Modello atomico di Bohr

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data-event-name="pinnable-header.vector-toc.unpin">nascondi</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">Inizio</div> </a> </li> <li id="toc-Stato_dell'arte_prima_di_Bohr" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Stato_dell'arte_prima_di_Bohr"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Stato dell'arte prima di Bohr</span> </div> </a> <button aria-controls="toc-Stato_dell'arte_prima_di_Bohr-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>Attiva/disattiva la sottosezione Stato dell'arte prima di Bohr</span> </button> <ul id="toc-Stato_dell'arte_prima_di_Bohr-sublist" class="vector-toc-list"> <li id="toc-Spettroscopia_atomica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spettroscopia_atomica"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Spettroscopia atomica</span> </div> </a> <ul id="toc-Spettroscopia_atomica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Primi_modelli_atomici" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Primi_modelli_atomici"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Primi modelli atomici</span> </div> </a> <ul id="toc-Primi_modelli_atomici-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-I_tre_postulati_di_Bohr" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#I_tre_postulati_di_Bohr"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>I tre postulati di Bohr</span> </div> </a> <ul id="toc-I_tre_postulati_di_Bohr-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantizzazione_delle_orbite_e_raggio_di_Bohr" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Quantizzazione_delle_orbite_e_raggio_di_Bohr"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Quantizzazione delle orbite e raggio di Bohr</span> </div> </a> <ul id="toc-Quantizzazione_delle_orbite_e_raggio_di_Bohr-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantizzazione_delle_energie_e_formula_di_Rydberg" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Quantizzazione_delle_energie_e_formula_di_Rydberg"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Quantizzazione delle energie e formula di Rydberg</span> </div> </a> <button aria-controls="toc-Quantizzazione_delle_energie_e_formula_di_Rydberg-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>Attiva/disattiva la sottosezione Quantizzazione delle energie e formula di Rydberg</span> </button> <ul id="toc-Quantizzazione_delle_energie_e_formula_di_Rydberg-sublist" class="vector-toc-list"> <li id="toc-Quantizzazione_delle_energie_orbitali" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantizzazione_delle_energie_orbitali"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Quantizzazione delle energie orbitali</span> </div> </a> <ul id="toc-Quantizzazione_delle_energie_orbitali-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivazione_della_formula_di_Rydberg" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivazione_della_formula_di_Rydberg"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Derivazione della formula di Rydberg</span> </div> </a> <ul id="toc-Derivazione_della_formula_di_Rydberg-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Correzione_per_la_massa_finita_del_nucleo" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Correzione_per_la_massa_finita_del_nucleo"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Correzione per la massa finita del nucleo</span> </div> </a> <ul id="toc-Correzione_per_la_massa_finita_del_nucleo-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Limiti_del_modello_di_Bohr" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Limiti_del_modello_di_Bohr"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Limiti del modello di Bohr</span> </div> </a> <ul id="toc-Limiti_del_modello_di_Bohr-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modello_di_Bohr-Sommerfeld" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Modello_di_Bohr-Sommerfeld"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Modello di Bohr-Sommerfeld</span> </div> </a> <ul id="toc-Modello_di_Bohr-Sommerfeld-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_progetti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Altri_progetti"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Altri progetti</span> </div> </a> <ul id="toc-Altri_progetti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Collegamenti_esterni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Collegamenti_esterni"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Collegamenti esterni</span> </div> </a> <ul id="toc-Collegamenti_esterni-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="Indice" 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="Mostra/Nascondi l'indice" > <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">Mostra/Nascondi l'indice</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">Modello atomico di Bohr</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="Vai a una voce in un'altra lingua. Disponibile in 69 lingue" > <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-69" 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">69 lingue</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Bohrsches_Atommodell" title="Bohrsches Atommodell - tedesco svizzero" lang="gsw" hreflang="gsw" data-title="Bohrsches Atommodell" data-language-autonym="Alemannisch" data-language-local-name="tedesco svizzero" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Modelo_atomico_de_Bohr" title="Modelo atomico de Bohr - aragonese" lang="an" hreflang="an" data-title="Modelo atomico de Bohr" data-language-autonym="Aragonés" data-language-local-name="aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D9%85%D9%88%D8%B0%D8%AC_%D8%A8%D9%88%D8%B1" title="نموذج بور - arabo" lang="ar" hreflang="ar" data-title="نموذج بور" data-language-autonym="العربية" data-language-local-name="arabo" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Modelu_at%C3%B3micu_de_Bohr" title="Modelu atómicu de Bohr - asturiano" lang="ast" hreflang="ast" data-title="Modelu atómicu de Bohr" data-language-autonym="Asturianu" data-language-local-name="asturiano" 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/Bor_modeli" title="Bor modeli - azerbaigiano" lang="az" hreflang="az" data-title="Bor modeli" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaigiano" 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%91%D0%BE%D1%80%D0%B0%D1%9E%D1%81%D0%BA%D0%B0%D1%8F_%D0%BC%D0%B0%D0%B4%D1%8D%D0%BB%D1%8C_%D0%B0%D1%82%D0%B0%D0%BC%D0%B0" title="Бораўская мадэль атама - bielorusso" lang="be" hreflang="be" data-title="Бораўская мадэль атама" data-language-autonym="Беларуская" data-language-local-name="bielorusso" 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%9C%D0%BE%D0%B4%D0%B5%D0%BB_%D0%BD%D0%B0_%D0%91%D0%BE%D1%80" title="Модел на Бор - bulgaro" lang="bg" hreflang="bg" data-title="Модел на Бор" data-language-autonym="Български" data-language-local-name="bulgaro" 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%AC%E0%A7%8B%E0%A6%B0_%E0%A6%AE%E0%A6%A1%E0%A7%87%E0%A6%B2" title="বোর মডেল - bengalese" lang="bn" hreflang="bn" data-title="বোর মডেল" data-language-autonym="বাংলা" data-language-local-name="bengalese" 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/Bohrov_model_atoma" title="Bohrov model atoma - bosniaco" lang="bs" hreflang="bs" data-title="Bohrov model atoma" data-language-autonym="Bosanski" data-language-local-name="bosniaco" 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/Model_at%C3%B2mic_de_Bohr" title="Model atòmic de Bohr - catalano" lang="ca" hreflang="ca" data-title="Model atòmic de Bohr" data-language-autonym="Català" data-language-local-name="catalano" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bohr%C5%AFv_model_atomu" title="Bohrův model atomu - ceco" lang="cs" hreflang="cs" data-title="Bohrův model atomu" data-language-autonym="Čeština" data-language-local-name="ceco" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Bohrs_atommodel" title="Bohrs atommodel - danese" lang="da" hreflang="da" data-title="Bohrs atommodel" data-language-autonym="Dansk" data-language-local-name="danese" 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/Bohrsches_Atommodell" title="Bohrsches Atommodell - tedesco" lang="de" hreflang="de" data-title="Bohrsches Atommodell" data-language-autonym="Deutsch" data-language-local-name="tedesco" 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%91%CF%84%CE%BF%CE%BC%CE%B9%CE%BA%CF%8C_%CF%80%CF%81%CF%8C%CF%84%CF%85%CF%80%CE%BF_%CF%84%CE%BF%CF%85_%CE%9C%CF%80%CE%BF%CF%81" title="Ατομικό πρότυπο του Μπορ - greco" lang="el" hreflang="el" data-title="Ατομικό πρότυπο του Μπορ" data-language-autonym="Ελληνικά" data-language-local-name="greco" 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/Bohr_model" title="Bohr model - inglese" lang="en" hreflang="en" data-title="Bohr model" data-language-autonym="English" data-language-local-name="inglese" 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/Modelo_at%C3%B3mico_de_Bohr" title="Modelo atómico de Bohr - spagnolo" lang="es" hreflang="es" data-title="Modelo atómico de Bohr" data-language-autonym="Español" data-language-local-name="spagnolo" 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/Bohri_aatomiteooria" title="Bohri aatomiteooria - estone" lang="et" hreflang="et" data-title="Bohri aatomiteooria" data-language-autonym="Eesti" data-language-local-name="estone" 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/Bohrren_eredu_atomikoa" title="Bohrren eredu atomikoa - basco" lang="eu" hreflang="eu" data-title="Bohrren eredu atomikoa" data-language-autonym="Euskara" data-language-local-name="basco" 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%AF%D9%84_%D8%A8%D9%88%D8%B1" title="مدل