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class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Aparença"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Aparença" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span 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mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Crea_compte&returnto=Unitat+at%C3%B2mica" title="Us animem a crear un compte i iniciar una sessió, encara que no és obligatori" class=""><span>Crea un compte</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Registre_i_entrada&returnto=Unitat+at%C3%B2mica" title="Us animem a registrar-vos, però no és obligatori [o]" accesskey="o" class=""><span>Inicia la sessió</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Més opcions" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Eines personals" 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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-Unitats_fonamentals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Unitats_fonamentals"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Unitats fonamentals</span> </div> </a> <ul id="toc-Unitats_fonamentals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algunes_unitats_derivades" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algunes_unitats_derivades"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Algunes unitats derivades</span> </div> </a> <ul id="toc-Algunes_unitats_derivades-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Comparació_amb_les_unitats_de_Planck" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Comparació_amb_les_unitats_de_Planck"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Comparació amb les unitats de Planck</span> </div> </a> <ul id="toc-Comparació_amb_les_unitats_de_Planck-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mecànica_i_electrodinàmica_quàntiques_simplificades" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mecànica_i_electrodinàmica_quàntiques_simplificades"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Mecànica i electrodinàmica quàntiques simplificades</span> </div> </a> <ul id="toc-Mecànica_i_electrodinàmica_quàntiques_simplificades-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Enllaços_externs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Enllaços_externs"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Enllaços externs</span> </div> </a> <ul id="toc-Enllaços_externs-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">Unitat atòmica</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 18 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-18" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">18 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/%D9%88%D8%AD%D8%AF%D8%A9_%D8%B0%D8%B1%D9%8A%D8%A9" title="وحدة ذرية - àrab" lang="ar" hreflang="ar" data-title="وحدة ذرية" data-language-autonym="العربية" data-language-local-name="àrab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Atomare_enheder" title="Atomare enheder - danès" lang="da" hreflang="da" data-title="Atomare enheder" data-language-autonym="Dansk" data-language-local-name="danès" 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/Atomare_Einheiten" title="Atomare Einheiten - alemany" lang="de" hreflang="de" data-title="Atomare Einheiten" data-language-autonym="Deutsch" data-language-local-name="alemany" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Atomic_units" title="Atomic units - anglès" lang="en" hreflang="en" data-title="Atomic units" data-language-autonym="English" data-language-local-name="anglès" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Unidades_at%C3%B3micas" title="Unidades atómicas - espanyol" lang="es" hreflang="es" data-title="Unidades atómicas" data-language-autonym="Español" data-language-local-name="espanyol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Syst%C3%A8me_d%27unit%C3%A9s_atomiques" title="Système d'unités atomiques - francès" lang="fr" hreflang="fr" data-title="Système d'unités