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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="Projekt"> <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="Obsah" 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">Obsah</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">přesunout do postranního panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">skrýt</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">(úvod)</div> </a> </li> <li id="toc-Odvození_rovnice" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Odvození_rovnice"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Odvození rovnice</span> </div> </a> <ul id="toc-Odvození_rovnice-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Obecné_vyjádření" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Obecné_vyjádření"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Obecné vyjádření</span> </div> </a> <ul id="toc-Obecné_vyjádření-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stacionární_Schrödingerova_rovnice" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Stacionární_Schrödingerova_rovnice"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Stacionární Schrödingerova rovnice</span> </div> </a> <button aria-controls="toc-Stacionární_Schrödingerova_rovnice-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>Přepnout podsekci Stacionární Schrödingerova rovnice</span> </button> <ul id="toc-Stacionární_Schrödingerova_rovnice-sublist" class="vector-toc-list"> <li id="toc-Stacionární_stav" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stacionární_stav"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Stacionární stav</span> </div> </a> <ul id="toc-Stacionární_stav-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Vlastnosti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vlastnosti"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Vlastnosti</span> </div> </a> <ul id="toc-Vlastnosti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Související_články" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Související_články"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Související články</span> </div> </a> <ul id="toc-Související_články-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Externí_odkazy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Externí_odkazy"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Externí odkazy</span> </div> </a> <ul id="toc-Externí_odkazy-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="Obsah" 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="Přepnout obsah" > <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">Přepnout obsah</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">Schrödingerova rovnice</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="Přejděte k článku v jiném jazyce. Je dostupný v 72 jazycích" > <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-72" 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">72 jazyků</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Schr%C3%B6dinger-vergelyking" title="Schrödinger-vergelyking – afrikánština" lang="af" hreflang="af" data-title="Schrödinger-vergelyking" data-language-autonym="Afrikaans" data-language-local-name="afrikánština" class="interlanguage-link-target"><span>Afrikaans</span></a></li><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_%D8%B4%D8%B1%D9%88%D8%AF%D9%86%D8%BA%D8%B1" title="معادلة شرودنغر – arabština" lang="ar" hreflang="ar" data-title="معادلة شرودنغر" data-language-autonym="العربية" data-language-local-name="arabština" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A9_%D8%B4%D8%B1%D9%88%D8%AF%D9%8A%D9%86%D8%AC%D8%B1" title="معادلة شرودينجر – arabština (egyptská)" lang="arz" hreflang="arz" data-title="معادلة شرودينجر" data-language-autonym="مصرى" data-language-local-name="arabština (egyptská)" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Ecuaci%C3%B3n_de_Schr%C3%B6dinger" title="Ecuación de Schrödinger – asturština" lang="ast" hreflang="ast" data-title="Ecuación de Schrödinger" data-language-autonym="Asturianu" data-language-local-name="asturština" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C5%9Eredinger_t%C9%99nliyi" title="Şredinger tənliyi – ázerbájdžánština" lang="az" hreflang="az" data-title="Şredinger tənliyi" data-language-autonym="Azərbaycanca" data-language-local-name="ázerbájdžánština" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D1%9E%D0%BD%D0%B5%D0%BD%D0%BD%D0%B5_%D0%A8%D1%80%D0%BE%D0%B4%D0%B7%D1%96%D0%BD%D0%B3%D0%B5%D1%80%D0%B0" title="Ураўненне Шродзінгера – běloruština" lang="be" hreflang="be" data-title="Ураўненне Шродзінгера" data-language-autonym="Беларуская" data-language-local-name="běloruština" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%BD%D0%B0_%D0%A8%D1%80%D1%8C%D0%BE%D0%B4%D0%B8%D0%BD%D0%B3%D0%B5%D1%80" title="Уравнение на Шрьодингер – bulharština" lang="bg" hreflang="bg" data-title="Уравнение на Шрьодингер" data-language-autonym="Български" data-language-local-name="bulharština" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B6%E0%A7%8D%E0%A6%B0%E0%A7%8B%E0%A6%A1%E0%A6%BF%E0%A6%99%E0%A6%BE%E0%A6%B0_%E0%A6%B8%E0%A6%AE%E0%A7%80%E0%A6%95%E0%A6%B0%E0%A6%A3" title="শ্রোডিঙার সমীকরণ – bengálština" lang="bn" hreflang="bn" data-title="শ্রোডিঙার সমীকরণ" data-language-autonym="বাংলা" data-language-local-name="bengálština" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Schr%C3%B6dingerova_jedna%C4%8Dina" title="Schrödingerova jednačina – bosenština" lang="bs" hreflang="bs" data-title="Schrödingerova jednačina" data-language-autonym="Bosanski" data-language-local-name="bosenština" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Equaci%C3%B3_de_Schr%C3%B6dinger" title="Equació de Schrödinger – katalánština" lang="ca" hreflang="ca" data-title="Equació de Schrödinger" data-language-autonym="Català" data-language-local-name="katalánština" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A8%D1%80%D1%91%D0%B4%D0%B8%D0%BD%D0%B3%D0%B5%D1%80_%D1%82%D0%B0%D0%BD%D0%BB%C4%83%D1%85%C4%95" title="Шрёдингер танлăхĕ – čuvaština" lang="cv" hreflang="cv" data-title="Шрёдингер танлăхĕ" data-language-autonym="Чӑвашла" data-language-local-name="čuvaština" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Schr%C3%B6dingers_ligning" title="Schrödingers ligning – dánština" lang="da" hreflang="da" data-title="Schrödingers ligning" data-language-autonym="Dansk" data-language-local-name="dánština" 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/Schr%C3%B6dingergleichung" title="Schrödingergleichung – němčina" lang="de" hreflang="de" data-title="Schrödingergleichung" data-language-autonym="Deutsch" data-language-local-name="němčina" 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%A3%CF%81%CE%AD%CE%BD%CF%84%CE%B9%CE%BD%CE%B3%CE%BA%CE%B5%CF%81" title="Εξίσωση Σρέντινγκερ – řečtina" lang="el" hreflang="el" data-title="Εξίσωση Σρέντινγκερ" data-language-autonym="Ελληνικά" data-language-local-name="řečtina" 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/Schr%C3%B6dinger_equation" title="Schrödinger equation – angličtina" lang="en" hreflang="en" data-title="Schrödinger equation" data-language-autonym="English" data-language-local-name="angličtina" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Ekvacio_de_Schr%C3%B6dinger" title="Ekvacio de Schrödinger – esperanto" lang="eo" hreflang="eo" data-title="Ekvacio de Schrödinger" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ecuaci%C3%B3n_de_Schr%C3%B6dinger" title="Ecuación de Schrödinger – španělština" lang="es" hreflang="es" data-title="Ecuación de Schrödinger" data-language-autonym="Español" data-language-local-name="španělština" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Schr%C3%B6dingeri_v%C3%B5rrand" title="Schrödingeri võrrand – estonština" lang="et" hreflang="et" data-title="Schrödingeri võrrand" data-language-autonym="Eesti" data-language-local-name="estonština" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Schr%C3%B6dingerren_ekuazioa" title="Schrödingerren ekuazioa – baskičtina" lang="eu" hreflang="eu" data-title="Schrödingerren ekuazioa" data-language-autonym="Euskara" data-language-local-name="baskičtina" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D9%87_%D8%B4%D8%B1%D9%88%D8%AF%DB%8C%D9%86%DA%AF%D8%B1" title="معادله شرودینگر – perština" lang="fa" hreflang="fa" data-title="معادله شرودینگر" data-language-autonym="فارسی" data-language-local-name="perština" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Schr%C3%B6dingerin_yht%C3%A4l%C3%B6" title="Schrödingerin yhtälö – finština" lang="fi" hreflang="fi" data-title="Schrödingerin yhtälö" data-language-autonym="Suomi" data-language-local-name="finština" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89quation_de_Schr%C3%B6dinger" title="Équation de Schrödinger – francouzština" lang="fr" hreflang="fr" data-title="Équation de Schrödinger" data-language-autonym="Français" data-language-local-name="francouzština" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Ecuaci%C3%B3n_de_Schr%C3%B6dinger" title="Ecuación de Schrödinger – galicijština" lang="gl" hreflang="gl" data-title="Ecuación de Schrödinger" data-language-autonym="Galego" data-language-local-name="galicijština" class="interlanguage-link-target"><span>Galego</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%A9%D7%A8%D7%93%D7%99%D7%A0%D7%92%D7%A8" title="משוואת שרדינגר – hebrejština" lang="he" hreflang="he" data-title="משוואת שרדינגר" data-language-autonym="עברית" data-language-local-name="hebrejština" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B6%E0%A5%8D%E0%A4%B0%E0%A5%8B%E0%A4%A1%E0%A4%BF%E0%A4%82%E0%A4%97%E0%A4%B0_%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3" title="श्रोडिंगर समीकरण – hindština" lang="hi" hreflang="hi" data-title="श्रोडिंगर समीकरण" data-language-autonym="हिन्दी" data-language-local-name="hindština" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Schr%C3%B6dingerova_jednad%C5%BEba" title="Schrödingerova jednadžba – chorvatština" lang="hr" hreflang="hr" data-title="Schrödingerova jednadžba" data-language-autonym="Hrvatski" data-language-local-name="chorvatština" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Schr%C3%B6dinger-egyenlet" title="Schrödinger-egyenlet – maďarština" lang="hu" hreflang="hu" data-title="Schrödinger-egyenlet" data-language-autonym="Magyar" data-language-local-name="maďarština" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%87%D6%80%D5%B5%D5%B8%D5%A4%D5%AB%D5%B6%D5%A3%D5%A5%D6%80%D5%AB_%D5%B0%D5%A1%D5%BE%D5%A1%D5%BD%D5%A1%D6%80%D5%B8%D6%82%D5%B4" title="Շրյոդինգերի հավասարում – arménština" lang="hy" hreflang="hy" data-title="Շրյոդինգերի հավասարում" data-language-autonym="Հայերեն" data-language-local-name="arménština" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Equation_de_Schr%C3%B6dinger" title="Equation de Schrödinger – interlingua" lang="ia" hreflang="ia" data-title="Equation de Schrödinger" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Persamaan_Schr%C3%B6dinger" title="Persamaan Schrödinger – indonéština" lang="id" hreflang="id" data-title="Persamaan Schrödinger" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonéština" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Jafna_Schr%C3%B6dingers" title="Jafna Schrödingers – islandština" lang="is" hreflang="is" data-title="Jafna Schrödingers" data-language-autonym="Íslenska" data-language-local-name="islandština" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Equazione_di_Schr%C3%B6dinger" title="Equazione di Schrödinger – italština" lang="it" hreflang="it" data-title="Equazione di Schrödinger" data-language-autonym="Italiano" data-language-local-name="italština" 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%B7%E3%83%A5%E3%83%AC%E3%83%BC%E3%83%87%E3%82%A3%E3%83%B3%E3%82%AC%E3%83%BC%E6%96%B9%E7%A8%8B%E5%BC%8F" title="シュレーディンガー方程式 – japonština" lang="ja" hreflang="ja" data-title="シュレーディンガー方程式" data-language-autonym="日本語" data-language-local-name="japonština" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A8%E1%83%A0%E1%83%94%E1%83%93%E1%83%98%E1%83%9C%E1%83%92%E1%83%94%E1%83%A0%E1%83%98%E1%83%A1_%E1%83%92%E1%83%90%E1%83%9C%E1%83%A2%E1%83%9D%E1%83%9A%E1%83%94%E1%83%91%E1%83%90" title="შრედინგერის განტოლება – gruzínština" lang="ka" hreflang="ka" data-title="შრედინგერის განტოლება" data-language-autonym="ქართული" data-language-local-name="gruzínština" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/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 – korejština" lang="ko" hreflang="ko" data-title="슈뢰딩거 방정식" data-language-autonym="한국어" data-language-local-name="korejština" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Schr%C3%B6dingervergelieking" title="Schrödingervergelieking – limburština" lang="li" hreflang="li" data-title="Schrödingervergelieking" data-language-autonym="Limburgs" data-language-local-name="limburština" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/%C5%A0r%C4%97dingerio_lygtis" title="Šrėdingerio lygtis – litevština" lang="lt" hreflang="lt" data-title="Šrėdingerio lygtis" data-language-autonym="Lietuvių" data-language-local-name="litevština" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/%C5%A0r%C4%93dingera_vien%C4%81dojums" title="Šrēdingera vienādojums – lotyština" lang="lv" hreflang="lv" data-title="Šrēdingera vienādojums" data-language-autonym="Latviešu" data-language-local-name="lotyština" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk badge-Q17437796 badge-featuredarticle mw-list-item" title="nejlepší článek"><a href="https://mk.wikipedia.org/wiki/%D0%A8%D1%80%D0%B5%D0%B4%D0%B8%D0%BD%D0%B3%D0%B5%D1%80%D0%BE%D0%B2%D0%B0_%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D0%BA%D0%B0" title="Шредингерова равенка – makedonština" lang="mk" hreflang="mk" data-title="Шредингерова равенка" data-language-autonym="Македонски" data-language-local-name="makedonština" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A8%D1%80%D0%B5%D0%B4%D0%B8%D0%BD%D0%B3%D0%B5%D1%80%D0%B8%D0%B9%D0%BD_%D1%82%D1%8D%D0%B3%D1%88%D0%B8%D1%82%D0%B3%D1%8D%D0%BB" title="Шредингерийн тэгшитгэл – mongolština" lang="mn" hreflang="mn" data-title="Шредингерийн тэгшитгэл" data-language-autonym="Монгол" data-language-local-name="mongolština" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Persamaan_Schr%C3%B6dinger" title="Persamaan Schrödinger – malajština" lang="ms" hreflang="ms" data-title="Persamaan Schrödinger" data-language-autonym="Bahasa Melayu" data-language-local-name="malajština" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Ekwazzjoni_ta%27_Schr%C3%B6dinger" title="Ekwazzjoni ta' Schrödinger – maltština" lang="mt" hreflang="mt" data-title="Ekwazzjoni ta' Schrödinger" data-language-autonym="Malti" data-language-local-name="maltština" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Schr%C3%B6dingervergelijking" title="Schrödingervergelijking – nizozemština" lang="nl" hreflang="nl" data-title="Schrödingervergelijking" data-language-autonym="Nederlands" data-language-local-name="nizozemština" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Schr%C3%B6dingerlikninga" title="Schrödingerlikninga – norština (nynorsk)" lang="nn" hreflang="nn" data-title="Schrödingerlikninga" data-language-autonym="Norsk nynorsk" data-language-local-name="norština (nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Schr%C3%B6dinger-ligning" title="Schrödinger-ligning – norština (bokmål)" lang="nb" hreflang="nb" data-title="Schrödinger-ligning" data-language-autonym="Norsk bokmål" data-language-local-name="norština (bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Equacion_de_Schr%C3%B6dinger" title="Equacion