بور - persiano" lang="fa" hreflang="fa" data-title="مدل بور" data-language-autonym="فارسی" data-language-local-name="persiano" 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/Bohrin_atomimalli" title="Bohrin atomimalli - finlandese" lang="fi" hreflang="fi" data-title="Bohrin atomimalli" data-language-autonym="Suomi" data-language-local-name="finlandese" 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/Mod%C3%A8le_de_Bohr" title="Modèle de Bohr - francese" lang="fr" hreflang="fr" data-title="Modèle de Bohr" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Samhail_Bohr" title="Samhail Bohr - irlandese" lang="ga" hreflang="ga" data-title="Samhail Bohr" data-language-autonym="Gaeilge" data-language-local-name="irlandese" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Modelo_at%C3%B3mico_de_Bohr" title="Modelo atómico de Bohr - galiziano" lang="gl" hreflang="gl" data-title="Modelo atómico de Bohr" data-language-autonym="Galego" data-language-local-name="galiziano" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%AC%E0%AB%89%E0%AA%B9%E0%AA%B0%E0%AA%A8%E0%AB%8B_%E0%AA%B8%E0%AA%BF%E0%AA%A6%E0%AB%8D%E0%AA%A7%E0%AA%BE%E0%AA%82%E0%AA%A4" title="બૉહરનો સિદ્ધાંત - gujarati" lang="gu" hreflang="gu" data-title="બૉહરનો સિદ્ધાંત" data-language-autonym="ગુજરાતી" data-language-local-name="gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%95%D7%93%D7%9C_%D7%94%D7%90%D7%98%D7%95%D7%9D_%D7%A9%D7%9C_%D7%91%D7%95%D7%94%D7%A8" title="מודל האטום של בוהר - ebraico" lang="he" hreflang="he" data-title="מודל האטום של בוהר" data-language-autonym="עברית" data-language-local-name="ebraico" 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%AC%E0%A5%8B%E0%A4%B0_%E0%A4%95%E0%A4%BE_%E0%A4%AA%E0%A4%B0%E0%A4%AE%E0%A4%BE%E0%A4%A3%E0%A5%81_%E0%A4%AE%E0%A5%89%E0%A4%A1%E0%A4%B2" title="बोर का परमाणु मॉडल - hindi" lang="hi" hreflang="hi" data-title="बोर का परमाणु मॉडल" data-language-autonym="हिन्दी" data-language-local-name="hindi" 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/Bohrov_model_atoma" title="Bohrov model atoma - croato" lang="hr" hreflang="hr" data-title="Bohrov model atoma" data-language-autonym="Hrvatski" data-language-local-name="croato" 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/Bohr-f%C3%A9le_atommodell" title="Bohr-féle atommodell - ungherese" lang="hu" hreflang="hu" data-title="Bohr-féle atommodell" data-language-autonym="Magyar" data-language-local-name="ungherese" 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/%D4%B2%D5%B8%D6%80%D5%AB_%D5%B4%D5%B8%D5%A4%D5%A5%D5%AC" title="Բորի մոդել - armeno" lang="hy" hreflang="hy" data-title="Բորի մոդել" data-language-autonym="Հայերեն" data-language-local-name="armeno" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Model_Bohr" title="Model Bohr - indonesiano" lang="id" hreflang="id" data-title="Model Bohr" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiano" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9C%E3%83%BC%E3%82%A2%E3%81%AE%E5%8E%9F%E5%AD%90%E6%A8%A1%E5%9E%8B" title="ボーアの原子模型 - giapponese" lang="ja" hreflang="ja" data-title="ボーアの原子模型" data-language-autonym="日本語" data-language-local-name="giapponese" 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%91%E1%83%9D%E1%83%A0%E1%83%98%E1%83%A1_%E1%83%9B%E1%83%9D%E1%83%93%E1%83%94%E1%83%9A%E1%83%98" title="ბორის მოდელი - georgiano" lang="ka" hreflang="ka" data-title="ბორის მოდელი" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D0%BE%D1%80_%D2%9B%D0%B0%D2%93%D0%B8%D0%B4%D0%B0%D0%BB%D0%B0%D1%80%D1%8B" title="Бор қағидалары - kazako" lang="kk" hreflang="kk" data-title="Бор қағидалары" data-language-autonym="Қазақша" data-language-local-name="kazako" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%B4%EC%96%B4_%EB%AA%A8%ED%98%95" title="보어 모형 - coreano" lang="ko" hreflang="ko" data-title="보어 모형" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Boro_teorija" title="Boro teorija - lituano" lang="lt" hreflang="lt" data-title="Boro teorija" data-language-autonym="Lietuvių" data-language-local-name="lituano" 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/Bora_atoma_uzb%C5%ABves_modelis" title="Bora atoma uzbūves modelis - lettone" lang="lv" hreflang="lv" data-title="Bora atoma uzbūves modelis" data-language-autonym="Latviešu" data-language-local-name="lettone" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D0%BE%D1%80%D0%BE%D0%B2_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB" title="Боров модел - macedone" lang="mk" hreflang="mk" data-title="Боров модел" data-language-autonym="Македонски" data-language-local-name="macedone" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AC%E0%B5%8B%E0%B5%BC_%E0%B4%AE%E0%B4%BE%E0%B4%A4%E0%B5%83%E0%B4%95" title="ബോർ മാതൃക - malayalam" lang="ml" hreflang="ml" data-title="ബോർ മാതൃക" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AC%E0%A5%8B%E0%A4%B0%E0%A4%9A%E0%A5%80_%E0%A4%85%E0%A4%A3%E0%A5%82%E0%A4%9A%E0%A5%80_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%83%E0%A4%A4%E0%A5%80" title="बोरची अणूची प्रतिकृती - marathi" lang="mr" hreflang="mr" data-title="बोरची अणूची प्रतिकृती" data-language-autonym="मराठी" data-language-local-name="marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Model_Bohr" title="Model Bohr - malese" lang="ms" hreflang="ms" data-title="Model Bohr" data-language-autonym="Bahasa Melayu" data-language-local-name="malese" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Atoommodel_van_Bohr" title="Atoommodel van Bohr - olandese" lang="nl" hreflang="nl" data-title="Atoommodel van Bohr" data-language-autonym="Nederlands" data-language-local-name="olandese" 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/Atommodell" title="Atommodell - norvegese nynorsk" lang="nn" hreflang="nn" data-title="Atommodell" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegese 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/Bohrs_atommodell" title="Bohrs atommodell - norvegese bokmål" lang="nb" hreflang="nb" data-title="Bohrs atommodell" data-language-autonym="Norsk bokmål" data-language-local-name="norvegese bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AC%E0%A9%8B%E0%A8%B9%E0%A8%B0_%E0%A8%AE%E0%A8%BE%E0%A8%A1%E0%A8%B2" title="ਬੋਹਰ ਮਾਡਲ - punjabi" lang="pa" hreflang="pa" data-title="ਬੋਹਰ ਮਾਡਲ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punjabi" 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/Model_atomu_Bohra" title="Model atomu Bohra - polacco" lang="pl" hreflang="pl" data-title="Model atomu Bohra" data-language-autonym="Polski" data-language-local-name="polacco" 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%A8%D9%88%DB%81%D8%B1_%D9%85%D8%A7%DA%88%D9%84" title="بوہر ماڈل - Western Punjabi" lang="pnb" hreflang="pnb" data-title="بوہر ماڈل" 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/%C3%81tomo_de_Bohr" title="Átomo de Bohr - portoghese" lang="pt" hreflang="pt" data-title="Átomo de Bohr" data-language-autonym="Português" data-language-local-name="portoghese" 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/Modelul_atomic_Bohr" title="Modelul atomic Bohr - rumeno" lang="ro" hreflang="ro" data-title="Modelul atomic Bohr" data-language-autonym="Română" data-language-local-name="rumeno" 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%91%D0%BE%D1%80%D0%BE%D0%B2%D1%81%D0%BA%D0%B0%D1%8F_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB%D1%8C_%D0%B0%D1%82%D0%BE%D0%BC%D0%B0" title="Боровская модель атома - russo" lang="ru" hreflang="ru" data-title="Боровская модель атома" data-language-autonym="Русский" data-language-local-name="russo" 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/Mud%C3%A8llu_at%C3%B2micu_di_Bohr" title="Mudèllu atòmicu di Bohr - siciliano" lang="scn" hreflang="scn" data-title="Mudèllu atòmicu di Bohr" data-language-autonym="Sicilianu" data-language-local-name="siciliano" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Bohr_model" title="Bohr model - scozzese" lang="sco" hreflang="sco" data-title="Bohr model" data-language-autonym="Scots" data-language-local-name="scozzese" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Bohrov_model_atoma" title="Bohrov model atoma - serbo-croato" lang="sh" hreflang="sh" data-title="Bohrov model atoma" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croato" 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/Bohr_model" title="Bohr model - Simple English" lang="en-simple" hreflang="en-simple" data-title="Bohr model" 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/Bohrov_model_at%C3%B3mu" title="Bohrov model atómu - slovacco" lang="sk" hreflang="sk" data-title="Bohrov model atómu" data-language-autonym="Slovenčina" data-language-local-name="slovacco" 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/Bohrov_model_atoma" title="Bohrov model atoma - sloveno" lang="sl" hreflang="sl" data-title="Bohrov model atoma" data-language-autonym="Slovenščina" data-language-local-name="sloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D0%BE%D1%80%D0%BE%D0%B2_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB_%D0%B0%D1%82%D0%BE%D0%BC%D0%B0" title="Боров модел атома - serbo" lang="sr" hreflang="sr" data-title="Боров модел атома" data-language-autonym="Српски / srpski" data-language-local-name="serbo" 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/Bohrs_atommodell" title="Bohrs atommodell - svedese" lang="sv" hreflang="sv" data-title="Bohrs atommodell" data-language-autonym="Svenska" data-language-local-name="svedese" 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%AA%E0%AF%8B%E0%AE%B0%E0%AF%8D_%E0%AE%85%E0%AE%A3%E0%AF%81_%E0%AE%AE%E0%AE%BE%E0%AE%A4%E0%AE%BF%E0%AE%B0%E0%AE%BF" title="போர் அணு மாதிரி - tamil" lang="ta" hreflang="ta" data-title="போர் அணு மாதிரி" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%AC%E0%B1%8B%E0%B0%B0%E0%B1%8D_%E0%B0%A8%E0%B0%AE%E0%B1%82%E0%B0%A8%E0%B0%BE" title="బోర్ నమూనా - telugu" lang="te" hreflang="te" data-title="బోర్ నమూనా" data-language-autonym="తెలుగు" data-language-local-name="telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%9A%E0%B8%9A%E0%B8%88%E0%B8%B3%E0%B8%A5%E0%B8%AD%E0%B8%87%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B9%82%E0%B8%9B%E0%B8%A3%E0%B9%8C" title="แบบจำลองของโปร์ - thailandese" lang="th" hreflang="th" data-title="แบบจำลองของโปร์" data-language-autonym="ไทย" data-language-local-name="thailandese" 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/Modelong_Bohr" title="Modelong Bohr - tagalog" lang="tl" hreflang="tl" data-title="Modelong Bohr" 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/Bohr_modeli" title="Bohr modeli - turco" lang="tr" hreflang="tr" data-title="Bohr modeli" data-language-autonym="Türkçe" data-language-local-name="turco" 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_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB%D1%8C_%D0%91%D0%BE%D1%80%D0%B0" title="Атомна модель Бора - ucraino" lang="uk" hreflang="uk" data-title="Атомна модель Бора" data-language-autonym="Українська" data-language-local-name="ucraino" 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/Bor_postulatlari" title="Bor postulatlari - uzbeco" lang="uz" hreflang="uz" data-title="Bor postulatlari" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeco" 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/M%C3%B4_h%C3%ACnh_Bohr" title="Mô hình Bohr - vietnamita" lang="vi" hreflang="vi" data-title="Mô hình Bohr" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" 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/%E7%8E%BB%E5%B0%94%E6%A8%A1%E5%9E%8B" title="玻尔模型 - wu" lang="wuu" hreflang="wuu" data-title="玻尔模型" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%8E%BB%E5%B0%94%E6%A8%A1%E5%9E%8B" title="玻尔模型 - cinese" lang="zh" hreflang="zh" data-title="玻尔模型" data-language-autonym="中文" data-language-local-name="cinese" 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/%E7%8E%BB%E7%88%BE%E6%A8%A1%E5%9E%8B" title="玻爾模型 - cinese classico" lang="lzh" hreflang="lzh" data-title="玻爾模型" data-language-autonym="文言" data-language-local-name="cinese classico" 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/%E7%8E%BB%E7%88%BE%E6%A8%A1%E5%9E%8B" title="玻爾模型 - cantonese" lang="yue" hreflang="yue" data-title="玻爾模型" data-language-autonym="粵語" data-language-local-name="cantonese" 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/Q172948#sitelinks-wikipedia" title="Modifica collegamenti interlinguistici" class="wbc-editpage">Modifica collegamenti</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bohr_atom_model_English.svg/langit-330px-Bohr_atom_model_English.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Bohr_atom_model_English.svg/langit-440px-Bohr_atom_model_English.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>Schema del modello atomico di Bohr. Il <a href="/wiki/Nucleo_atomico" title="Nucleo atomico">nucleo atomico</a> centrale contiene <a href="/wiki/Protone" title="Protone">protoni</a> (carichi positivamente) e <a href="/wiki/Neutrone" title="Neutrone">neutroni</a> (elettricamente neutri), mentre gli <a href="/wiki/Elettrone" title="Elettrone">elettroni</a> (carichi negativamente) ruotano intorno ad esso su orbite quantizzate, con livelli di energia crescenti.</figcaption></figure> <p>Il <b>modello atomico di Bohr</b>, proposto dal fisico <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a> nel 1913 e successivamente ampliato da <a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Arnold Sommerfeld</a> nel 1916, fu il primo modello <a href="/wiki/Atomo" title="Atomo">atomico</a> ad utilizzare la <a href="/wiki/Quantizzazione_(fisica)" title="Quantizzazione (fisica)">quantizzazione</a> dell'<a href="/wiki/Energia" title="Energia">energia</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>Nota 1<span class="cite-bracket">]</span></a></sup> Proposto inizialmente solo per l'<a href="/wiki/Atomo" title="Atomo">atomo</a> d'<a href="/wiki/Idrogeno" title="Idrogeno">idrogeno</a>, riusciva a spiegarne le caratteristiche <a href="/wiki/Linea_spettrale" title="Linea spettrale">righe</a> dello <a href="/wiki/Spettro_dell%27atomo_di_idrogeno" title="Spettro dell'atomo di idrogeno">spettro di emissione</a> teorizzando per l'unico elettrone orbite con valori discreti di energia. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Stato_dell'arte_prima_di_Bohr"><span id="Stato_dell.27arte_prima_di_Bohr"></span>Stato dell'arte prima di Bohr</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_atomico_di_Bohr&veaction=edit&section=1" title="Modifica la sezione Stato dell'arte prima di Bohr" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_atomico_di_Bohr&action=edit&section=1" title="Edit section's source code: Stato dell'arte prima di Bohr"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Spettroscopia_atomica">Spettroscopia atomica</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_atomico_di_Bohr&veaction=edit&section=2" title="Modifica la sezione Spettroscopia atomica" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_atomico_di_Bohr&action=edit&section=2" title="Edit section's source code: Spettroscopia atomica"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>All'inizio del <a href="/wiki/XX_secolo" title="XX secolo">XX secolo</a> lo studio della <a href="/wiki/Spettroscopia" title="Spettroscopia">spettroscopia</a> era giunto a un buon livello. Erano noti infatti moltissimi <a href="/wiki/Spettro_di_emissione" title="Spettro di emissione">spettri di emissione</a> provenienti da vari atomi, caratterizzati da <a href="/wiki/Linea_spettrale" title="Linea spettrale">righe luminose discrete</a> in una successione ben precisa. Mancava tuttavia un inquadramento teorico del fenomeno dell'emissione a righe da parte dei vari elementi chimici. Una prima sistematizzazione dei dati avvenne nel 1885 quando <a href="/wiki/Johann_Jakob_Balmer" title="Johann Jakob Balmer">Johann Jakob Balmer</a>, insegnante <a href="/wiki/Svizzera" title="Svizzera">svizzero</a>, fece notare come le lunghezze d'onda nella <a href="/wiki/Spettro_visibile" title="Spettro visibile">parte visibile all'occhio umano</a> (intervallo compreso fra 380 e 750 nm)<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> dello spettro dell'<a href="/wiki/Idrogeno" title="Idrogeno">idrogeno</a> potevano essere rappresentate con grande precisione da una formula empirica che le correlava a dei numeri interi: </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Spektrum.Balmer-Serie.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Spektrum.Balmer-Serie.png/310px-Spektrum.Balmer-Serie.png" decoding="async" width="310" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Spektrum.Balmer-Serie.png/465px-Spektrum.Balmer-Serie.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Spektrum.Balmer-Serie.png/620px-Spektrum.Balmer-Serie.png 2x" data-file-width="1600" data-file-height="1024" /></a><figcaption>Le prime cinque righe della serie di Balmer.</figcaption></figure> <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 \ =B\,{\frac {m^{2}}{m^{2}-n^{2}}}=B\,{\frac {m^{2}}{m^{2}-4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mtext> </mtext> <mo>=</mo> <mi>B</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>B</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \ =B\,{\frac {m^{2}}{m^{2}-n^{2}}}=B\,{\frac {m^{2}}{m^{2}-4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef4fad0c1d1010ef5e6913d498eefe7d26c4aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.589ex; height:6.176ex;" alt="{\displaystyle \lambda \ =B\,{\frac {m^{2}}{m^{2}-n^{2}}}=B\,{\frac {m^{2}}{m^{2}-4}}}"></span></dd></dl> <p>dove </p> <ul><li>λ lunghezza d'onda della luce emessa</li> <li><i>B</i> limite di Balmer, pari a 3,6456 × 10<sup>-7</sup> m o 364,56 nm o 3645,6 Å</li> <li><i>n</i> = 2</li> <li><i>m</i> intero con <i>m</i> > <i>n</i></li></ul> <p>L'insieme delle <a href="/wiki/Linea_spettrale" title="Linea spettrale">righe spettroscopiche</a> ottenibili con la formula di Balmer prende il nome di <a href="/wiki/Serie_di_Balmer" title="Serie di Balmer">serie di Balmer</a>. Sostituendo <i>m</i> = 3 nella formula si ottiene la lunghezza d'onda della riga rossa (λ = 656 nm), per <i>m</i> = 4 della riga celeste (λ = 486 nm), per <i>m</i> = 5 della riga blu (λ = 434 nm) e per <i>m</i> = 6 della riga violetta (λ = 410 nm). Basandosi sulla sua formula, Balmer predisse per <i>m</i> = 7 l'esistenza di un'altra riga spettroscopica (λ = 397 nm). Venne poi a sapere che <a href="/wiki/Anders_Jonas_%C3%85ngstr%C3%B6m" title="Anders Jonas Ångström">Ångström</a> aveva in effetti da poco osservato tale riga. </p><p>Con le sostituzioni successive (<i>m</i> > 8) si ottengono lunghezze d'onda proprie dei <a href="/wiki/Raggi_UV" class="mw-redirect" title="Raggi UV">raggi UV</a>, non osservarvabili ad <a href="/wiki/Occhio_nudo" title="Occhio nudo">occhio nudo</a>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Spectral_lines_of_the_hydrogen_atom.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Spectral_lines_of_the_hydrogen_atom.svg/310px-Spectral_lines_of_the_hydrogen_atom.svg.png" decoding="async" width="310" height="355" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Spectral_lines_of_the_hydrogen_atom.svg/465px-Spectral_lines_of_the_hydrogen_atom.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Spectral_lines_of_the_hydrogen_atom.svg/620px-Spectral_lines_of_the_hydrogen_atom.svg.png 2x" data-file-width="911" data-file-height="1043" /></a><figcaption>Serie delle righe spettrali dell'atomo d'idrogeno.