atomiques" data-language-autonym="Français" data-language-local-name="francès" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Unit%C3%A0_atomiche" title="Unità atomiche - italià" lang="it" hreflang="it" data-title="Unità atomiche" 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/%E5%8E%9F%E5%AD%90%E5%8D%98%E4%BD%8D%E7%B3%BB" title="原子単位系 - japonès" lang="ja" hreflang="ja" data-title="原子単位系" 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%9B%90%EC%9E%90_%EB%8B%A8%EC%9C%84%EA%B3%84" title="원자 단위계 - coreà" lang="ko" hreflang="ko" data-title="원자 단위계" data-language-autonym="한국어" data-language-local-name="coreà" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Atomaire_eenheden" title="Atomaire eenheden - neerlandès" lang="nl" hreflang="nl" data-title="Atomaire eenheden" data-language-autonym="Nederlands" data-language-local-name="neerlandès" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Jednostki_atomowe" title="Jednostki atomowe - polonès" lang="pl" hreflang="pl" data-title="Jednostki atomowe" 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/Unidades_at%C3%B4micas" title="Unidades atômicas - portuguès" lang="pt" hreflang="pt" data-title="Unidades atômicas" 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%90%D1%82%D0%BE%D0%BC%D0%BD%D0%B0%D1%8F_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%B5%D0%B4%D0%B8%D0%BD%D0%B8%D1%86" title="Атомная система единиц - rus" lang="ru" hreflang="ru" data-title="Атомная система единиц" data-language-autonym="Русский" data-language-local-name="rus" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Unitati_atomiche" title="Unitati atomiche - sicilià" lang="scn" hreflang="scn" data-title="Unitati atomiche" data-language-autonym="Sicilianu" data-language-local-name="sicilià" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Atomske_enote" title="Atomske enote - eslovè" lang="sl" hreflang="sl" data-title="Atomske enote" data-language-autonym="Slovenščina" data-language-local-name="eslovè" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Atomik_birimler" title="Atomik birimler - turc" lang="tr" hreflang="tr" data-title="Atomik birimler" 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%90%D1%82%D0%BE%D0%BC%D0%BD%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%BE%D0%B4%D0%B8%D0%BD%D0%B8%D1%86%D1%8C" title="Атомна система одиниць - ucraïnès" lang="uk" hreflang="uk" data-title="Атомна система одиниць" 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%8E%9F%E5%AD%90%E5%8D%95%E4%BD%8D%E5%88%B6" title="原子单位制 - xinès" lang="zh" hreflang="zh" data-title="原子单位制" data-language-autonym="中文" data-language-local-name="xinès" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div 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title="Modifica el codi font d'aquesta pàgina [e]" accesskey="e"><span>Modifica</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Unitat_at%C3%B2mica&action=history"><span>Mostra l'historial</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Especial:Enlla%C3%A7os/Unitat_at%C3%B2mica" title="Una llista de totes les pàgines wiki que enllacen amb aquesta [j]" accesskey="j"><span>Què hi enllaça</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:Seguiment/Unitat_at%C3%B2mica" rel="nofollow" title="Canvis recents a pàgines enllaçades des d'aquesta pàgina [k]" accesskey="k"><span>Canvis relacionats</span></a></li><li id="t-specialpages" class="mw-list-item"><a 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per a impressora</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> En altres projectes </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q757568" title="Enllaç a l'element del repositori de dades connectat [g]" accesskey="g"><span>Element a Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Eines de la pàgina"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aparença"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aparença</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.