de Schrödinger – okcitánština" lang="oc" hreflang="oc" data-title="Equacion de Schrödinger" data-language-autonym="Occitan" data-language-local-name="okcitánština" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A8%BC%E0%A9%8D%E0%A8%B0%E0%A9%8B%E0%A8%A1%E0%A8%BF%E0%A9%B0%E0%A8%9C%E0%A8%B0_%E0%A8%87%E0%A8%95%E0%A9%81%E0%A8%8F%E0%A8%B8%E0%A8%BC%E0%A8%A8" title="ਸ਼੍ਰੋਡਿੰਜਰ ਇਕੁਏਸ਼ਨ – paňdžábština" lang="pa" hreflang="pa" data-title="ਸ਼੍ਰੋਡਿੰਜਰ ਇਕੁਏਸ਼ਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="paňdžábština" 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_Schr%C3%B6dingera" title="Równanie Schrödingera – polština" lang="pl" hreflang="pl" data-title="Równanie Schrödingera" data-language-autonym="Polski" data-language-local-name="polština" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%B4%D8%B1%D9%88%DA%88%D9%86%DA%AF%D8%B1_%D9%85%D8%B3%D8%A7%D9%88%D8%A7%D8%AA" title="شروڈنگر مساوات – Western Punjabi" lang="pnb" hreflang="pnb" data-title="شروڈنگر مساوات" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</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_Schr%C3%B6dinger" title="Equação de Schrödinger – portugalština" lang="pt" hreflang="pt" data-title="Equação de Schrödinger" data-language-autonym="Português" data-language-local-name="portugalština" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Ecua%C8%9Bia_lui_Schr%C3%B6dinger" title="Ecuația lui Schrödinger – rumunština" lang="ro" hreflang="ro" data-title="Ecuația lui Schrödinger" data-language-autonym="Română" data-language-local-name="rumunština" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%A8%D1%80%D1%91%D0%B4%D0%B8%D0%BD%D0%B3%D0%B5%D1%80%D0%B0" title="Уравнение Шрёдингера – ruština" lang="ru" hreflang="ru" data-title="Уравнение Шрёдингера" data-language-autonym="Русский" data-language-local-name="ruština" 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/Schr%C3%B6dingerova_jedna%C4%8Dina" title="Schrödingerova jednačina – srbochorvatština" lang="sh" hreflang="sh" data-title="Schrödingerova jednačina" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srbochorvatština" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Schrödinger equation" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Schr%C3%B6dingerova_rovnica" title="Schrödingerova rovnica – slovenština" lang="sk" hreflang="sk" data-title="Schrödingerova rovnica" data-language-autonym="Slovenčina" data-language-local-name="slovenština" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Schr%C3%B6dingerjeva_ena%C4%8Dba" title="Schrödingerjeva enačba – slovinština" lang="sl" hreflang="sl" data-title="Schrödingerjeva enačba" data-language-autonym="Slovenščina" data-language-local-name="slovinština" 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_i_Shrodingerit" title="Ekuacioni i Shrodingerit – albánština" lang="sq" hreflang="sq" data-title="Ekuacioni i Shrodingerit" data-language-autonym="Shqip" data-language-local-name="albánština" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%C5%A0redingerova_jedna%C4%8Dina" title="Šredingerova jednačina – srbština" lang="sr" hreflang="sr" data-title="Šredingerova jednačina" data-language-autonym="Српски / srpski" data-language-local-name="srbština" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Schr%C3%B6dingerekvationen" title="Schrödingerekvationen – švédština" lang="sv" hreflang="sv" data-title="Schrödingerekvationen" data-language-autonym="Svenska" data-language-local-name="švédština" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AF%81%E0%AE%B0%E0%AF%8B%E0%AE%9F%E0%AE%BF%E0%AE%99%E0%AF%8D%E0%AE%95%E0%AE%B0%E0%AF%8D_%E0%AE%9A%E0%AE%AE%E0%AE%A9%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81" title="சுரோடிங்கர் சமன்பாடு – tamilština" lang="ta" hreflang="ta" data-title="சுரோடிங்கர் சமன்பாடு" data-language-autonym="தமிழ்" data-language-local-name="tamilština" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AA%E0%B8%A1%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%8A%E0%B9%80%E0%B8%A3%E0%B8%AD%E0%B8%94%E0%B8%B4%E0%B8%87%E0%B9%80%E0%B8%87%E0%B8%AD%E0%B8%A3%E0%B9%8C" title="สมการชเรอดิงเงอร์ – thajština" lang="th" hreflang="th" data-title="สมการชเรอดิงเงอร์" data-language-autonym="ไทย" data-language-local-name="thajština" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Ekwasyong_Schr%C3%B6dinger" title="Ekwasyong Schrödinger – tagalog" lang="tl" hreflang="tl" data-title="Ekwasyong Schrödinger" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Schr%C3%B6dinger_denklemi" title="Schrödinger denklemi – turečtina" lang="tr" hreflang="tr" data-title="Schrödinger denklemi" data-language-autonym="Türkçe" data-language-local-name="turečtina" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A8%D1%80%D3%A9%D0%B4%D0%B8%D0%BD%D0%B3%D0%B5%D1%80_%D1%82%D0%B8%D0%B3%D0%B5%D0%B7%D0%BB%D3%99%D0%BC%D3%99%D1%81%D0%B5" title="Шрөдингер тигезләмәсе – tatarština" lang="tt" hreflang="tt" data-title="Шрөдингер тигезләмәсе" data-language-autonym="Татарча / tatarça" data-language-local-name="tatarština" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D1%96%D0%B2%D0%BD%D1%8F%D0%BD%D0%BD%D1%8F_%D0%A8%D1%80%D0%B5%D0%B4%D1%96%D0%BD%D0%B3%D0%B5%D1%80%D0%B0" title="Рівняння Шредінгера – ukrajinština" lang="uk" hreflang="uk" data-title="Рівняння Шредінгера" data-language-autonym="Українська" data-language-local-name="ukrajinština" 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/Shredinger_tenglamasi" title="Shredinger tenglamasi – uzbečtina" lang="uz" hreflang="uz" data-title="Shredinger tenglamasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbečtina" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C6%B0%C6%A1ng_tr%C3%ACnh_Schr%C3%B6dinger" title="Phương trình Schrödinger – vietnamština" lang="vi" hreflang="vi" data-title="Phương trình Schrödinger" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamština" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%96%9B%E5%AE%9A%E8%B0%94%E6%96%B9%E7%A8%8B" title="薛定谔方程 – čínština (dialekty Wu)" lang="wuu" hreflang="wuu" data-title="薛定谔方程" data-language-autonym="吴语" data-language-local-name="čínština (dialekty Wu)" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A9%D7%A8%D7%A2%D7%93%D7%99%D7%A0%D7%92%D7%A2%D7%A8_%D7%92%D7%9C%D7%99%D7%99%D7%9B%D7%95%D7%A0%D7%92" title="שרעדינגער גלייכונג – jidiš" lang="yi" hreflang="yi" data-title="שרעדינגער גלייכונג" data-language-autonym="ייִדיש" data-language-local-name="jidiš" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh badge-Q17437798 badge-goodarticle mw-list-item" title="dobrý článek"><a href="https://zh.wikipedia.org/wiki/%E8%96%9B%E5%AE%9A%E8%B0%94%E6%96%B9%E7%A8%8B" title="薛定谔方程 – čínština" lang="zh" hreflang="zh" data-title="薛定谔方程" data-language-autonym="中文" data-language-local-name="čínština" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E8%96%9B%E5%AE%9A%E8%AB%A4%E6%96%B9%E7%A8%8B" title="薛定諤方程 – čínština (klasická)" lang="lzh" hreflang="lzh" data-title="薛定諤方程" data-language-autonym="文言" data-language-local-name="čínština (klasická)" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%96%9B%E5%AE%9A%E8%AB%A4%E6%96%B9%E7%A8%8B%E5%BC%8F" title="薛定諤方程式 – kantonština" lang="yue" hreflang="yue" data-title="薛定諤方程式" data-language-autonym="粵語" data-language-local-name="kantonština" class="interlanguage-link-target"><span>粵語</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/Q165498#sitelinks-wikipedia" title="Editovat 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data-event-name="pinnable-header.vector-appearance.unpin">skrýt</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Z Wikipedie, otevřené encyklopedie</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="cs" dir="ltr"><p><b>Schrödingerova rovnice</b> je <a href="/wiki/Pohybov%C3%A1_rovnice" title="Pohybová rovnice">pohybová