</figcaption></figure> <p>Nel <a href="/wiki/1889" title="1889">1889</a> il fisico <a href="/wiki/Svezia" title="Svezia">svedese</a> <a href="/wiki/Johannes_Rydberg" title="Johannes Rydberg">Johannes Rydberg</a> generalizzò, con la <a href="/wiki/Equazione_di_Rydberg" class="mw-redirect" title="Equazione di Rydberg">formula di Rydberg</a>, quella di Balmer per tutte le transizioni dell'idrogeno (non solo la <a href="/wiki/Serie_di_Balmer" title="Serie di Balmer">serie di Balmer</a> L (<i>n</i> = 2), parzialmente nello <a href="/wiki/Spettro_visibile" title="Spettro visibile">spettro visibile</a>, ma anche la <a href="/wiki/Serie_di_Lyman" title="Serie di Lyman">serie di Lyman</a> K (<i>n</i> = 1) nell'ultravioletto e quelle di <a href="/wiki/Serie_di_Paschen" title="Serie di Paschen">Paschen</a> M (<i>n</i> = 3), <a href="/wiki/Serie_di_Brackett" title="Serie di Brackett">Brackett</a> N (<i>n</i> = 4), <a href="/wiki/Serie_di_Pfund" title="Serie di Pfund">Pfund</a> O (<i>n</i> = 5) e <a href="/wiki/Serie_di_Humphreys" title="Serie di Humphreys">Humphreys</a> P (<i>n</i> = 6) nell'infrarosso): </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}{\lambda }}={\frac {4}{B}}\left({\frac {1}{n^{2}}}-{\frac {1}{m^{2}}}\right)=R_{\mathrm {H} }\left({\frac {1}{n^{2}}}-{\frac {1}{m^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mi>B</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\lambda }}={\frac {4}{B}}\left({\frac {1}{n^{2}}}-{\frac {1}{m^{2}}}\right)=R_{\mathrm {H} }\left({\frac {1}{n^{2}}}-{\frac {1}{m^{2}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae30486ae6ad438902b932c69efbf555424daeb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.946ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\lambda }}={\frac {4}{B}}\left({\frac {1}{n^{2}}}-{\frac {1}{m^{2}}}\right)=R_{\mathrm {H} }\left({\frac {1}{n^{2}}}-{\frac {1}{m^{2}}}\right)}"></span></dd></dl> <p>con </p> <ul><li>λ lunghezza d'onda della radiazione emessa</li> <li><i>R</i><sub>H</sub> = 4/<i>B</i> <a href="/wiki/Costante_di_Rydberg" title="Costante di Rydberg">costante di Rydberg</a> dell'idrogeno, pari a circa (1,097 x 10<sup>7</sup>) m<sup>-1</sup></li> <li><i>n</i> ed <i>m</i> numeri interi positivi con <i>m</i> > <i>n</i></li></ul> <p>I due termini, la cui differenza dà una riga spettrale, rappresentano i livelli energetici atomici della transizione. </p><p>Per <i>n</i> = 2 si ritrova la serie di Balmer: </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}{\lambda }}={\frac {4}{B}}\left({\frac {1}{4}}-{\frac {1}{m^{2}}}\right)=R_{\mathrm {H} }\left({\frac {1}{4}}-{\frac {1}{m^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mi>B</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\lambda }}={\frac {4}{B}}\left({\frac {1}{4}}-{\frac {1}{m^{2}}}\right)=R_{\mathrm {H} }\left({\frac {1}{4}}-{\frac {1}{m^{2}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e03c8a39ffff4c01a17dd958a30a8d14c382bae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.373ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\lambda }}={\frac {4}{B}}\left({\frac {1}{4}}-{\frac {1}{m^{2}}}\right)=R_{\mathrm {H} }\left({\frac {1}{4}}-{\frac {1}{m^{2}}}\right)}"></span></dd></dl> <p>con <i>m</i> = 3, 4, 5, 6, 7... </p><p>Per i numeri quantici <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/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> fisso ed <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> variabile si trovano le diverse serie spettroscopiche dell'<a href="/wiki/Idrogeno" title="Idrogeno">idrogeno</a>: </p> <table class="wikitable" style="text-align: center"> <caption>Serie spettroscopiche dell'idrogeno </caption> <tbody><tr> <th>Orbita</th> <th>Numero quantico <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/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></th> <th>Numero quantico <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></th> <th>Serie spettroscopica</th> <th>Lunghezza d'onda minima</th> <th>Lunghezza d'onda massima </th></tr> <tr> <td></td> <td></td> <td></td> <td></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 \lambda _{min}=n^{2}/R_{H}=n^{2}\,91}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mn>91</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{min}=n^{2}/R_{H}=n^{2}\,91}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1681e0bbef4e77958ec0ec11cd42b4ec817d7d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.009ex; height:3.176ex;" alt="{\displaystyle \lambda _{min}=n^{2}/R_{H}=n^{2}\,91}"></span> nm</td> <td>Riga <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m-n=1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>m</mi> <mo>−</mo> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (m-n=1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02bf5ea22ce751e6bc6f95749ce396e54262ad38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.346ex; height:2.843ex;" alt="{\displaystyle (m-n=1)}"></span> </td></tr> <tr> <td>K</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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></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 2\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbcdf702257a9ab3c4fccf4e13c23943cc0ba617" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.1ex; height:2.176ex;" alt="{\displaystyle 2\rightarrow \infty }"></span></td> <td><a href="/wiki/Serie_di_Lyman" title="Serie di Lyman">Lyman</a></td> <td>91 nm</td> <td>121 nm </td></tr> <tr> <td>L</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 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span></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 3\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a23a52ef75f7e22fbc2458b9a8555de5a4e7bfc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.1ex; height:2.176ex;" alt="{\displaystyle 3\rightarrow \infty }"></span></td> <td><a href="/wiki/Serie_di_Balmer" title="Serie di Balmer">Balmer</a></td> <td>365 nm</td> <td>656 nm </td></tr> <tr> <td>M</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 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}"></span></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 4\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/272c0cf1570abd21d8601fd6bf9a980f77362338" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.1ex; height:2.176ex;" alt="{\displaystyle 4\rightarrow \infty }"></span></td> <td><a href="/wiki/Serie_di_Paschen" title="Serie di Paschen">Paschen</a></td> <td>820 nm</td> <td>1.874 nm </td></tr> <tr> <td>N</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 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295b4bf1de7cd3500e740e0f4f0635db22d87b42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 4}"></span></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 5\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c9292c1eb03a5e18411a2dbd7c79ecaf3b8083a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.1ex; height:2.176ex;" alt="{\displaystyle 5\rightarrow \infty }"></span></td> <td><a href="/wiki/Serie_di_Brackett" title="Serie di Brackett">Brackett</a></td> <td>1.458 nm</td> <td>4.051 nm </td></tr> <tr> <td>O</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 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}"></span></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 6\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eabf357e3e47a591f9047a9b14aa89c637c18de8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.1ex; height:2.176ex;" alt="{\displaystyle 6\rightarrow \infty }"></span></td> <td><a href="/wiki/Serie_di_Pfund" title="Serie di Pfund">Pfund</a></td> <td>2.278 nm</td> <td>7.456 nm </td></tr> <tr> <td>P</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 6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39d81124420a058a7474dfeda48228fb6ee1e253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 6}"></span></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 7\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7</mn> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0b37952d1ba6632657f0d705512e46c62a55c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.1ex; height:2.176ex;" alt="{\displaystyle 7\rightarrow \infty }"></span></td> <td><a href="/wiki/Serie_di_Humphreys" title="Serie di Humphreys">Humphreys</a></td> <td>3.281 nm</td> <td>12.365 nm </td></tr></tbody></table> <p>La <a href="/wiki/Equazione_di_Rydberg" class="mw-redirect" title="Equazione di Rydberg">formula di Rydberg</a> mostra che la <a href="/wiki/Lunghezza_d%27onda" title="Lunghezza d'onda">lunghezza d'onda</a>, la <a href="/wiki/Frequenza" title="Frequenza">frequenza</a> e quindi anche l'<a href="/wiki/Energia" title="Energia">energia</a> della <a href="/wiki/Radiazione" title="Radiazione">radiazione</a> elettromagnetica emessa dall'idrogeno sono <a href="/wiki/Quantizzazione_(fisica)" title="Quantizzazione (fisica)">quantizzate</a>, cioè non continue. La ragione della quantizzazione delle <a href="/wiki/Linea_spettrale" title="Linea spettrale">righe spettrali</a> e il motivo per cui tale formula empirica riproduce con grande accuratezza i <a href="/wiki/Numero_d%27onda" title="Numero d'onda">numeri d'onda</a> delle <a href="/wiki/Spettro_dell%27atomo_di_idrogeno" title="Spettro dell'atomo di idrogeno">serie spettroscopiche dell'idrogeno</a> non verranno compresi fino al <a href="/wiki/1913" title="1913">1913</a>, anno in cui <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a> pubblicherà il suo modello atomico quantizzato. </p> <div class="mw-heading mw-heading3"><h3 id="Primi_modelli_atomici">Primi modelli atomici</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_atomico_di_Bohr&veaction=edit&section=3" title="Modifica la sezione Primi modelli atomici" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_atomico_di_Bohr&action=edit&section=3" title="Edit section's source code: Primi modelli atomici"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La scoperta nel 1897 dell'<a href="/wiki/Elettrone" title="Elettrone">elettrone</a>, prima <a href="/wiki/Particella_(fisica)" title="Particella (fisica)">particella</a> subatomica isolata in laboratorio, da parte di <a href="/wiki/Joseph_John_Thomson" title="Joseph John Thomson">J. J. Thomson</a> segnò la fine dell'antico concetto di <a href="/wiki/Atomo" title="Atomo">atomo</a> (dal <a href="/wiki/Lingua_greca_antica" title="Lingua greca antica">greco</a> ἄτομος <i>átomos</i>: <i>indivisibile</i>) come struttura indivisibile e l'inizio di un periodo di ricerca di nuovi modelli, che includessero gli elettroni come componenti della struttura atomica. Prima di Bohr si era tentato di spiegare la struttura dell'atomo con modelli proposti da </p> <ul><li><a href="/wiki/Jean_Baptiste_Perrin" title="Jean Baptiste Perrin">Perrin</a> (1901);</li> <li><a href="/wiki/Modello_atomico_di_Thomson" title="Modello atomico di Thomson">Thomson</a> (1904);</li> <li><a href="/wiki/Modello_atomico_di_Nagaoka" title="Modello atomico di Nagaoka">Nagaoka</a> (1904);</li> <li><a href="/wiki/Modello_atomico_di_Rutherford" title="Modello atomico di Rutherford">Rutherford</a> (1911).