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</div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Especial:Contribucions_pr%C3%B2pies" title="Una llista de les modificacions fetes des d'aquesta adreça IP [y]" accesskey="y"><span>Contribucions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Especial:Discussi%C3%B3_personal" title="Discussió sobre les edicions per aquesta adreça ip. [n]" accesskey="n"><span>Discussió per aquest IP</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Lloc"> <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="Contingut" 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">Contingut</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">amaga</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">Inici</div> </a> </li> <li id="toc-Longitud_d'ona_de_Compton_reduïda" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Longitud_d'ona_de_Compton_reduïda"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Longitud d'ona de Compton reduïda</span> </div> </a> <ul id="toc-Longitud_d'ona_de_Compton_reduïda-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Funció_en_equacions_per_partícules_massives" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Funció_en_equacions_per_partícules_massives"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Funció en equacions per partícules massives</span> </div> </a> <ul id="toc-Funció_en_equacions_per_partícules_massives-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referències" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referències"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Referències</span> </div> </a> <ul id="toc-Referències-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="Contingut" 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="Commuta la taula de continguts." > <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">Commuta la taula de continguts.</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">Longitud d'ona Compton</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="Vés a un article en una altra llengua. Disponible en 22 llengües" > <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-22" 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">22 llengües</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B7%D9%88%D9%84_%D9%85%D9%88%D8%AC%D8%A9_%D9%83%D9%88%D9%85%D8%A8%D8%AA%D9%88%D9%86" title="طول موجة كومبتون - àrab" lang="ar" hreflang="ar" data-title="طول موجة كومبتون" data-language-autonym="العربية" data-language-local-name="àrab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D1%82%D0%B0%D0%BD%D0%B0%D1%9E%D1%81%D0%BA%D0%B0%D1%8F_%D0%B4%D0%B0%D1%9E%D0%B6%D1%8B%D0%BD%D1%8F_%D1%85%D0%B2%D0%B0%D0%BB%D1%96" title="Комптанаўская даўжыня хвалі - belarús" lang="be" hreflang="be" data-title="Комптанаўская даўжыня хвалі" data-language-autonym="Беларуская" data-language-local-name="belarús" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Compton_wavelength" title="Compton wavelength - anglès" lang="en" hreflang="en" data-title="Compton wavelength" data-language-autonym="English" data-language-local-name="anglès" 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/Ondolongo_de_Komptono" title="Ondolongo de Komptono - esperanto" lang="eo" hreflang="eo" data-title="Ondolongo de Komptono" 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/Longitud_de_onda_Compton" title="Longitud de onda Compton - espanyol" lang="es" hreflang="es" data-title="Longitud de onda Compton" data-language-autonym="Español" data-language-local-name="espanyol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Compton_uhin-luzera" title="Compton uhin-luzera - basc" lang="eu" hreflang="eu" data-title="Compton uhin-luzera" data-language-autonym="Euskara" data-language-local-name="basc" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B7%D9%88%D9%84_%D9%85%D9%88%D8%AC_%DA%A9%D8%A7%D9%85%D9%BE%D8%AA%D9%88%D9%86" title="طول موج کامپتون - persa" lang="fa" hreflang="fa" data-title="طول موج کامپتون" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Longueur_d%27onde_de_Compton" title="Longueur d'onde de Compton - francès" lang="fr" hreflang="fr" data-title="Longueur d'onde de Compton" data-language-autonym="Français" data-language-local-name="francès" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%95%D7%A8%D7%9A_%D7%92%D7%9C_%D7%A7%D7%95%D7%9E%D7%A4%D7%98%D7%95%D7%9F" title="אורך גל קומפטון - hebreu" lang="he" hreflang="he" data-title="אורך גל קומפטון" data-language-autonym="עברית" data-language-local-name="hebreu" 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%95%E0%A5%89%E0%A4%AE%E0%A5%8D%E0%A4%AA%E0%A4%9F%E0%A4%A8_%E0%A4%A4%E0%A4%B0%E0%A4%82%E0%A4%97%E0%A4%A6%E0%A5%88%E0%A4%B0%E0%A5%8D%E0%A4%98%E0%A5%8D%E0%A4%AF" title="कॉम्पटन तरंगदैर्घ्य - hindi" lang="hi" hreflang="hi" data-title="कॉम्पटन तरंगदैर्घ्य" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Panjang_gelombang_Compton" title="Panjang gelombang Compton - indonesi" lang="id" hreflang="id" data-title="Panjang gelombang Compton" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesi" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Lunghezza_d%27onda_Compton" title="Lunghezza d'onda Compton - italià" lang="it" hreflang="it" data-title="Lunghezza d'onda Compton" data-language-autonym="Italiano" data-language-local-name="italià" 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%B3%E3%83%B3%E3%83%97%E3%83%88%E3%83%B3%E6%B3%A2%E9%95%B7" title="コンプトン波長 - japonès" lang="ja" hreflang="ja" data-title="コンプトン波長" data-language-autonym="日本語" data-language-local-name="japonès" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%BD%A4%ED%94%84%ED%84%B4_%ED%8C%8C%EC%9E%A5" title="콤프턴 파장 - coreà" lang="ko" hreflang="ko" data-title="콤프턴 파장" data-language-autonym="한국어" data-language-local-name="coreà" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/D%C5%82ugo%C5%9B%C4%87_fali_Comptona" title="Długość fali Comptona - polonès" lang="pl" hreflang="pl" data-title="Długość fali Comptona" data-language-autonym="Polski" data-language-local-name="polonès" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Comprimento_de_onda_Compton" title="Comprimento de onda Compton - portuguès" lang="pt" hreflang="pt" data-title="Comprimento de onda Compton" data-language-autonym="Português" data-language-local-name="portuguès" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D1%82%D0%BE%D0%BD%D0%BE%D0%B2%D1%81%D0%BA%D0%B0%D1%8F_%D0%B4%D0%BB%D0%B8%D0%BD%D0%B0_%D0%B2%D0%BE%D0%BB%D0%BD%D1%8B" title="Комптоновская длина волны - rus" lang="ru" hreflang="ru" data-title="Комптоновская длина волны" data-language-autonym="Русский" data-language-local-name="rus" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Comptonova_valovna_dol%C5%BEina" title="Comptonova valovna dolžina - eslovè" lang="sl" hreflang="sl" data-title="Comptonova valovna dolžina" data-language-autonym="Slovenščina" data-language-local-name="eslovè" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%A2%E0%B8%B2%E0%B8%A7%E0%B8%84%E0%B8%A5%E0%B8%B7%E0%B9%88%E0%B8%99%E0%B8%84%E0%B8%AD%E0%B8%A1%E0%B8%9B%E0%B9%8C%E0%B8%95%E0%B8%B1%E0%B8%99" title="ความยาวคลื่นคอมป์ตัน - tai" lang="th" hreflang="th" data-title="ความยาวคลื่นคอมป์ตัน" data-language-autonym="ไทย" data-language-local-name="tai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Compton_dalga_boyu" title="Compton dalga boyu - turc" lang="tr" hreflang="tr" data-title="Compton dalga boyu" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D1%82%D0%BE%D0%BD%D1%96%D0%B2%D1%81%D1%8C%D0%BA%D0%B0_%D0%B4%D0%BE%D0%B2%D0%B6%D0%B8%D0%BD%D0%B0_%D1%85%D0%B2%D0%B8%D0%BB%D1%96" title="Комптонівська довжина хвилі - ucraïnès" lang="uk" hreflang="uk" data-title="Комптонівська довжина хвилі" data-language-autonym="Українська" data-language-local-name="ucraïnès" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%BA%B7%E6%99%AE%E9%A0%93%E6%B3%A2%E9%95%B7" title="康普頓波長 - xinès" lang="zh" hreflang="zh" data-title="康普頓波長" data-language-autonym="中文" data-language-local-name="xinès" class="interlanguage-link-target"><span>中文</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/Q1145377#sitelinks-wikipedia" title="Modifica enllaços interlingües" class="wbc-editpage">Modifica els enllaços</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espais de noms"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Longitud_d%27ona_Compton" title="Vegeu el contingut de la pàgina [c]" accesskey="c"><span>Pàgina</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Discussi%C3%B3:Longitud_d%27ona_Compton&action=edit&redlink=1" rel="discussion" class="new" title="Discussió sobre el contingut d'aquesta pàgina (encara no existeix) [t]" accesskey="t"><span>Discussió</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown 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aria-label="Eines" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Eines</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Eines</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" 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</div> <div id="siteSub" class="noprint">De la Viquipèdia, l'enciclopèdia lliure</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="ca" dir="ltr"><p>La <b>longitud d'ona Compton</b> és una propietat mecànica quàntica d'una partícula. Fou introduïda per <a href="/wiki/Arthur_Holly_Compton" title="Arthur Holly Compton">Arthur Compton</a> a la seva explicació de la dispersió de <a href="/wiki/Fot%C3%B3" title="Fotó">fotons</a> per <a href="/wiki/Electr%C3%B3" title="Electró">electrons</a> (procés conegut com a <a