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-appearance.unpin">amaga</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">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>Les <b>unitats atòmiques</b> (<b>ua</b>) són un <a href="/wiki/Sistema_d%27unitats" title="Sistema d'unitats">sistema d'unitats</a> adequat per a la <a href="/wiki/F%C3%ADsica_at%C3%B2mica" title="Física atòmica">física atòmica</a>, l'<a href="/wiki/Electromagnetisme" title="Electromagnetisme">electromagnetisme</a> i l'<a href="/wiki/Electrodin%C3%A0mica_qu%C3%A0ntica" title="Electrodinàmica quàntica">electrodinàmica quàntica</a>, especialment perquè està enfocat a les propietats dels <a href="/wiki/Electrons" class="mw-redirect" title="Electrons">electrons</a>. El <a href="/wiki/Sistema_Internacional_d%27Unitats" title="Sistema Internacional d'Unitats">Sistema Internacional d'Unitats</a> és un mitjà de comunicació universal però segueix una lògica a l'escala humana que no és la més adient per a certs dominis científics. El sistema d'unitats atòmiques es basa en el valor de sis <a href="/wiki/Constant_f%C3%ADsica" title="Constant física">constants físiques</a>: </p> <ul><li>Dues propietats de l'electró: la seva <a href="/wiki/Massa_en_rep%C3%B2s" title="Massa en repòs">massa</a> i la seva <a href="/wiki/C%C3%A0rrega_elemental" class="mw-redirect" title="Càrrega elemental">càrrega</a></li> <li>Dues propietats del <a href="/wiki/Proti" title="Proti">proti</a>: el seu <a href="/wiki/Radi_de_Bohr" title="Radi de Bohr">radi de Bohr</a> i el <a href="/wiki/Valor_absolut" title="Valor absolut">valor absolut</a> de la seva energia potencial elèctrica en estat fonamental.</li> <li>Dues constants físiques: la <a href="/wiki/Constant_de_Dirac" class="mw-redirect" title="Constant de Dirac">constant de Dirac</a> i la <a href="/wiki/Llei_de_Coulomb" title="Llei de Coulomb">llei de Coulomb</a>.</li></ul> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Unitats_fonamentals">Unitats fonamentals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unitat_at%C3%B2mica&action=edit&section=1" title="Modifica la secció: Unitats fonamentals"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable"> <tbody><tr> <th colspan="5"><font size="+0">Unitats atòmiques fonamentals</font> </th></tr> <tr> <td><b>Unitat</b> </td> <td><b>Nom</b> </td> <td><b>Símbol</b> </td> <td><b>Valor al <a href="/wiki/Sistema_Internacional_d%27Unitats" title="Sistema Internacional d'Unitats">SI</a></b> </td> <td><b>Escala d'<a href="/wiki/Unitats_de_Planck" title="Unitats de Planck">unitats de Planck</a></b> </td></tr> <tr> <td><a href="/wiki/Longitud" title="Longitud">longitud</a></td> <td><a href="/wiki/Radi_de_Bohr" title="Radi de Bohr">radi de Bohr</a></td> <td><i>a</i><sub>0</sub></td> <td>5.291 772 108(18)×10<sup>-11</sup> m</td> <td>10<sup>-35</sup> m </td></tr> <tr> <td><a href="/wiki/Massa" title="Massa">massa</a></td> <td><a href="/wiki/Massa_en_rep%C3%B2s" title="Massa en repòs">massa en repòs</a> de l'<a href="/wiki/Electr%C3%B3" title="Electró">electró</a></td> <td><i>m</i><sub>e</sub></td> <td>9.109 3826(16)×10<sup>-31</sup> kg</td> <td>10<sup>-8</sup> kg </td></tr> <tr> <td><a href="/wiki/C%C3%A0rrega_el%C3%A8ctrica" title="Càrrega elèctrica">càrrega elèctrica</a></td> <td><a href="/wiki/C%C3%A0rrega_elemental" class="mw-redirect" title="Càrrega elemental">càrrega elemental</a></td> <td><i>e</i></td> <td>1.602 176 53(14)×10<sup>-19</sup> C</td> <td>10<sup>-18</sup> C </td></tr> <tr> <td><a href="/wiki/Moment_angular" title="Moment angular">moment angular</a></td> <td><a href="/wiki/Constant_de_Dirac" class="mw-redirect" title="Constant de Dirac">constant de Dirac</a></td> <td><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 =h/2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar =h/2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f836f3a2de168369eb9f80f4057518222ecde88b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.401ex; height:2.843ex;" alt="{\displaystyle \hbar =h/2\pi }"></span></td> <td>1.054 571 68(18)×10<sup>-34</sup> J s</td> <td>(igual) </td></tr> <tr> <td><a href="/wiki/Energia" title="Energia">energia</a></td> <td><a