</li></ul> <p>Nel 1901 <a href="/wiki/Jean_Baptiste_Perrin" title="Jean Baptiste Perrin">Perrin</a> aveva ipotizzato che l'atomo fosse un sistema solare in miniatura, in cui gli elettroni si muovono come pianeti attorno a uno o più nuclei carichi positivamente, nei quali è concentrata la quasi totalità della <a href="/wiki/Massa_(fisica)" title="Massa (fisica)">massa</a> atomica. <a href="/wiki/Joseph_John_Thomson" title="Joseph John Thomson">Joseph John Thomson</a>, lo scopritore dell'elettrone, immaginò l'atomo come un corpo compatto (<i>plumcake</i> o panettone) con carica positiva diffusa, contenente al suo interno gli elettroni (canditi o uvette) con carica negativa. Questo modello si basava solo sulla presenza di forze elettrostatiche e non era in grado di spiegare come il sistema rimanesse in uno stato d'equilibrio.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Nagaoka_Hantar%C5%8D" title="Nagaoka Hantarō">Nagaoka</a> suppose invece che la carica positiva fosse concentrata in un'unica, grande e massiccia sfera centrale, il <a href="/wiki/Nucleo_atomico" title="Nucleo atomico">nucleo</a>, circondata dagli elettroni disposti su anelli simili alle fasce del pianeta <a href="/wiki/Saturno_(astronomia)" title="Saturno (astronomia)">Saturno</a> (da cui il nome di <i>modello saturniano</i>). La stabilità dell'atomo risultava anche in questo caso un problema irrisolto, che portò presto all'abbandono del modello saturniano. </p> <style data-mw-deduplicate="TemplateStyles:r130657691">body:not(.skin-minerva) .mw-parser-output .vedi-anche{font-size:95%}</style><style data-mw-deduplicate="TemplateStyles:r139142988">.mw-parser-output .hatnote-content{align-items:center;display:flex}.mw-parser-output .hatnote-icon{flex-shrink:0}.mw-parser-output .hatnote-icon img{display:flex}.mw-parser-output .hatnote-text{font-style:italic}body:not(.skin-minerva) .mw-parser-output .hatnote{border:1px solid #CCC;display:flex;margin:.5em 0;padding:.2em .5em}body:not(.skin-minerva) .mw-parser-output .hatnote-text{padding-left:.5em}body.skin-minerva .mw-parser-output .hatnote-icon{padding-right:8px}body.skin-minerva .mw-parser-output .hatnote-icon img{height:auto;width:16px}body.skin--responsive .mw-parser-output .hatnote a.new{color:#d73333}body.skin--responsive .mw-parser-output .hatnote a.new:visited{color:#a55858}</style> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Esperimento_di_Rutherford" title="Esperimento di Rutherford">Esperimento di Rutherford</a></b> e <b><a href="/wiki/Scattering_Rutherford" title="Scattering Rutherford">Scattering Rutherford</a></b>.</span></div> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Scatteringrutherford.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Scatteringrutherford.jpg/310px-Scatteringrutherford.jpg" decoding="async" width="310" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Scatteringrutherford.jpg/465px-Scatteringrutherford.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/1/1a/Scatteringrutherford.jpg 2x" data-file-width="514" data-file-height="329" /></a><figcaption><a href="/wiki/Esperimento_di_Rutherford" title="Esperimento di Rutherford">Esperimento di Geiger e Marsdsen.</a></figcaption></figure> <p>Tra il 1908 e il 1913 <a href="/wiki/Hans_Wilhelm_Geiger" title="Hans Wilhelm Geiger">Hans Wilhelm Geiger</a> e <a href="/wiki/Ernest_Marsden" title="Ernest Marsden">Ernest Marsden</a>, sotto la supervisione di <a href="/wiki/Ernest_Rutherford" title="Ernest Rutherford">Ernest Rutherford</a>, realizzarono <a href="/wiki/Esperimento_di_Rutherford" title="Esperimento di Rutherford">esperimenti</a> importantissimi per la comprensione della struttura dell'atomo: bombardando una sottile lamina d'<a href="/wiki/Oro" title="Oro">oro</a> con <a href="/wiki/Particella_%CE%B1" title="Particella α">particelle alfa</a> notarono che, mentre la maggior parte di esse subiva deviazioni nulle o minime dalla traiettoria iniziale, una minima parte veniva deviata in misura considerevole o respinta dalla lamina.<sup id="cite_ref-a_4-0" class="reference"><a href="#cite_note-a-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Nell'interpretare i dati sperimentali, Rutherford confermò nel 1911 l'esistenza di un nucleo atomico, anche se estremamente piccolo rispetto all'atomo stesso, circondato dalle cariche negative degli elettroni.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Nonostante Rutherford non si fosse sbilanciato sull'eventuale moto degli elettroni, si prese a rappresentarli in orbita attorno al nucleo. Il <a href="/wiki/Modello_atomico_di_Rutherford" title="Modello atomico di Rutherford">modello atomico planetario</a> così concepito soffriva però d'instabilità di natura elettromagnetica. L'elettrone infatti, nel suo moto accelerato intorno al nucleo, avrebbe dovuto irradiare <a href="/wiki/Energia_del_campo_elettromagnetico" title="Energia del campo elettromagnetico">energia elettromagnetica</a> fino a ricadere sul nucleo stesso con un moto a <a href="/wiki/Spirale" title="Spirale">spirale</a>. Inoltre, a prescindere dall'irraggiamento, nel caso di atomi più pesanti con tanti elettroni in orbita, una qualsiasi perturbazione esterna sarebbe stata sufficiente ad alterare pesantemente la distribuzione degli stessi, resi instabili dalla forza elettrostatica repulsiva.<sup id="cite_ref-a_4-1" class="reference"><a href="#cite_note-a-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Fu <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a> a risolvere, nel 1913, le difficoltà del modello di Rutherford, spiegando anche la struttura dello <a href="/wiki/Spettro_atomico" title="Spettro atomico">spettro atomico</a> dell'<a href="/wiki/Idrogeno" title="Idrogeno">idrogeno</a>. </p> <div class="mw-heading mw-heading2"><h2 id="I_tre_postulati_di_Bohr">I tre postulati di Bohr</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_atomico_di_Bohr&veaction=edit&section=4" title="Modifica la sezione I tre postulati di Bohr" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_atomico_di_Bohr&action=edit&section=4" title="Edit section's source code: I tre postulati di Bohr"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bohr, che aveva lavorato con <a href="/wiki/Ernest_Rutherford" title="Ernest Rutherford">Rutherford</a> a <a href="/wiki/Manchester" title="Manchester">Manchester</a> per cinque mesi nel 1912, propose nel 1913 un modello che, applicando al <a href="/wiki/Modello_atomico_di_Rutherford" title="Modello atomico di Rutherford">modello atomico planetario</a> di Rutherford la quantizzazione dell'energia introdotta da <a href="/wiki/Max_Planck" title="Max Planck">Planck</a> nel 1900, riusciva a giustificare lo <a href="/wiki/Spettro_atomico" title="Spettro atomico">spettro atomico</a> dell'<a href="/wiki/Idrogeno" title="Idrogeno">idrogeno</a>. </p><p>Il modello atomico di Bohr era fondato su tre <a href="/wiki/Postulato" title="Postulato">postulati</a>, due dei quali non classici e imposti <i>ad hoc</i> come vincoli addizionali al modello planetario classico di Rutherford:<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <ol><li>Gli elettroni si muovono attorno al nucleo su orbite <a href="/wiki/Ellisse" title="Ellisse">ellittiche</a> stazionarie, con energia definita e costante;</li> <li>Gli elettroni che si muovono su tali orbite non <a href="/wiki/Irraggiamento" title="Irraggiamento">irraggiano</a> e quindi non emettono energia in modo continuo (contrariamente a quanto previsto dall'<a href="/wiki/Interazione_elettromagnetica" title="Interazione elettromagnetica">elettromagnetismo</a> classico);</li> <li>Un elettrone atomico <a href="/wiki/Irraggiamento" title="Irraggiamento">irraggia</a> energia soltanto quando passa (mediante un <i>salto quantico discontinuo</i>) da un'orbita stazionaria a un'altra. Deve passare ad un'orbita stazionaria più esterna nel caso di <a href="/wiki/Assorbimento_(ottica)" title="Assorbimento (ottica)">assorbimento</a> di <a href="/wiki/Interazione_elettromagnetica" title="Interazione elettromagnetica">radiazione elettromagnetica</a>, mentre salta ad una più interna nel caso di <a href="/wiki/Emissione_spontanea" title="Emissione spontanea">emissione</a>.