rovnice</a> nerelativistické <a href="/wiki/Kvantov%C3%A1_fyzika" title="Kvantová fyzika">kvantové teorie</a>. V roce <a href="/wiki/1925" title="1925">1925</a> ji formuloval <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a>. Popisuje časový a prostorový vývoj <a href="/wiki/Vlnov%C3%A1_funkce" title="Vlnová funkce">vlnové funkce</a> <a href="/wiki/%C4%8C%C3%A1stice" title="Částice">částice</a>, která se <a href="/wiki/Mechanick%C3%BD_pohyb" title="Mechanický pohyb">pohybuje</a> v <a href="/wiki/Fyzik%C3%A1ln%C3%AD_pole" title="Fyzikální pole">poli sil</a>. Tato rovnice má v <a href="/wiki/Kvantov%C3%A1_mechanika" title="Kvantová mechanika">kvantové mechanice</a> stejné postavení jako <a href="/wiki/Newtonovy_pohybov%C3%A9_z%C3%A1kony" title="Newtonovy pohybové zákony">druhý Newtonův zákon</a> v <a href="/wiki/Klasick%C3%A1_mechanika" title="Klasická mechanika">klasické mechanice</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Odvození_rovnice"><span id="Odvozen.C3.AD_rovnice"></span>Odvození rovnice</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6dingerova_rovnice&veaction=edit&section=1" title="Editace sekce: Odvození rovnice" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Schr%C3%B6dingerova_rovnice&action=edit&section=1" title="Editovat zdrojový kód sekce Odvození rovnice"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Schrödingerova rovnice ve své době přirozeně vyplynula z předchozích výzkumů. </p><p>V roce <a href="/wiki/1905" title="1905">1905</a> došel <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> při studiu <a href="/wiki/Fotoelektrick%C3%BD_jev" title="Fotoelektrický jev">fotoelektrického jevu</a> ke vztahu </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=hf\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>h</mi> <mi>f</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=hf\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6293994cd4801bc2e58aa92ab156216de6946002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.137ex; height:2.509ex;" alt="{\displaystyle E=hf\;}"></span>,</dd></dl> <p>který vyjadřuje vztah mezi <a href="/wiki/Energie" title="Energie">energií</a> <i>E</i> a <a href="/wiki/Frekvence" title="Frekvence">frekvencí</a> <i>f</i> <a href="/wiki/Kvantum" title="Kvantum">kvanta</a> <a href="/wiki/Elektromagnetick%C3%A9_z%C3%A1%C5%99en%C3%AD" title="Elektromagnetické záření">elektromagnetického záření</a> (<a href="/wiki/Foton" title="Foton">fotonu</a>), přičemž <i>h</i> označuje <a href="/wiki/Planckova_konstanta" title="Planckova konstanta">Planckovu konstantu</a>. </p><p>V roce <a href="/wiki/1924" title="1924">1924</a> přišel <a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">Louis de Broglie</a> s <a href="/wiki/Hypot%C3%A9za" title="Hypotéza">hypotézou</a>, podle které přísluší <i>všem</i> <a href="/wiki/%C4%8C%C3%A1stice" title="Částice">částicím</a> (nejen fotonům) <a href="/wiki/Vlnov%C3%A1_funkce" title="Vlnová funkce">vlnová funkce</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span>, přičemž vztah mezi <a href="/wiki/Hybnost" title="Hybnost">hybností</a> částice a <a href="/wiki/Vlnov%C3%A1_d%C3%A9lka" title="Vlnová délka">vlnovou délkou</a> vlny, která byla částici přiřazena (tzv. <b><a href="/wiki/De_Broglieova_vlna" title="De Broglieova vlna">de Broglieho vlna</a></b>) vyjádřil vztahem </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 p=h/\lambda \;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>λ</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=h/\lambda \;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90965a07ea3255c9d83487f92d61a19744301056" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.859ex; height:2.843ex;" alt="{\displaystyle p=h/\lambda \;}"></span>.</dd></dl> <p>De Broglie pomocí těchto <a href="/wiki/Vln%C4%9Bn%C3%AD" title="Vlnění">vln</a> také ukázal, že Einsteinův vztah <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=hf}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>h</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=hf}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f39fac3593bb1e2dec0282c112c4dff7a99007f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.492ex; height:2.509ex;" alt="{\displaystyle E=hf}"></span> platí nejen pro fotony, ale pro všechny částice. </p><p>Pro energii a hybnost lze pomocí <a href="/wiki/%C3%9Ahlov%C3%A1_frekvence" title="Úhlová frekvence">úhlové frekvence</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =2\pi f\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =2\pi f\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b110a0b8fdb5dde2ddd3cb47479d82b56e17fc79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.962ex; height:2.509ex;" alt="{\displaystyle \omega =2\pi f\;}"></span> a <a href="/wiki/Vlnov%C3%A9_%C4%8D%C3%ADslo" title="Vlnové číslo">vlnového čísla</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=2\pi /\lambda \;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>λ</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=2\pi /\lambda \;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b35033f8f2c6f20dbd39d8d18306d2fc044777dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.967ex; height:2.843ex;" alt="{\displaystyle k=2\pi /\lambda \;}"></span>, kde <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }"></span> je <a href="/wiki/Planckova_konstanta" title="Planckova konstanta">redukovaná Planckova konstanta</a> získat vztahy </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=\hbar \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=\hbar \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb16565f02349106457258633097e0d0414a8e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.626ex; height:2.176ex;" alt="{\displaystyle E=\hbar \omega }"></span></dd></dl> <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} =\hbar \mathbf {k} \;}"> <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 class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =\hbar \mathbf {k} \;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73189af80c4decdca96caf2a71724dcbd3e678ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.946ex; height:2.509ex;" alt="{\displaystyle \mathbf {p} =\hbar \mathbf {k} \;}"></span>.</dd></dl> <p><br /> Schrödinger vyšel z předpokladu, že pohyb částice můžeme spojovat s de Broglieho vlnou. Vlnu šířící se ve směru osy <i>x</i> lze popsat <a href="/wiki/Vlnov%C3%A1_rovnice" title="Vlnová rovnice">vlnovou rovnicí</a>, jejíž řešení lze vyjádřit jako </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 \Psi =A\mathrm {e} ^{-\mathrm {i} \omega \left(t-{\frac {x}{v}}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo>=</mo> <mi>A</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>ω</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>v</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi =A\mathrm {e} ^{-\mathrm {i} \omega \left(t-{\frac {x}{v}}\right)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/521365ad84a2fc8c322e15c2cbba4c097e545cb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.65ex; height:3.009ex;" alt="{\displaystyle \Psi =A\mathrm {e} ^{-\mathrm {i} \omega \left(t-{\frac {x}{v}}\right)}}"></span>,</dd></dl> <p>kde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> je <a href="/wiki/%C3%9Ahlov%C3%A1_frekvence" title="Úhlová frekvence">úhlová frekvence</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 v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> je <a href="/wiki/F%C3%A1zov%C3%A1_rychlost" title="Fázová rychlost">fázová rychlost</a> a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> je <a href="/wiki/Integr%C3%A1l" title="Integrál">integrační konstanta</a>. Toto řešení lze také přepsat do tvaru </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 \Psi =A\mathrm {e} ^{-{\frac {\mathrm {i} }{\hbar }}\left(Et-px\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo>=</mo> <mi>A</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>E</mi> <mi>t</mi> <mo>−</mo> <mi>p</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi =A\mathrm {e} ^{-{\frac {\mathrm {i} }{\hbar }}\left(Et-px\right)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2f175df5e3ce51edf63a0b74ad0f1929a86fefc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.953ex; height:3.509ex;" alt="{\displaystyle \Psi =A\mathrm {e} ^{-{\frac {\mathrm {i} }{\hbar }}\left(Et-px\right)}}"></span>.