href="/wiki/Energia_de_Hartree" title="Energia de Hartree">energia de Hartree</a></td> <td><i>E</i><sub>h</sub></td> <td>4.359 744 17(75)×10<sup>-18</sup> J</td> <td>10⁹ J </td></tr> <tr> <td><a href="/wiki/Llei_de_Coulomb" title="Llei de Coulomb">constant electroestàtica</a></td> <td><a href="/wiki/Llei_de_Coulomb" title="Llei de Coulomb">constant de Coulomb</a></td> <td>1/(4πε<sub>0</sub>)</td> <td>8.9875516×10⁹ C<sup>-2</sup> N m²</td> <td>(igual) </td></tr></tbody></table> <p>Aquest sis valors no són independents, per tal de normalitzar-los a 1 n'hi ha prou amb normalitzar quatre, perquè l'energia de Hartree i la constant electroestàtica (o de Coulomb) són una conseqüència de normalitzar les altres. </p> <div class="mw-heading mw-heading2"><h2 id="Algunes_unitats_derivades">Algunes unitats derivades</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unitat_at%C3%B2mica&action=edit&section=2" title="Modifica la secció: Algunes unitats derivades"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable"> <tbody><tr> <th colspan="4"><font size="+0">Unitats atòmiques derivades</font> </th></tr> <tr> <td><b>Unitat</b> </td> <td><b>Expressió</b> </td> <td><b>Valor al <a href="/wiki/Sistema_Internacional_d%27Unitats" title="Sistema Internacional d'Unitats">SI</a></b> </td> <td><b>Escala d'<a href="/wiki/Unitats_de_Planck" title="Unitats de Planck">Unitats de Planck</a></b> </td></tr> <tr> <td><a href="/wiki/Temps" title="Temps">temps</a></td> <td><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 {\hbar }{E_{h}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\hbar }{E_{h}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46e97867d657baff7864bef608c214c03a519a5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:3.73ex; height:5.676ex;" alt="{\displaystyle {\frac {\hbar }{E_{h}}}}"></span></td> <td>2.418 884 326 505(16)×10<sup>-17</sup> s</td> <td>10<sup>-43</sup> s </td></tr> <tr> <td><a href="/wiki/Velocitat" title="Velocitat">velocitat</a></td> <td><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 {a_{0}E_{h}}{\hbar }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a_{0}E_{h}}{\hbar }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2444d3eec6ba3a36677022994fe8d036c2eaa26d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.014ex; height:5.509ex;" alt="{\displaystyle {\frac {a_{0}E_{h}}{\hbar }}}"></span></td> <td>2.187 691 2633(73)×10⁶ m s<sup>-1</sup></td> <td>10<sup>8</sup> m s<sup>-1</sup> </td></tr> <tr> <td><a href="/wiki/For%C3%A7a" title="Força">força</a></td> <td><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 {E_{h}}{a_{o}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {E_{h}}{a_{o}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c023f4d868c9a87f9bd1ce073e8ca6f38aea3437" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:3.73ex; height:5.509ex;" alt="{\displaystyle {\frac {E_{h}}{a_{o}}}}"></span></td> <td>8.238 7225(14)×10<sup>-8</sup> N</td> <td>10<sup>44</sup> N </td></tr> <tr> <td><a href="/wiki/Corrent_el%C3%A8ctric" title="Corrent elèctric">corrent elèctric</a></td> <td><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 {eE_{h}}{\hbar }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>e</mi> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {eE_{h}}{\hbar }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc3aba4e80971d84c4f6d5b13c2e28ce5ebdc6be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.814ex; height:5.343ex;" alt="{\displaystyle {\frac {eE_{h}}{\hbar }}}"></span></td> <td>6.623 617 82(57)×10<sup>-3</sup> A</td> <td>10<sup>26</sup> A </td></tr> <tr> <td><a href="/wiki/Temperatura" title="Temperatura">temperatura</a></td> <td><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 {E_{h}}{k_{B}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {E_{h}}{k_{B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aca23f28bd39fdf8f579686f98f331583bd052dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:3.73ex; height:5.676ex;" alt="{\displaystyle {\frac {E_{h}}{k_{B}}}}"></span></td> <td>3.157 7464(55)×10⁵ K</td> <td>10³² K </td></tr> <tr> <td><a href="/wiki/Pressi%C3%B3" title="Pressió">pressió</a></td> <td><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 {E_{h}}{{a_{o}}^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {E_{h}}{{a_{o}}^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc2f15b3fd3b27fc2edaa3a2586c7c0fc8ee422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:4.15ex; height:5.843ex;" alt="{\displaystyle {\frac {E_{h}}{{a_{o}}^{3}}}}"></span></td> <td>2.9421912(19)×10<sup>13</sup> N m<sup>-2</sup></td> <td>10<sup>114</sup> Pa </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Comparació_amb_les_unitats_de_Planck"><span id="Comparaci.C3.B3_amb_les_unitats_de_Planck"></span>Comparació amb les unitats de Planck</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unitat_at%C3%B2mica&action=edit&section=3" title="Modifica la secció: Comparació amb les unitats de Planck"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tant les <b>unitats atòmiques</b> com les <a href="/wiki/Unitats_de_Planck" title="Unitats de Planck">unitats de Planck</a> deriven de certes propietats fonamentals del món físic, lliures de consideracions <a href="/wiki/Antropocentrisme" title="Antropocentrisme">antropocèntriques</a>. Per a facilitar la comparació entre els dos sistemes d'unitats, les taules anteriors mostren l'<a href="/wiki/Ordre_de_magnitud" title="Ordre de magnitud">ordre de magnitud</a>, en unitats del <a href="/wiki/Sistema_Internacional_d%27Unitats" title="Sistema Internacional d'Unitats">Sistema Internacional</a>, de la unitat de Planck que correspon a cada unitat atòmica. Generalment, quan una unitat atòmica és "gran" en termes del SI, la corresponent unitat de Planck és "petita", i viceversa. Cal tenir en consideració que les unitats atòmiques van ser dissenyades per a fer càlculs a escala atòmica a l'<a href="/wiki/Univers" title="Univers">Univers</a> actual, mentre que les <a href="/wiki/Unitats_de_Planck" title="Unitats de Planck">unitats de Planck</a> són més adequades per la <a href="/wiki/Gravetat_qu%C3%A0ntica" class="mw-redirect" title="Gravetat quàntica">gravetat quàntica</a> i la <a href="/wiki/Cosmologia" title="Cosmologia">cosmologia</a> de l'Univers primitiu. </p><p>Tant les unitats atòmiques com les unitats de Planck normalitzen la constant de Dirac i la constant electroestàtica a 1. Més enllà d'això, les unitats de Planck normalitzen a 1 dues constants fonamentals de la <a href="/wiki/Relativitat_general" title="Relativitat general">relativitat general</a> i la cosmologia: la <a href="/wiki/Constant_gravitacional" class="mw-redirect" title="Constant gravitacional">constant gravitacional</a> <i>G</i> i la <a href="/wiki/Velocitat_de_la_llum" title="Velocitat de la llum">velocitat de la llum</a> al <a href="/wiki/Buit" title="Buit">buit</a>, <i>c</i>. Essent α la notació de la <a href="/wiki/Constant_d%27estructura_fina" title="Constant d'estructura fina">constant d'estructura fina</a>, el valor en unitats atòmiques de <i>c</i> és α<sup>-1</sup> ≈ 137.036. </p><p>Les unitats atòmiques, per contra, normalitzen a 1 la massa i la càrrega de l'electró, i <i>a</i><sub>0</sub>, el radi de Bohr del proti. Normalitzar <i>a</i><sub>0</sub> a 1 comporta la normalització de la <a href="/wiki/Constant_de_Rydberg" title="Constant de Rydberg">constant de Rydberg</a>, <i>R</i><sub>∞</sub>, a 4π/α = 4π<i>c</i>. Donat en unitats atòmiques, el <a href="/wiki/Magnet%C3%B3_de_Bohr" title="Magnetó de Bohr">magnetó de Bohr</a> seria μ<sub>B</sub>=1/2, el valor corresponent en unitats de Planck és <i>e</i>/2<i>m</i><sub>e</sub>. Finalment, les unitats atòmiques normalitzen a 1 la unitat d'energia atòmica, mentre les unitats de Planck normalitzen a 1 la <a href="/wiki/Constant_de_Boltzmann" title="Constant de Boltzmann">constant de Boltzmann</a> <i>k</i>, que relaciona <a href="/wiki/Energia" title="Energia">energia</a> i <a href="/wiki/Temperatura" title="Temperatura">temperatura</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Mecànica_i_electrodinàmica_quàntiques_simplificades"><span