&section=7" title="부분 편집: 같이 보기"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%EC%82%AC%EC%9D%B8-%EA%B3%A0%EB%93%A0_%EB%B0%A9%EC%A0%95%EC%8B%9D" 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