</dd></dl> <p>Tento vztah popisuje částici s celkovou energií <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> a hybností <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>, která se pohybuje ve směru osy <i>x</i>. Označujeme ji také jako <i>vlnovou funkci <a href="/wiki/Voln%C3%A1_%C4%8D%C3%A1stice" title="Volná částice">volné částice</a></i>. Tento vztah však také představuje řešení Schrödingerovy rovnice, jejíž tvar z něj můžeme získat. </p><p>Celkovou energii (nerelativistické) částice v <a href="/wiki/Fyzik%C3%A1ln%C3%AD_pole" title="Fyzikální pole">potenciálním poli</a> lze zapsat jako </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=E_{k}+E_{p}={\frac {p^{2}}{2m}}+V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=E_{k}+E_{p}={\frac {p^{2}}{2m}}+V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f4b45e30f2edc6ac4be5108ac8a01cefba5290" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.058ex; height:5.676ex;" alt="{\displaystyle E=E_{k}+E_{p}={\frac {p^{2}}{2m}}+V}"></span>,</dd></dl> <p>kde <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_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7587849b44d775263271e89499f4327eeac5dc81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.804ex; height:2.509ex;" alt="{\displaystyle E_{k}}"></span> je <a href="/wiki/Kinetick%C3%A1_energie" title="Kinetická energie">kinetická energie</a> částice, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=E_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=E_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d18299703e61d86d8f24c11290dfc8230195e5fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.66ex; height:2.843ex;" alt="{\displaystyle V=E_{p}}"></span> je <a href="/wiki/Potenci%C3%A1ln%C3%AD_energie" title="Potenciální energie">potenciální energie</a> částice (v kvantové mechanice je zvykem potenciální energii značit jako <i>V</i>, kinetickou energii jako <i>T</i>), <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> je <a href="/wiki/Hybnost" title="Hybnost">hybnost</a> 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 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> je <a href="/wiki/Hmotnost" title="Hmotnost">hmotnost</a> částice. </p><p>Derivací vlnové funkce volné částice získáme následující vztahy </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 {\partial ^{2}\Psi }{\partial x^{2}}}=-{\frac {p^{2}}{\hbar ^{2}}}\Psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Ψ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</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> </mfrac> </mrow> <mi mathvariant="normal">Ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}\Psi }{\partial x^{2}}}=-{\frac {p^{2}}{\hbar ^{2}}}\Psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aa814f7f78b33cf43414d1dc174122ef4a53f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.969ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}\Psi }{\partial x^{2}}}=-{\frac {p^{2}}{\hbar ^{2}}}\Psi }"></span></dd></dl> <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 {\partial \Psi }{\partial t}}=-\mathrm {i} {\frac {E}{\hbar }}\Psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Ψ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mi mathvariant="normal">Ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \Psi }{\partial t}}=-\mathrm {i} {\frac {E}{\hbar }}\Psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/455815215826858b6cc146cc2093b67d03d44726" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.936ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \Psi }{\partial t}}=-\mathrm {i} {\frac {E}{\hbar }}\Psi }"></span>.</dd></dl> <p>Dosazením do výrazu pro celkovou energii získáme </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\Psi =\mathrm {i} \hbar {\frac {\partial \Psi }{\partial t}}={\frac {p^{2}}{2m}}\Psi +V\Psi =-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\Psi }{\partial x^{2}}}+V\Psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mi mathvariant="normal">Ψ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Ψ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mi mathvariant="normal">Ψ</mi> <mo>+</mo> <mi>V</mi> <mi mathvariant="normal">Ψ</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Ψ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>V</mi> <mi mathvariant="normal">Ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\Psi =\mathrm {i} \hbar {\frac {\partial \Psi }{\partial t}}={\frac {p^{2}}{2m}}\Psi +V\Psi =-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\Psi }{\partial x^{2}}}+V\Psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d4f8e7dc3020b527451904f84dd160a900cbca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:48.402ex; height:6.009ex;" alt="{\displaystyle E\Psi =\mathrm {i} \hbar {\frac {\partial \Psi }{\partial t}}={\frac {p^{2}}{2m}}\Psi +V\Psi =-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\Psi }{\partial x^{2}}}+V\Psi }"></span>.</dd></dl> <p>Časově závislý tvar jednorozměrné Schrödingerovy rovnice lze tedy zapsat jako </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 \mathrm {i} \hbar {\frac {\partial \Psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\Psi }{\partial x^{2}}}+V\Psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Ψ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Ψ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>V</mi> <mi mathvariant="normal">Ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {i} \hbar {\frac {\partial \Psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\Psi }{\partial x^{2}}}+V\Psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eee59934184198c929d5ba7447d0344c5156361" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:26.339ex; height:6.009ex;" alt="{\displaystyle \mathrm {i} \hbar {\frac {\partial \Psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\Psi }{\partial x^{2}}}+V\Psi }"></span>.</dd></dl> <p><br /> V trojrozměrném prostoru má časová Schrödingerova rovnice tvar </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 \mathrm {i} \hbar {\frac {\partial \psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}{\nabla ^{2}}\psi +V\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ψ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>ψ</mi> <mo>+</mo> <mi>V</mi> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {i} \hbar {\frac {\partial \psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}{\nabla ^{2}}\psi +V\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2b6e41cbabcdc6d24fdd007a88308b0e3fa8b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.21ex; height:5.843ex;" alt="{\displaystyle \mathrm {i} \hbar {\frac {\partial \psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}{\nabla ^{2}}\psi +V\psi }"></span>,</dd></dl> <p>kde <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 \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"></span> je <a href="/wiki/Laplace%C5%AFv_oper%C3%A1tor" title="Laplaceův operátor">Laplaceův operátor</a>. </p><p><br /> Schrödinger pomocí této rovnice spočítal <a href="/wiki/Spektr%C3%A1ln%C3%AD_%C4%8D%C3%A1ra" title="Spektrální čára">spektrální čáry</a> <a href="/wiki/Vod%C3%ADk" title="Vodík">vodíku</a>, kdy