id="Mec.C3.A0nica_i_electrodin.C3.A0mica_qu.C3.A0ntiques_simplificades"></span>Mecànica i electrodinàmica quàntiques simplificades</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unitat_at%C3%B2mica&action=edit&section=4" title="Modifica la secció: Mecànica i electrodinàmica quàntiques simplificades"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>L'<a href="/wiki/Equaci%C3%B3_de_Schr%C3%B6dinger" title="Equació de Schrödinger">equació de Schrödinger</a> (no relativista) per a un electró en unitats del SI és: </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 {\hbar ^{2}}{2m_{e}}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>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>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mi>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>i</mi> <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 stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2m_{e}}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ec3a568c9bffe6e667fdc2f9c3d0d3e8ff7f997" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:43.63ex; height:6.009ex;" alt="{\displaystyle -{\frac {\hbar ^{2}}{2m_{e}}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)}"></span>.</dd></dl> <p>La mateixa <a href="/wiki/Equaci%C3%B3" title="Equació">equació</a> en <b>unitats atòmiques</b> és </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {1}{2}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)=i{\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>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>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mi>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>i</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 stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {1}{2}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)=i{\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44f5f366822ef89cc819903c0d37f4cb1d38fbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:39.285ex; height:5.676ex;" alt="{\displaystyle -{\frac {1}{2}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)=i{\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)}"></span>.</dd></dl> <p>Per al cas especial de l'electró entorn d'un <a href="/wiki/Proti" title="Proti">proti</a>, l'<a href="/wiki/Operador_hamiltoni%C3%A0" class="mw-redirect" title="Operador hamiltonià">operador hamiltonià</a> és en unitats del SI: </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}}=-{{{\hbar ^{2}} \over {2m_{e}}}\nabla ^{2}}-{1 \over {4\pi \epsilon _{0}}}{{e^{2}} \over {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> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <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 class="MJX-TeXAtom-ORD"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}=-{{{\hbar ^{2}} \over {2m_{e}}}\nabla ^{2}}-{1 \over {4\pi \epsilon _{0}}}{{e^{2}} \over {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbdd6e253f8b85da61fd027e187158f9be73da5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.141ex; height:6.176ex;" alt="{\displaystyle {\hat {H}}=-{{{\hbar ^{2}} \over {2m_{e}}}\nabla ^{2}}-{1 \over {4\pi \epsilon _{0}}}{{e^{2}} \over {r}}}"></span></dd></dl> <p>mentre que les unitats atòmiques transformen l'equació precedent en </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}}=-{{{1} \over {2}}\nabla ^{2}}-{{1} \over {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> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}=-{{{1} \over {2}}\nabla ^{2}}-{{1} \over {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e520121ae76ba49437572174f12664cb18cebf74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.798ex; height:5.176ex;" alt="{\displaystyle {\hat {H}}=-{{{1} \over {2}}\nabla ^{2}}-{{1} \over {r}}}"></span>.