href="/wiki/Efecte_Compton" title="Efecte Compton">efecte Compton</a>). La longitud d'ona Compton d'una <a href="/wiki/Part%C3%ADcula_elemental" title="Partícula elemental">partícula</a> és equivalent a la <a href="/wiki/Longitud_d%27ona" title="Longitud d'ona">longitud d'ona</a> d'un fotó l'energia del qual és la mateixa que la massa de la partícula. </p><p>La longitud d'ona Compton estàndard, <span class="texhtml">λ</span>, d'una partícula ve donada per </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 \lambda ={\frac {h}{mc}},\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mi>m</mi> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {h}{mc}},\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f4ee55cbb36b5fb08df82a2b1232951e252fcd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.952ex; height:5.343ex;" alt="{\displaystyle \lambda ={\frac {h}{mc}},\ }"></span></dd></dl> <p>On <span class="texhtml"><i>h</i></span> és la <a href="/wiki/Constant_de_Planck" title="Constant de Planck">constant de Planck</a>, <span class="texhtml"><i>m</i></span> és la <a href="/wiki/Massa" title="Massa">massa</a> de la partícula, i <span class="texhtml"><i>c</i></span> és la <a href="/wiki/Velocitat_de_la_llum" title="Velocitat de la llum">velocitat de la llum</a>. La importància d'aquesta fórmula es mostra en la derivació de la fórmula de canvi de Compton. </p><p>El valor <a href="/wiki/CODATA" title="CODATA">CODATA</a> 2014 per la longitud d'ona Compton de l'electró és 2,4263102367 (11) × 10<sup>-12</sup> m.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Altres partícules tenen diferents longituds d'ona Compton. </p> <div class="mw-heading mw-heading2"><h2 id="Longitud_d'ona_de_Compton_reduïda"><span id="Longitud_d.27ona_de_Compton_redu.C3.AFda"></span>Longitud d'ona de Compton reduïda</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Longitud_d%27ona_Compton&action=edit&section=1" title="Modifica la secció: Longitud d'ona de Compton reduïda"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div><p> Quan la longitud d'ona Compton es divideix per 2π s'obté la longitud d'ona Compton "reduïda" (lambda barrada), és a dir, la longitud de Compton per 1 radian enlloc de 2π radians.</p><div style="font-size: 1.45em;"><span class="texhtml">ƛ = <span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; padding:0 0.1em;">λ</span><span style="display:none;">/</span><span style="display:block; line-height:1em; padding:0 0.1em; border-top:1px solid;">2<i>π</i></span></span> = <span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><span style="display:block; line-height:1em; padding:0 0.1em;"><i>ħ</i></span><span style="display:none;">/</span><span style="display:block; line-height:1em; padding:0 0.1em; border-top:1px solid;"><i>mc</i></span></span>,</span></div><p>On <span class="texhtml"><i>ħ</i></span> és la constant de Planck "reduïda". </p><div class="mw-heading mw-heading2"><h2 id="Funció_en_equacions_per_partícules_massives"><span id="Funci.C3.B3_en_equacions_per_part.C3.ADcules_massives"></span>Funció en equacions per partícules massives</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Longitud_d%27ona_Compton&action=edit&section=2" title="Modifica la secció: Funció en equacions per partícules massives"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La longitud d'ona Compton reduïda és una representació natural de la massa en l'escala quàntica, i com a tal, apareix en moltes de les equacions fonamentals de la mecànica quàntica. La longitud d'ona Compton reduïda apareix en l'equació relativista Klein-Gordon per una partícula lliure: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\nabla } ^{2}\psi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\psi =\left({\frac {mc}{\hbar }}\right)^{2}\psi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>c</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\nabla } ^{2}\psi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\psi =\left({\frac {mc}{\hbar }}\right)^{2}\psi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98ce32b8f1ff2b6f931cdf6b80d9efa48884e376" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.774ex; height:6.009ex;" alt="{\displaystyle \mathbf {\nabla } ^{2}\psi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\psi =\left({\frac {mc}{\hbar }}\right)^{2}\psi .