popsal <a href="/wiki/Elektron" title="Elektron">elektron</a> jako vlnu nacházející se v <a href="/wiki/Potenci%C3%A1lov%C3%A1_j%C3%A1ma" title="Potenciálová jáma">potenciálové jámě</a> vytvořené <a href="/wiki/Proton" title="Proton">protonem</a> (tedy <a href="/wiki/Atomov%C3%A9_j%C3%A1dro" title="Atomové jádro">jádrem atomu</a>). Tento výpočet souhlasil s <a href="/wiki/Experiment" title="Experiment">experimenty</a>, výsledky <a href="/wiki/Bohr%C5%AFv_model_atomu" title="Bohrův model atomu">Bohrova modelu atomu</a> a také s <a href="/wiki/Maticov%C3%A1_kvantov%C3%A1_mechanika" title="Maticová kvantová mechanika">maticovou mechanikou</a> <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Wernera Heisenberga</a>, přičemž Schrödinger nepotřeboval uvažovat s <a href="/w/index.php?title=Nekomutativnost&action=edit&redlink=1" class="new" title="Nekomutativnost (stránka neexistuje)">nekomutativností</a> <a href="/w/index.php?title=Pozorovateln%C3%A1_veli%C4%8Dina&action=edit&redlink=1" class="new" title="Pozorovatelná veličina (stránka neexistuje)">pozorovatelných</a>, jak tomu bylo právě v maticové mechanice. Schrödinger svou práci o vlnové funkci a spektrálních čarách publikoval v roce <a href="/wiki/1926" title="1926">1926</a>. </p><p>Schrödingerova rovnice určuje chování vlnové funkce, avšak neurčuje, co vlastně vlnová funkce je. Interpretaci vlnové funkce jako <a href="/wiki/Amplituda_pravd%C4%9Bpodobnosti" title="Amplituda pravděpodobnosti">amplitudy pravděpodobnosti</a> předložil v roce <a href="/wiki/1926" title="1926">1926</a> <a href="/wiki/Max_Born" title="Max Born">Max Born</a>. Jsou však i jiné <a href="/wiki/Interpretace_kvantov%C3%A9_mechaniky" title="Interpretace kvantové mechaniky">interpretace kvantové mechaniky</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Obecné_vyjádření"><span id="Obecn.C3.A9_vyj.C3.A1d.C5.99en.C3.AD"></span>Obecné vyjádření</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6dingerova_rovnice&veaction=edit&section=2" title="Editace sekce: Obecné vyjádření" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Schr%C3%B6dingerova_rovnice&action=edit&section=2" title="Editovat zdrojový kód sekce Obecné vyjádření"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>V obecném tvaru se Schrödingerova rovnice zapisuje jako </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 {\hat {H}}(t)\Psi (\mathbf {r} ,t)=\mathrm {i} \hbar {\frac {\partial \Psi (\mathbf {r} ,t)}{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}(t)\Psi (\mathbf {r} ,t)=\mathrm {i} \hbar {\frac {\partial \Psi (\mathbf {r} ,t)}{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd3601232884cfe7c182bedb9faa906717766877" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.105ex; height:5.843ex;" alt="{\displaystyle {\hat {H}}(t)\Psi (\mathbf {r} ,t)=\mathrm {i} \hbar {\frac {\partial \Psi (\mathbf {r} ,t)}{\partial t}}}"></span>,</dd></dl> <p>kde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb06de5217295d7fbdbf68fb9c5309a513fc99e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.843ex;" alt="{\displaystyle {\hat {H}}}"></span> je časově závislý <a href="/wiki/Hamilton%C5%AFv_oper%C3%A1tor" title="Hamiltonův operátor">Hamiltonův operátor (hamiltonián)</a> popisující pohyb částice v časově závislých vnějších polích. Ten vyjadřuje ve formě <a href="/wiki/Oper%C3%A1tor" title="Operátor">operátoru</a> celkovou <a href="/wiki/Energie" title="Energie">energii</a> částice jako součet <a href="/wiki/Kinetick%C3%A1_energie" title="Kinetická energie">kinetické</a> a <a href="/wiki/Potenci%C3%A1ln%C3%AD_energie" title="Potenciální energie">potenciální</a> energie. Výraz na pravé straně vyjadřuje časovou změnu vlnové funkce. Tato obecná Schrödingerova rovnice bývá také označována jako <b>časová</b> nebo <b>nestacionární</b>. </p><p>Obecné nestacionární řešení časové Schrödingerovy rovnice s časově nezávislým hamiltoniánem lze vyjádřit prostřednictvím rozvoje do ortonormálních stacionárních stavů, tzn. </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 \Psi (\mathbf {r} ,t)=\sum _{n}c_{n}\psi _{n}(\mathbf {r} )\mathrm {e} ^{-{\frac {\mathrm {i} }{\hbar }}E_{n}t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (\mathbf {r} ,t)=\sum _{n}c_{n}\psi _{n}(\mathbf {r} )\mathrm {e} ^{-{\frac {\mathrm {i} }{\hbar }}E_{n}t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba753f31f92f9962812cd8440f38719a831ad75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.203ex; height:6.176ex;" alt="{\displaystyle \Psi (\mathbf {r} ,t)=\sum _{n}c_{n}\psi _{n}(\mathbf {r} )\mathrm {e} ^{-{\frac {\mathrm {i} }{\hbar }}E_{n}t}}"></span>,</dd></dl> <p>kde <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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b7e944bcb1be88e9a6a940638f2adce0ec4211a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.009ex;" alt="{\displaystyle c_{n}}"></span> jsou časově nezávislá <a href="/wiki/Komplexn%C3%AD_%C4%8D%C3%ADslo" title="Komplexní číslo">komplexní čísla</a> určená počáteční podmínkou <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (\mathbf {r} ,t=t_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (\mathbf {r} ,t=t_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f60bf8e051c2d9900f8882fe7943c6d3090656" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.585ex; height:2.843ex;" alt="{\displaystyle \Psi (\mathbf {r} ,t=t_{0})}"></span>. <a href="/wiki/St%C5%99edn%C3%AD_hodnota" title="Střední hodnota">Střední hodnota</a> energie těchto stavů je na čase nezávislá. </p> <div class="mw-heading mw-heading2"><h2 id="Stacionární_Schrödingerova_rovnice"><span id="Stacion.C3.A1rn.C3.AD_Schr.C3.B6dingerova_rovnice"></span>Stacionární Schrödingerova rovnice</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6dingerova_rovnice&veaction=edit&section=3" title="Editace sekce: Stacionární Schrödingerova rovnice" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Schr%C3%B6dingerova_rovnice&action=edit&section=3" title="Editovat zdrojový kód sekce Stacionární Schrödingerova rovnice"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Zvláštním případem Schrödingerovy rovnice je tzv. <b>stacionární</b> (<b>časově nezávislá</b>, <b>bezčasová</b> nebo <b>nečasová</b>) <b>Schrödingerova rovnice</b>, kterou lze získat za předpokladu, že vývoj systému je popsán Schrödingerovou rovnicí, v níž je <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb06de5217295d7fbdbf68fb9c5309a513fc99e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.843ex;" alt="{\displaystyle {\hat {H}}}"></span> <a href="/wiki/%C4%8Cas" title="Čas">časově</a> nezávislý hamiltonián popisující pohyb částice v časově nezávislých vnějších polích. </p><p>V takovém případě lze provést <a href="/wiki/Separace_prom%C4%9Bnn%C3%BDch" title="Separace proměnných">separaci proměnných</a> a hledat vlnovou funkci ve tvaru </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 \Psi (\mathbf {r} ,t)=\psi (\mathbf {r} )\varphi (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (\mathbf {r} ,t)=\psi (\mathbf {r} )\varphi (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d13a2e1dcf350309077eab21e5b2f64a6c9ed39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.285ex; height:2.843ex;" alt="{\displaystyle \Psi (\mathbf {r} ,t)=\psi (\mathbf {r} )\varphi (t)}"></span>.</dd></dl> <p>S tímto předpokladem dostaneme po dosazení do Schrödingerovy rovnice: </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 \mathrm {i} \hbar {\frac {\frac {\mathrm {d} \varphi (t)}{\mathrm {d} t}}{\varphi (t)}}={\frac {{\hat {H}}\psi (\mathbf {r} )}{\psi (\mathbf {r} )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {i} \hbar {\frac {\frac {\mathrm {d} \varphi (t)}{\mathrm {d} t}}{\varphi (t)}}={\frac {{\hat {H}}\psi (\mathbf {r} )}{\psi (\mathbf {r} )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dfa4a41fbb7578b4d3495e2439a0cb026a9f8f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.91ex; height:7.843ex;" alt="{\displaystyle \mathrm {i} \hbar {\frac {\frac {\mathrm {d} \varphi (t)}{\mathrm {d} t}}{\varphi (t)}}={\frac {{\hat {H}}\psi (\mathbf {r} )}{\psi (\mathbf {r} )}}}"></span>.