</dd></dl> <p>Finalment, les <a href="/wiki/Equacions_de_Maxwell" title="Equacions de Maxwell">equacions de Maxwell</a> prenen una forma elegant en unitats atòmiques: </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 \nabla \cdot \mathbf {E} =4\pi \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mn>4</mn> <mi>π</mi> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {E} =4\pi \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2038cb70658f28c2850fbbf5c97c27cea43c7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.167ex; height:2.676ex;" alt="{\displaystyle \nabla \cdot \mathbf {E} =4\pi \rho }"></span></dd> <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 \nabla \cdot \mathbf {B} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {B} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16ee950683349dacdd9e9c262ff6133812747edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.777ex; height:2.176ex;" alt="{\displaystyle \nabla \cdot \mathbf {B} =0}"></span></dd> <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 \nabla \times \mathbf {E} =-\alpha {\frac {\partial \mathbf {B} }{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {E} =-\alpha {\frac {\partial \mathbf {B} }{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2078ba45a842beae30917bf0552c3ed66aeb9cd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.983ex; height:5.509ex;" alt="{\displaystyle \nabla \times \mathbf {E} =-\alpha {\frac {\partial \mathbf {B} }{\partial t}}}"></span></dd> <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 \nabla \times \mathbf {B} =\alpha \left({\frac {\partial \mathbf {E} }{\partial t}}+4\pi \mathbf {J} \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mi>α</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>4</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {B} =\alpha \left({\frac {\partial \mathbf {E} }{\partial t}}+4\pi \mathbf {J} \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/333d934af708e95c963e0b4b0cb7828eae569dc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.699ex; height:6.176ex;" alt="{\displaystyle \nabla \times \mathbf {B} =\alpha \left({\frac {\partial \mathbf {E} }{\partial t}}+4\pi \mathbf {J} \right)}"></span></dd></dl> <p>(Actualment hi ha una certa ambigüitat en la definició de la unitat de <a href="/wiki/Camp_magn%C3%A8tic" title="Camp magnètic">camp magnètic</a>. Les equacions de Maxwell anteriors utilitzen la convenció <a href="/wiki/Llei_de_Gauss" title="Llei de Gauss">gaussiana</a>, segons la qual una <a href="/wiki/Ona" title="Ona">ona</a> plana té un camp elèctric i magnètic d'idèntica magnitud. A la convenció de la <a href="/wiki/For%C3%A7a_de_Lorentz" title="Força de Lorentz">força de Lorentz</a>, el factor α és absorbit a <b>B</b>.) </p> <div class="mw-heading mw-heading2"><h2 id="Enllaços_externs"><span id="Enlla.C3.A7os_externs"></span>Enllaços externs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unitat_at%C3%B2mica&action=edit&section=5" title="Modifica la secció: Enllaços externs"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.codata.org/taskgroups/TGfundconst/index.html">Constants físiques fonamentals</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20071027092745/http://www.codata.org/taskgroups/TGfundconst/index.html">Arxivat</a> 2007-10-27 a <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. <style 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aquesta plantilla" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Commons-emblem-notice.svg/18px-Commons-emblem-notice.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Commons-emblem-notice.svg/27px-Commons-emblem-notice.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Commons-emblem-notice.svg/36px-Commons-emblem-notice.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></li></ul></div><div id="Unitats_del_Sistema_Internacional" style="font-size:114%;margin:0 4em"><a href="/wiki/Sistema_Internacional_d%27Unitats" title="Sistema Internacional d'Unitats">Unitats del Sistema Internacional</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#b0d1ad"><a href="/wiki/Unitat_b%C3%A0sica_del_SI" class="mw-redirect" title="Unitat bàsica del SI">Unitats bàsiques</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Ampere" title="Ampere">Ampere</a></li> <li><a href="/wiki/Candela" title="Candela">Candela</a></li> <li><a href="/wiki/Kelvin" title="Kelvin">Kelvin</a></li> <li><a href="/wiki/Metre" title="Metre">Metre</a></li> <li><a href="/wiki/Mol" title="Mol">Mol</a></li> <li><a href="/wiki/Quilogram" title="Quilogram">Quilogram</a></li> <li><a href="/wiki/Segon" title="Segon">Segon</a></li></ul> </div></td><td class="navbox-image" rowspan="4" style="width:1px;padding:0px 0px 0px 2px"><div><span typeof="mw:File"><a href="/wiki/Fitxer:SI_base_unit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/SI_base_unit.svg/75px-SI_base_unit.svg.