}"></span></dd></dl> <p>Apareix en l'<a href="/wiki/Equaci%C3%B3_de_Dirac" title="Equació de Dirac">equació de Dirac</a> (la següent és una forma explícitament <a href="/wiki/Covari%C3%A0ncia" title="Covariància">covariant</a> que fa servir la convenció de suma d'Einstein): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -i\gamma ^{\mu }\partial _{\mu }\psi +\left({\frac {mc}{\hbar }}\right)\psi =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>ψ</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>c</mi> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>ψ</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i\gamma ^{\mu }\partial _{\mu }\psi +\left({\frac {mc}{\hbar }}\right)\psi =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91cba1560cf5d4e07d9c51d21016ab4036b9beaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.392ex; height:5.009ex;" alt="{\displaystyle -i\gamma ^{\mu }\partial _{\mu }\psi +\left({\frac {mc}{\hbar }}\right)\psi =0.}"></span></dd></dl> <p>La longitud d'ona Compton reduïda també apareix en <a href="/wiki/Equaci%C3%B3_de_Schr%C3%B6dinger" title="Equació de Schrödinger">l'equació de Schrödinger</a>, encara que la seva presència s'amaga en representacions tradicionals de l'equació. La següent és la representació tradicional de l'equació de Schrödinger per un electró en un àtom similar a l'hidrogen: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi -{\frac {1}{4\pi \epsilon _{0}}}{\frac {Ze^{2}}{r}}\psi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>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> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>r</mi> </mfrac> </mrow> <mi>ψ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi -{\frac {1}{4\pi \epsilon _{0}}}{\frac {Ze^{2}}{r}}\psi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac3d44204d1fb02f289661419563c6f4ff37df61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.048ex; height:6.176ex;" alt="{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi -{\frac {1}{4\pi \epsilon _{0}}}{\frac {Ze^{2}}{r}}\psi .}"></span></dd></dl> <p>Dividint per <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 c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d07975b905245e67bb9a6ee394e55e98431d85a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.313ex; height:2.176ex;" alt="{\displaystyle \hbar c}"></span>, i reescrivint termes de la constant d'estructura fina, s'obté: </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 {i}{c}}{\frac {\partial }{\partial t}}\psi =-{\frac {1}{2}}\left({\frac {\hbar }{mc}}\right)\nabla ^{2}\psi -{\frac {\alpha Z}{r}}\psi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi>c</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mrow> <mi>m</mi> <mi>c</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>α</mi> <mi>Z</mi> </mrow> <mi>r</mi> </mfrac> </mrow> <mi>ψ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {i}{c}}{\frac {\partial }{\partial t}}\psi =-{\frac {1}{2}}\left({\frac {\hbar }{mc}}\right)\nabla ^{2}\psi -{\frac {\alpha Z}{r}}\psi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/718a7bb5e8ee215f973b461b894e0136b2a301cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.842ex; height:6.176ex;" alt="{\displaystyle {\frac {i}{c}}{\frac {\partial }{\partial t}}\psi =-{\frac {1}{2}}\left({\frac {\hbar }{mc}}\right)\nabla ^{2}\psi -{\frac {\alpha Z}{r}}\psi .}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Referències"><span id="Refer.C3.A8ncies"></span>Referències</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Longitud_d%27ona_Compton&action=edit&section=3" title="Modifica la secció: Referències"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist {{#if: | references-column-count references-column-count-{{{col}}}" style="list-style-type: decimal;"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span class="citation" style="font-style:normal">«<a rel="nofollow" class="external text" href="https://physics.nist.gov/cuu/Constants/">Fundamental Physical Constants from NIST</a>». [Consulta: 22 octubre 2017].</span></span> </li> </ol></div></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="">Obtingut de «<a dir="ltr" href="https://ca.wikipedia.org/w/index.php?title=Longitud_d%27ona_Compton&oldid=32081486">https://ca.wikipedia.org/w/index.php?title=Longitud_d%27ona_Compton&oldid=32081486</a>»</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categories</a>: <ul><li><a href="/wiki/Categoria:F%C3%ADsica_de_part%C3%ADcules" title="Categoria:Física de partícules">Física de partícules</a></li><li><a href="/wiki/Categoria:F%C3%ADsica_at%C3%B2mica" title="Categoria:Física atòmica">Física atòmica</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> La pàgina va ser modificada per darrera vegada el 28 juny 2023 a les 10:32.</li> <li id="footer-info-copyright">El text està disponible sota la <a href="/wiki/Viquip%C3%A8dia:Text_de_la_llic%C3%A8ncia_de_Creative_Commons_Reconeixement-Compartir_Igual_4.0_No_adaptada" title="Viquipèdia:Text de la llicència de Creative Commons Reconeixement-Compartir Igual 4.0 No adaptada"> Llicència de Creative Commons Reconeixement i Compartir-Igual</a>; es poden aplicar termes addicionals. 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