</li></ol> <p>Nell'ipotesi semplificativa che l'elettrone dell'atomo d'idrogeno si muova su un'orbita circolare (un caso particolare di orbita ellittica, con <a href="/wiki/Eccentricit%C3%A0_(matematica)" title="Eccentricità (matematica)">eccentricità</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9850169d70a5ab7df71c2126441a86cec93eec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle e=0}"></span>), il secondo postulato si traduce in una condizione di quantizzazione per il modulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> del <a href="/wiki/Momento_angolare" title="Momento angolare">momento angolare</a> dell'elettrone che ruota intorno al nucleo, che deve essere un multiplo intero della <a href="/wiki/Costante_di_Planck_ridotta" class="mw-redirect" title="Costante di Planck ridotta">costante di Planck ridotta</a>: </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 L_{n}=m\,v\,r=n\,\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mspace width="thinmathspace" /> <mi>v</mi> <mspace width="thinmathspace" /> <mi>r</mi> <mo>=</mo> <mi>n</mi> <mspace width="thinmathspace" /> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{n}=m\,v\,r=n\,\hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83eaed4e791a9859ccfcee2c3198ec8392bca260" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.077ex; height:2.509ex;" alt="{\displaystyle L_{n}=m\,v\,r=n\,\hbar }"></span></dd></dl> <p>dove <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> e <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> sono la massa e la velocità dell'elettrone orbitale, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> il raggio dell'orbita, <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> la costante di Planck ridotta ed <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/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> il <a href="/wiki/Numero_quantico_principale" title="Numero quantico principale">numero quantico principale</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 (n=1,2,3...).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3...</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n=1,2,3...).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d2ecd29e6405ff02df0402be8573d4641737158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.445ex; height:2.843ex;" alt="{\displaystyle (n=1,2,3...).}"></span> Il valore del momento angolare orbitale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{1}=\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{1}=\hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1eddcb937a658119385996308d79a4ab45d4c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.042ex; height:2.509ex;" alt="{\displaystyle L_{1}=\hbar }"></span> previsto per la prima orbita <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"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> </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/863277921bb6a619f684b3dee34605e3dc132fcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.465ex; height:2.843ex;" alt="{\displaystyle (n=1)}"></span> risulterà sbagliato. Il valore sperimentalmente accertato è invece <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{1}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{1}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5c0370fb28e15d1921cb0c7c0f072793da1c1fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.898ex; height:2.509ex;" alt="{\displaystyle L_{1}=0}"></span>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Electrons-standing-waves-orbits.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Electrons-standing-waves-orbits.svg/260px-Electrons-standing-waves-orbits.svg.png" decoding="async" width="260" height="248" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Electrons-standing-waves-orbits.svg/390px-Electrons-standing-waves-orbits.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Electrons-standing-waves-orbits.svg/520px-Electrons-standing-waves-orbits.svg.png 2x" data-file-width="335" data-file-height="319" /></a><figcaption>L'onda elettronica in viola è stazionaria e genera quindi un'orbita atomica stazionaria. Quella in verde non è stazionaria; ad essa non può corrispondere un'orbita stazionaria.</figcaption></figure> <p>La condizione di quantizzazione di Bohr sarà ricavata in modo semplice nel 1924 da <a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">Louis de Broglie</a>, a partire dall'<a href="/wiki/Ipotesi_di_de_Broglie" title="Ipotesi di de Broglie">ipotesi</a> del <a href="/wiki/Dualismo_onda-particella" title="Dualismo onda-particella">dualismo onda-particella</a>. Secondo de Broglie, sono stazionarie le orbite circolari (con <a href="/wiki/Circonferenza" title="Circonferenza">circonferenza</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 2\pi r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71e811131a9c6c5f45e6657e0fc506e7e2a37f06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.543ex; height:2.176ex;" alt="{\displaystyle 2\pi r}"></span>) che corrispondono a multipli interi della lunghezza d'onda <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> dell'elettrone: </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 2\pi \,r=n\lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mspace width="thinmathspace" /> <mi>r</mi> <mo>=</mo> <mi>n</mi> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi \,r=n\lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47ff58dfcb8e1e204875fa8c4a55af2c65459bb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.779ex; height:2.176ex;" alt="{\displaystyle 2\pi \,r=n\lambda }"></span></dd></dl> <p>Questa condizione significa che ad <a href="/wiki/Onda_stazionaria" title="Onda stazionaria">onde elettroniche stazionarie</a> corrispondono orbite atomiche circolari e stazionarie. </p><p>Siccome la lunghezza d'onda di de Broglie associata all'elettrone è </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}{mv}}}"> <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>v</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {h}{mv}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94df6d9afabe2a7d8f7d95e00d508f0f80eab2ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.458ex; height:5.343ex;" alt="{\displaystyle \lambda ={\frac {h}{mv}}}"></span></dd></dl> <p>per sostituzione nella formula precedente si ottiene </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 2\pi \,r=n\,\lambda =n\,{\frac {h}{mv}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mspace width="thinmathspace" /> <mi>r</mi> <mo>=</mo> <mi>n</mi> <mspace width="thinmathspace" /> <mi>λ</mi> <mo>=</mo> <mi>n</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mi>m</mi> <mi>v</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi \,r=n\,\lambda =n\,{\frac {h}{mv}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddad50f36858b6cdd131df7ea9bd4be10a561677" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.05ex; height:5.343ex;" alt="{\displaystyle 2\pi \,r=n\,\lambda =n\,{\frac {h}{mv}}}"></span></dd></dl> <p>da cui </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 m\,v\,r=n\,{\frac {h}{2\pi }}=n\,\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mspace width="thinmathspace" /> <mi>v</mi> <mspace width="thinmathspace" /> <mi>r</mi> <mo>=</mo> <mi>n</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>n</mi> <mspace width="thinmathspace" /> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\,v\,r=n\,{\frac {h}{2\pi }}=n\,\hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8381410d888a35d08ee1bb4d58c06a3673ff461b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.388ex; height:5.343ex;" alt="{\displaystyle m\,v\,r=n\,{\frac {h}{2\pi }}=n\,\hbar }"></span></dd></dl> <p>che coincide con la condizione di quantizzazione del modulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> del momento angolare dell'elettrone. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Condizione_della_frequenza_di_Bohr" title="Condizione della frequenza di Bohr">Condizione della frequenza di Bohr</a></b>.</span></div> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Bohr_atom_model.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Bohr_atom_model.svg/260px-Bohr_atom_model.svg.png" decoding="async" width="260" height="226" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Bohr_atom_model.svg/390px-Bohr_atom_model.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Bohr_atom_model.svg/520px-Bohr_atom_model.svg.png 2x" data-file-width="310" data-file-height="270" /></a><figcaption>Transizione di un elettrone tra due orbite atomiche, con emissione di radiazione.</figcaption></figure> <p>Per il terzo postulato, l'elettrone effettua una transizione da un'orbita stazionaria a un'altra solamente quando avviene un assorbimento o un'emissione di radiazione elettromagnetica, ovvero d'energia.