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/SI_base_unit.svg/113px-SI_base_unit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/SI_base_unit.svg/150px-SI_base_unit.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#b0d1ad"><a href="/wiki/Unitat_derivada_del_SI" class="mw-redirect" title="Unitat derivada del SI">Unitats derivades</a></th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Becquerel_(unitat)" title="Becquerel (unitat)">Becquerel</a></li> <li><a href="/wiki/Coulomb" title="Coulomb">Coulomb</a></li> <li><a href="/wiki/Grau_Celsius" title="Grau Celsius">Grau Celsius</a></li> <li><a href="/wiki/Estereoradian" title="Estereoradian">Estereoradian</a></li> <li><a href="/wiki/Farad" title="Farad">Farad</a></li> <li><a href="/wiki/Gray_(unitat)" title="Gray (unitat)">Gray</a></li> <li><a href="/wiki/Henry_(unitat)" title="Henry (unitat)">Henry</a></li> <li><a href="/wiki/Hertz" title="Hertz">Hertz</a></li> <li><a href="/wiki/Joule" title="Joule">Joule</a></li> <li><a href="/wiki/Katal" title="Katal">Katal</a></li> <li><a href="/wiki/Lumen" title="Lumen">Lumen</a></li> <li><a href="/wiki/Lux" title="Lux">Lux</a></li> <li><a href="/wiki/Newton_(unitat)" title="Newton (unitat)">Newton</a></li> <li><a href="/wiki/Ohm_(unitat)" title="Ohm (unitat)">Ohm</a></li> <li><a href="/wiki/Pascal_(unitat)" title="Pascal (unitat)">Pascal</a></li> <li><a href="/wiki/Radian" title="Radian">Radian</a></li> <li><a href="/wiki/Siemens_(unitat)" title="Siemens (unitat)">Siemens</a></li> <li><a href="/wiki/Sievert" title="Sievert">Sievert</a></li> <li><a href="/wiki/Tesla_(unitat)" title="Tesla (unitat)">Tesla</a></li> <li><a href="/wiki/Volt" title="Volt">Volt</a></li> <li><a href="/wiki/Watt" title="Watt">Watt</a></li> <li><a href="/wiki/Weber_(unitat)" title="Weber (unitat)">Weber</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#b0d1ad"><a href="/wiki/Unitats_que_no_formen_part_del_SI_per%C3%B2_s%27hi_mencionen" title="Unitats que no formen part del SI però s'hi mencionen">Unitats <br />acceptades</a></th><td class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Unitat_de_massa_at%C3%B2mica" title="Unitat de massa atòmica">Unitat de massa atòmica</a></li> <li><a href="/wiki/Unitat_astron%C3%B2mica" title="Unitat astronòmica">Unitat astronòmica</a></li> <li><a href="/wiki/Dia" title="Dia">Dia</a></li> <li><a href="/wiki/Decibel" title="Decibel">Decibel</a></li> <li><a href="/wiki/Grau_Celsius" title="Grau Celsius">Grau Celsius</a></li> <li><a href="/wiki/Grau_sexagesimal" title="Grau sexagesimal">Grau sexagesimal</a></li> <li><a href="/wiki/Electr%C3%B3-volt" title="Electró-volt">Electró-volt</a></li> <li><a href="/wiki/Hect%C3%A0rea" title="Hectàrea">Hectàrea</a></li> <li><a href="/wiki/Hora" title="Hora">Hora</a></li> <li><a href="/wiki/Litre" title="Litre">Litre</a></li> <li><a href="/wiki/Minut" title="Minut">Minut</a></li> <li><a href="/wiki/Minut_d%27arc" title="Minut d'arc">Minut d'arc</a></li> <li><a href="/wiki/Neper" title="Neper">Neper</a></li> <li><a href="/wiki/Segon_d%27arc" class="mw-redirect" title="Segon d'arc">Segon d'arc</a></li> <li><a href="/wiki/Tona_(unitat)" title="Tona (unitat)">Tona</a> <br /> <a class="mw-selflink selflink">Unitats atòmiques</a></li> <li><a href="/wiki/Unitat_natural" title="Unitat natural">Unitats naturals</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#b0d1ad">Vegeu també</th><td class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> 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