</dd></dl> <p>Obě strany výsledné rovnice se musí rovnat <a href="/wiki/Konstanta" title="Konstanta">konstantě</a>, kterou označíme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span>. Tato konstanta má <a href="/wiki/Fyzik%C3%A1ln%C3%AD_rozm%C4%9Br_veli%C4%8Diny" title="Fyzikální rozměr veličiny">rozměr</a> <a href="/wiki/Energie" title="Energie">energie</a>. Za uvedených předpokladů tak dostáváme dvě rovnice, přičemž první z nich se označuje jako <i>stacionární Schrödingerova rovnice</i> </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 {\hat {H}}\psi (\mathbf {r} )=E\psi (\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}\psi (\mathbf {r} )=E\psi (\mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c69ca8d7736d45f762198b31b31155747d7011a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.787ex; height:3.343ex;" alt="{\displaystyle {\hat {H}}\psi (\mathbf {r} )=E\psi (\mathbf {r} )}"></span></dd></dl> <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 \mathrm {i} \hbar {\frac {\mathrm {d} \varphi (t)}{\mathrm {d} t}}=E\varphi (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>E</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {i} \hbar {\frac {\mathrm {d} \varphi (t)}{\mathrm {d} t}}=E\varphi (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcdaa1c612f6a3c0f207052bee30535266629dcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.294ex; height:5.843ex;" alt="{\displaystyle \mathrm {i} \hbar {\frac {\mathrm {d} \varphi (t)}{\mathrm {d} t}}=E\varphi (t)}"></span>.</dd></dl> <p>Rozepsáním hamiltoniánu lze získat: </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 \Delta \Psi +{\frac {2m}{\hbar ^{2}}}(E-V)\Psi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi mathvariant="normal">Ψ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo>−</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Ψ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \Psi +{\frac {2m}{\hbar ^{2}}}(E-V)\Psi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8c7955afffa8bd87a03e0d56a38a4f94485d431" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.905ex; height:5.509ex;" alt="{\displaystyle \Delta \Psi +{\frac {2m}{\hbar ^{2}}}(E-V)\Psi =0}"></span>.</dd></dl> <p><br /> Vzhledem k tomu, že časově nezávislý hamiltonián se vyskytuje např. u popisu chování <a href="/wiki/Elektron" title="Elektron">elektronu</a> v <a href="/wiki/Atom" title="Atom">atomu</a>, představuje stacionární Schrödingerova rovnice velmi významnou rovnici kvantové mechaniky. </p> <div class="mw-heading mw-heading3"><h3 id="Stacionární_stav"><span id="Stacion.C3.A1rn.C3.AD_stav"></span>Stacionární stav</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6dingerova_rovnice&veaction=edit&section=4" title="Editace sekce: Stacionární stav" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Schr%C3%B6dingerova_rovnice&action=edit&section=4" title="Editovat zdrojový kód sekce Stacionární stav"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Podle stacionární rovnice jsou energie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> <a href="/wiki/Vlastn%C3%AD_vektory_a_vlastn%C3%AD_%C4%8D%C3%ADsla" title="Vlastní vektory a vlastní čísla">vlastními čísly</a> hamiltoniánu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb06de5217295d7fbdbf68fb9c5309a513fc99e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.843ex;" alt="{\displaystyle {\hat {H}}}"></span> (hovoří se též o <b>vlastních energiích</b>). K určení vlastních energií lze <a href="/wiki/Integr%C3%A1l" title="Integrál">integrovat</a> druhou rovnici, čímž získáme </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 \varphi _{n}(t)=N\mathrm {e} ^{-{\frac {\mathrm {i} }{\hbar }}E_{n}t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>N</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{n}(t)=N\mathrm {e} ^{-{\frac {\mathrm {i} }{\hbar }}E_{n}t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7194989e2c3cfe34a1746997d498e1e4c973001f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.45ex; height:4.009ex;" alt="{\displaystyle \varphi _{n}(t)=N\mathrm {e} ^{-{\frac {\mathrm {i} }{\hbar }}E_{n}t}}"></span>,</dd></dl> <p>kde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> je normovací konstanta, kterou lze obvykle položit <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=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85982022b9eb1f295b44de55023687a490db0a39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.325ex; height:2.176ex;" alt="{\displaystyle N=1}"></span>. </p><p>Stavy s vlastními energiemi <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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6b82f2a00af6c9efd4c16d4e99329605645c0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.934ex; height:2.509ex;" alt="{\displaystyle E_{n}}"></span> lze tedy popsat vlnovými funkcemi </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 \Psi _{n}(\mathbf {r} ,t)=\psi _{n}(\mathbf {r} )\mathrm {e} ^{-{\frac {\mathrm {i} }{\hbar }}E_{n}t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{n}(\mathbf {r} ,t)=\psi _{n}(\mathbf {r} )\mathrm {e} ^{-{\frac {\mathrm {i} }{\hbar }}E_{n}t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54cefec1b0ed0e5becf5b6ca9dd3ab1378cbf1ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.454ex; height:4.009ex;" alt="{\displaystyle \Psi _{n}(\mathbf {r} ,t)=\psi _{n}(\mathbf {r} )\mathrm {e} ^{-{\frac {\mathrm {i} }{\hbar }}E_{n}t}}"></span>.</dd></dl> <p>Takové stavy se označují jako <b>stacionární stavy</b>. Stacionární stavy jsou zvláštností kvantové fyziky. V <a href="/wiki/Klasick%C3%A1_mechanika" title="Klasická mechanika">klasické mechanice</a> se sice také vyskytují (např. nehybný <a href="/wiki/Hmotn%C3%BD_bod" title="Hmotný bod">hmotný bod</a>), jedná se však vždy o případy z hlediska klasické mechaniky nezajímavé. </p><p><a href="/wiki/Hustota_pravd%C4%9Bpodobnosti" title="Hustota pravděpodobnosti">Hustota pravděpodobnosti</a> stacionárního stavu na čase nezávisí, tzn. </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 {\left|\Psi _{n}(\mathbf {r} ,t)\right|}^{2}={\left|\psi _{n}(\mathbf {r} )\right|}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mrow> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\left|\Psi _{n}(\mathbf {r} ,t)\right|}^{2}={\left|\psi _{n}(\mathbf {r} )\right|}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8789714f70329a9eab147343e3f8120bc21fd9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.249ex; height:3.343ex;" alt="{\displaystyle {\left|\Psi _{n}(\mathbf {r} ,t)\right|}^{2}={\left|\psi _{n}(\mathbf {r} )\right|}^{2}}"></span>.</dd></dl> <p><a href="/wiki/St%C5%99edn%C3%AD_hodnota" title="Střední hodnota">Střední hodnota</a> libovolného časově nezávislého <a href="/wiki/Oper%C3%A1tor" title="Operátor">operátoru</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 {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f595a6c73d1183d6a1b2ac21fe47ac28c1483821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.843ex;" alt="{\displaystyle {\hat {A}}}"></span> ve stacionárních stavech <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 _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e979a931bf82db45f2c63353f9bc8bbbbee102b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle \Psi _{n}}"></span> nezávisí na čase, tedy </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 {\hat {A}}\rangle _{n}=\int \Psi _{n}^{\star }(\mathbf {r} ,t){\hat {A}}\Psi _{n}(\mathbf {r} ,t)\mathrm {d} V=\int \psi _{n}^{\star }(\mathbf {r} ){\hat {A}}\psi _{n}(\mathbf {r} )\mathrm {d} V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo>∫</mo> <msubsup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>=</mo> <mo>∫</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⋆</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {\hat {A}}\rangle _{n}=\int \Psi _{n}^{\star }(\mathbf {r} ,t){\hat {A}}\Psi _{n}(\mathbf {r} ,t)\mathrm {d} V=\int \psi _{n}^{\star }(\mathbf {r} ){\hat {A}}\psi _{n}(\mathbf {r} )\mathrm {d} V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/196fec6ae5c96d36213f57f460db6085e837dffe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:52.782ex; height:5.676ex;" alt="{\displaystyle \langle {\hat {A}}\rangle _{n}=\int \Psi _{n}^{\star }(\mathbf {r} ,t){\hat {A}}\Psi _{n}(\mathbf {r} ,t)\mathrm {d} V=\int \psi _{n}^{\star }(\mathbf {r} ){\hat {A}}\psi _{n}(\mathbf {r} )\mathrm {d} V}"></span>.</dd></dl> <p>Pro stacionární stavy je také <a href="/wiki/Hustota_toku_pravd%C4%9Bpodobnosti" title="Hustota toku pravděpodobnosti">hustota toku pravděpodobnosti</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span> nezávislá na čase. </p> <div class="mw-heading mw-heading2"><h2 id="Vlastnosti">Vlastnosti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6dingerova_rovnice&veaction=edit&section=5" title="Editace sekce: Vlastnosti" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Schr%C3%B6dingerova_rovnice&action=edit&section=5" title="Editovat zdrojový kód sekce Vlastnosti"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Protože Schrödingerova rovnice obsahuje na jedné straně první <a href="/wiki/Parci%C3%A1ln%C3%AD_derivace" title="Parciální derivace">parciální derivace</a> vlnové funkce podle <a href="/wiki/%C4%8Cas" title="Čas">času</a> a na druhé straně druhé derivace podle prostorových souřadnic (<a href="/wiki/Laplace%C5%AFv_oper%C3%A1tor" title="Laplaceův operátor">Laplaceův operátor</a>), není tato rovnice <a href="/wiki/Invariance" title="Invariance">invariantní</a> vůči <a href="/wiki/Lorentzova_transformace" title="Lorentzova transformace">Lorentzově transformaci</a>. Není tedy v souladu se <a href="/wiki/Speci%C3%A1ln%C3%AD_teorie_relativity" title="Speciální teorie relativity">speciální teorií relativity</a>. Nejedná se tedy o <a href="/wiki/Teorie_relativity" title="Teorie relativity">relativistickou</a> rovnici. Relativistickou obdobou Schrödingerovy rovnice jsou např. <a href="/wiki/Diracova_rovnice" title="Diracova rovnice">Diracova rovnice</a> nebo <a href="/wiki/Kleinova%E2%80%93Gordonova_rovnice" title="Kleinova–Gordonova rovnice">Kleinova–Gordonova rovnice</a>. </p><p>Schrödingerova rovnice umožňuje jednoduše formulovat a vyřešit v <a href="/wiki/Kvantov%C3%A1_mechanika" title="Kvantová mechanika">kvantové mechanice</a> problémy jako <a href="/wiki/Kvantov%C3%BD_harmonick%C3%BD_oscil%C3%A1tor" title="Kvantový harmonický oscilátor">lineární harmonický oscilátor</a>, částice v <a href="/wiki/Potenci%C3%A1lov%C3%A1_j%C3%A1ma" title="Potenciálová jáma">potenciálové jámě</a> nebo <a href="/wiki/Vod%C3%ADk" title="Vodík">vodíku</a> podobný <a href="/wiki/Atom" title="Atom">atom</a>. Vysvětluje <a href="/wiki/Stabilita" title="Stabilita">stabilitu</a> atomů, která byla pro <a href="/wiki/Klasick%C3%A1_fyzika" title="Klasická fyzika">klasickou fyziku</a> <a href="/wiki/Z%C3%A1hada" title="Záhada">záhadou</a>. Umožnila pevné propojení <a href="/wiki/Fyzika" title="Fyzika">fyziky</a> s <a href="/wiki/Chemie" title="Chemie">chemií</a>, protože vysvětlila nejen <a href="/wiki/Ioniza%C4%8Dn%C3%AD_potenci%C3%A1l" title="Ionizační potenciál">ionizační energie</a> <a href="/wiki/Chemick%C3%BD_prvek" title="Chemický prvek">prvků</a>, ale i různorodost jejich chemického chování pomocí <a href="/wiki/Atomov%C3%BD_orbital" title="Atomový orbital">orbitalů</a> tvořících atomový obal. Tyto poznatky umožnily vysvětlit <a href="/wiki/Spektr%C3%A1ln%C3%AD_%C4%8D%C3%A1ra" title="Spektrální čára">čáry</a> ve <a href="/wiki/Spektr%C3%A1ln%C3%AD_klasifikace" title="Spektrální klasifikace">spektru</a> zářících těles a pochopit tak stavbu a vývoj <a href="/wiki/Hv%C4%9Bzda" title="Hvězda">hvězd</a> analýzou jejich <a href="/wiki/Sv%C4%9Btlo" title="Světlo">světla</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Související_články"><span id="Souvisej.C3.ADc.C3.AD_.C4.8Dl.C3.A1nky"></span>Související články</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6dingerova_rovnice&veaction=edit&section=6" title="Editace sekce: Související články" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Schr%C3%B6dingerova_rovnice&action=edit&section=6" title="Editovat zdrojový kód sekce Související články"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Vlnov%C3%A1_rovnice" title="Vlnová rovnice">Vlnová rovnice</a></li> <li><a href="/wiki/Kvantov%C3%A9_%C4%8D%C3%ADslo" title="Kvantové číslo">Kvantové číslo</a></li> <li><a href="/wiki/Diracova_rovnice" title="Diracova rovnice">Diracova rovnice</a></li> <li><a href="/wiki/Kleinova%E2%80%93Gordonova_rovnice" title="Kleinova–Gordonova rovnice">Kleinova–Gordonova rovnice</a></li> <li><a href="/wiki/Schr%C3%B6dingerova_ko%C4%8Dka" title="Schrödingerova kočka">Schrödingerova kočka</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Externí_odkazy"><span id="Extern.C3.AD_odkazy"></span>Externí odkazy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schr%C3%B6dingerova_rovnice&veaction=edit&section=7" title="Editace sekce: Externí odkazy" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Schr%C3%B6dingerova_rovnice&action=edit&section=7" title="Editovat zdrojový kód sekce Externí odkazy"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="wd"><span class="sisterproject sisterproject-commons"><span class="sisterproject_image"><span typeof="mw:File"><a href="/wiki/Wikimedia_Commons" title="Wikimedia Commons"><img alt="Logo Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></span> <span class="sisterproject_text">Obrázky, zvuky či videa k tématu <span 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href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007560881205171">987007560881205171</a></span></span></li></ul> </div></td></tr></tbody></table></div>    </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="">Citováno z „<a dir="ltr" href="https://cs.wikipedia.org/w/index.php?title=Schrödingerova_rovnice&oldid=24172121">https://cs.wikipedia.org/w/index.php?title=Schrödingerova_rovnice&oldid=24172121</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/N%C3%A1pov%C4%9Bda:Kategorie" title="Nápověda:Kategorie">Kategorie</a>: <ul><li><a href="/wiki/Kategorie:Kvantov%C3%A1_fyzika" title="Kategorie:Kvantová fyzika">Kvantová fyzika</a></li><li><a href="/wiki/Kategorie:Kvantov%C3%A1_mechanika" title="Kategorie:Kvantová mechanika">Kvantová mechanika</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Skryté kategorie: <ul><li><a href="/wiki/Kategorie:Monitoring:%C4%8Cl%C3%A1nky_s_identifik%C3%A1torem_NKC" title="Kategorie:Monitoring:Články s identifikátorem NKC">Monitoring:Články s identifikátorem NKC</a></li><li><a href="/wiki/Kategorie:Monitoring:%C4%8Cl%C3%A1nky_s_identifik%C3%A1torem_PSH" title="Kategorie:Monitoring:Články s identifikátorem PSH">Monitoring:Články s identifikátorem PSH</a></li><li><a href="/wiki/Kategorie:Monitoring:%C4%8Cl%C3%A1nky_s_identifik%C3%A1torem_BNF" title="Kategorie:Monitoring:Články s identifikátorem BNF">Monitoring:Články s identifikátorem BNF</a></li><li><a href="/wiki/Kategorie:Monitoring:%C4%8Cl%C3%A1nky_s_identifik%C3%A1torem_GND" title="Kategorie:Monitoring:Články s identifikátorem GND">Monitoring:Články s identifikátorem GND</a></li><li><a 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