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>Nota 2<span class="cite-bracket">]</span></a></sup> La frequenza <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 \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> della radiazione è legata all'energia dei livelli atomici <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> ed <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/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, ovvero all'energia del livello di partenza e di quello di arrivo dell'elettrone, dalla relazione: </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 \nu ={\frac {E_{m}-E_{n}}{h}}={\frac {|E_{f}-E_{i}|}{h}}\;\;\;\;\;\;\;\;(m>n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mi>h</mi> </mfrac> </mrow> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>m</mi> <mo>></mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu ={\frac {E_{m}-E_{n}}{h}}={\frac {|E_{f}-E_{i}|}{h}}\;\;\;\;\;\;\;\;(m>n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d15cfebae731ec0163e10f5ddc1a2f0ff9d878e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:41.27ex; height:5.843ex;" alt="{\displaystyle \nu ={\frac {E_{m}-E_{n}}{h}}={\frac {|E_{f}-E_{i}|}{h}}\;\;\;\;\;\;\;\;(m>n)}"></span></dd></dl> <p>dove <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_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12ce471791c1904f7401133fb39c86f5a088f205" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.852ex; height:2.843ex;" alt="{\displaystyle E_{f}}"></span> ed <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}"></span> sono le energie connesse alle orbite finale ed iniziale. È presente il modulo in quanto la frequenza è sempre un numero positivo, mentre <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 E=(E_{f}-E_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>E</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta E=(E_{f}-E_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52e7be612798557ea183320085b151af6e0cdcc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.826ex; height:3.009ex;" alt="{\displaystyle \Delta E=(E_{f}-E_{i})}"></span> può essere negativo, se l'elettrone emette radiazione di frequenza <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 \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span>. L'energia che l'atomo scambia con il campo elettromagnetico soddisfa sia il principio della <a href="/wiki/Conservazione_dell%27energia" class="mw-redirect" title="Conservazione dell'energia">conservazione dell'energia</a>, sia la relazione tra energia e frequenza introdotta da Planck. </p> <div class="mw-heading mw-heading2"><h2 id="Quantizzazione_delle_orbite_e_raggio_di_Bohr">Quantizzazione delle orbite e raggio di Bohr</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_atomico_di_Bohr&veaction=edit&section=5" title="Modifica la sezione Quantizzazione delle orbite e raggio di Bohr" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_atomico_di_Bohr&action=edit&section=5" title="Edit section's source code: Quantizzazione delle orbite e raggio di Bohr"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Raggio_di_Bohr" title="Raggio di Bohr">Raggio di Bohr</a></b>.</span></div> </div> <p>Nel modello di Bohr l'<a href="/wiki/Energia_potenziale" title="Energia potenziale">energia potenziale</a> di un elettrone a <a href="/wiki/Distanza_(matematica)" title="Distanza (matematica)">distanza</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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> dal nucleo è di tipo elettrostatico: </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 U=-k{\frac {Ze^{2}}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=-k{\frac {Ze^{2}}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e5d7e8bf98e3a79889e2ff192747ae65564adb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.555ex; height:5.676ex;" alt="{\displaystyle U=-k{\frac {Ze^{2}}{r}}}"></span></dd></dl> <p>e la <a href="/wiki/Forza_di_Coulomb" title="Forza di Coulomb">forza attrattiva coulombiana</a> è di conseguenza </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 F_{e}=-k{\frac {Ze^{2}}{r^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>k</mi> <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> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{e}=-k{\frac {Ze^{2}}{r^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9519a907eaf9c5e201e36815ccb7cee216464f83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.265ex; height:6.009ex;" alt="{\displaystyle F_{e}=-k{\frac {Ze^{2}}{r^{2}}}}"></span></dd></dl> <p>dove <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> è la <a href="/wiki/Forza_di_Coulomb" title="Forza di Coulomb">costante di Coulomb</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 Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span> il <a href="/wiki/Numero_atomico" title="Numero atomico">numero atomico</a> del nucleo ed <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/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> la carica dell'elettrone. </p><p>Uguagliando tale forza con quella <a href="/wiki/Forza_centripeta" title="Forza centripeta">centripeta</a>: </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 F_{c}=-m{\frac {v^{2}}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{c}=-m{\frac {v^{2}}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0cc3f54c6cba920fc66bc2ec113011904906026" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.404ex; height:5.676ex;" alt="{\displaystyle F_{c}=-m{\frac {v^{2}}{r}}}"></span></dd></dl> <p>si ricava la velocità dell'elettrone </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 v={\sqrt {\frac {kZe^{2}}{mr}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>k</mi> <mi>Z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>m</mi> <mi>r</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\sqrt {\frac {kZe^{2}}{mr}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7097d55a72a17e09781019efcc3d5d3a05ed078" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.415ex; height:6.176ex;" alt="{\displaystyle v={\sqrt {\frac {kZe^{2}}{mr}}}}"></span></dd></dl> <p>Sostituendo la velocità nella formula della quantizzazione del momento angolare </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 L_{n}=m\,v\,r=n\,\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mspace width="thinmathspace" /> <mi>v</mi> <mspace width="thinmathspace" /> <mi>r</mi> <mo>=</mo> <mi>n</mi> <mspace width="thinmathspace" /> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{n}=m\,v\,r=n\,\hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83eaed4e791a9859ccfcee2c3198ec8392bca260" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.077ex; height:2.509ex;" alt="{\displaystyle L_{n}=m\,v\,r=n\,\hbar }"></span></dd></dl> <p>si ottengono i raggi quantizzati delle orbite permesse dal modello atomico di Bohr:<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </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 r_{n}={\frac {n^{2}}{Z}}{\frac {\hbar ^{2}}{kme^{2}}}={\frac {n^{2}}{Z}}{\frac {\varepsilon _{0}h^{2}}{\pi me^{2}}}={\frac {n^{2}}{Z}}a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>Z</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>k</mi> <mi>m</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>Z</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>π</mi> <mi>m</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>Z</mi> </mfrac> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}={\frac {n^{2}}{Z}}{\frac {\hbar ^{2}}{kme^{2}}}={\frac {n^{2}}{Z}}{\frac {\varepsilon _{0}h^{2}}{\pi me^{2}}}={\frac {n^{2}}{Z}}a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25329dc0dce1ab9ca0e75433b938d8a6c51a6388" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:36.274ex; height:6.009ex;" alt="{\displaystyle r_{n}={\frac {n^{2}}{Z}}{\frac {\hbar ^{2}}{kme^{2}}}={\frac {n^{2}}{Z}}{\frac {\varepsilon _{0}h^{2}}{\pi me^{2}}}={\frac {n^{2}}{Z}}a_{0}}"></span></dd></dl> <p>ove <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 \varepsilon _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acb0a8377db20e42274444cb181d51b5532b5844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{0}}"></span> è la <a href="/wiki/Permittivit%C3%A0_elettrica_del_vuoto" class="mw-redirect" title="Permittività elettrica del vuoto">permittività elettrica del vuoto</a> e </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 a_{0}={\frac {\varepsilon _{0}h^{2}}{\pi me^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>π</mi> <mi>m</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}={\frac {\varepsilon _{0}h^{2}}{\pi me^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d27cafc02f821ed0311a90fbeb16a2c210d80ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.729ex; height:6.009ex;" alt="{\displaystyle a_{0}={\frac {\varepsilon _{0}h^{2}}{\pi me^{2}}}}"></span></dd></dl> <p>è il <a href="/wiki/Raggio_di_Bohr" title="Raggio di Bohr">raggio di Bohr</a> del livello fondamentale dell'atomo di idrogeno. In accordo con i valori <a href="/wiki/CODATA" class="mw-redirect" title="CODATA">CODATA</a> del 2010, il raggio di Bohr vale </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 a_{0}=5,291\,777\,721\,092\cdot 10^{-11}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mn>291</mn> <mspace width="thinmathspace" /> <mn>777</mn> <mspace width="thinmathspace" /> <mn>721</mn> <mspace width="thinmathspace" /> <mn>092</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>11</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}=5,291\,777\,721\,092\cdot 10^{-11}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c650f94ff01efa166ed57375a4e697f5b8159206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.848ex; height:3.009ex;" alt="{\displaystyle a_{0}=5,291\,777\,721\,092\cdot 10^{-11}}"></span> m <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 =0,53}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>53</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =0,53}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8dffe48881a395f772886dcf31fec6adb643527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.975ex; height:2.509ex;" alt="{\displaystyle =0,53}"></span> Å</dd></dl> <p>dove Å sta per <a href="/wiki/%C3%85ngstr%C3%B6m" title="Ångström">Ångström</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>La formula derivata da Bohr per i raggi atomici risulta accurata solo per l'<a href="/wiki/Idrogeno" title="Idrogeno">idrogeno</a>, e in buon accordo coi dati sperimentali per gli <a href="/wiki/Atomo_idrogenoide" title="Atomo idrogenoide">atomi idrogenoidi</a> (quelli con un solo elettrone nel guscio<sup id="cite_ref-Bohr_1922_11-0" class="reference"><a href="#cite_note-Bohr_1922-11"><span class="cite-bracket">[</span>Nota 3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-guscio_12-0" class="reference"><a href="#cite_note-guscio-12"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> più esterno). Fissato il nucleo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84eb0c0feacfe730c6e1b53a4892259ccabd8b88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.49ex; height:2.843ex;" alt="{\displaystyle (Z)}"></span> a cui si fa riferimento, i raggi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57784cdf7f49f2c46baad4eb7b6a7d5c14eb5fa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.267ex; height:2.009ex;" alt="{\displaystyle r_{n}}"></span> delle orbite elettroniche quantizzate dipendono solo dal valore del <a href="/wiki/Numero_quantico_principale" title="Numero quantico principale">numero quantico principale</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 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/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>. </p> <table class="wikitable" style="text-align: center"> <caption>Raggi ed energie delle orbite dell'atomo d'idrogeno </caption> <tbody><tr> <th>Numero quantico <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/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></th> <th>Raggio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57784cdf7f49f2c46baad4eb7b6a7d5c14eb5fa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.267ex; height:2.009ex;" alt="{\displaystyle r_{n}}"></span> dell'orbita</th> <th>Energia <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> dell'orbita </th></tr> <tr> <td></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 r_{n}=n^{2}a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}=n^{2}a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/612805342cd3e771d8967cf10c107fed69dda34b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.099ex; height:3.009ex;" alt="{\displaystyle r_{n}=n^{2}a_{0}}"></span></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 E_{n}=-E_{0}/n^{2}}"> <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> <mo>=</mo> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}=-E_{0}/n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/588462836e7b6c590052ed7c998750aa64db4e0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.221ex; height:3.176ex;" alt="{\displaystyle E_{n}=-E_{0}/n^{2}}"></span> </td></tr> <tr> <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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></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 a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}"></span></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 -E_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -E_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/593d931f5802f0e617287be0a49e767ad52205e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.578ex; height:2.509ex;" alt="{\displaystyle -E_{0}}"></span> </td></tr> <tr> <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 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span></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 4\,a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mspace width="thinmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\,a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd6ca638dbc09b2c82f7fce7d98399866ed08050" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.834ex; height:2.509ex;" alt="{\displaystyle 4\,a_{0}}"></span></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 -E_{0}/4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -E_{0}/4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ffa5cad65288a56486cc2fee1e60ee94094dc2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.903ex; height:2.843ex;" alt="{\displaystyle -E_{0}/4}"></span> </td></tr> <tr> <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 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}"></span></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 9\,a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> <mspace width="thinmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9\,a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea83e6f3246d500ab66a9a9ce52d146f7a4a6dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.834ex; height:2.509ex;" alt="{\displaystyle 9\,a_{0}}"></span></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 -E_{0}/9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -E_{0}/9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f39891949b746a361902cb9510937c2047e66b90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.903ex; height:2.843ex;" alt="{\displaystyle -E_{0}/9}"></span> </td></tr> <tr> <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 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295b4bf1de7cd3500e740e0f4f0635db22d87b42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 4}"></span></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 16\,a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <mspace width="thinmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16\,a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0618e96cd4b3a849168b9709f2bb4d60921a533d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.996ex; height:2.509ex;" alt="{\displaystyle 16\,a_{0}}"></span></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 -E_{0}/16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -E_{0}/16}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c86e3a38140b17a8f6db3aadfca8b74d040a83f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.065ex; height:2.843ex;" alt="{\displaystyle -E_{0}/16}"></span> </td></tr> <tr> <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 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}"></span></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 25\,a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>25</mn> <mspace width="thinmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 25\,a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e283e521537a6cc7e9ffd7c4305e560da7f82cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.996ex; height:2.509ex;" alt="{\displaystyle 25\,a_{0}}"></span></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 -E_{0}/25}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>25</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -E_{0}/25}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35a65bafbcac0673a95fe9f5ae433266d398cd08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.065ex; height:2.843ex;" alt="{\displaystyle -E_{0}/25}"></span> </td></tr> <tr> <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 6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39d81124420a058a7474dfeda48228fb6ee1e253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 6}"></span></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 36\,a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>36</mn> <mspace width="thinmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 36\,a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7223c674714e4aaa573ebe27cf164841e24af10a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.996ex; height:2.509ex;" alt="{\displaystyle 36\,a_{0}}"></span></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 -E_{0}/36}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>36</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -E_{0}/36}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/102fe2050efdb9079c5ba1d107c551803de2fd69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.065ex; height:2.843ex;" alt="{\displaystyle -E_{0}/36}"></span> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Quantizzazione_delle_energie_e_formula_di_Rydberg">Quantizzazione delle energie e formula di Rydberg</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_atomico_di_Bohr&veaction=edit&section=6" title="Modifica la sezione Quantizzazione delle energie e formula di Rydberg" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_atomico_di_Bohr&action=edit&section=6" title="Edit section's source code: Quantizzazione delle energie e formula di Rydberg"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Quantizzazione_delle_energie_orbitali">Quantizzazione delle energie orbitali</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_atomico_di_Bohr&veaction=edit&section=7" title="Modifica la sezione Quantizzazione delle energie orbitali" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_atomico_di_Bohr&action=edit&section=7" title="Edit section's source code: Quantizzazione delle energie orbitali"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Hydrogen_energy_levels.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Hydrogen_energy_levels.svg/220px-Hydrogen_energy_levels.svg.png" decoding="async" width="220" height="531" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Hydrogen_energy_levels.svg/330px-Hydrogen_energy_levels.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Hydrogen_energy_levels.svg/440px-Hydrogen_energy_levels.svg.png 2x" data-file-width="354" data-file-height="854" /></a><figcaption>Livelli energetici dell'idrogeno secondo il modello atomico di Bohr.</figcaption></figure> <p>L'energia totale di un elettrone, nell'ipotesi semplificativa che si muova su un'orbita circolare con <a href="/wiki/Velocit%C3%A0" title="Velocità">velocità</a> <i>v</i>, è data dalla somma della sua energia cinetica </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 T={\frac {1}{2}}mv^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {1}{2}}mv^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab449fe0e35d97a2c7983b73fe2d4504d4e3a514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.956ex; height:5.176ex;" alt="{\displaystyle T={\frac {1}{2}}mv^{2}}"></span></dd></dl> <p>con l'energia potenziale elettrostatica </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 U=-{\frac {kZe^{2}}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>Z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=-{\frac {kZe^{2}}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cca99a79b418909e1ddc6c753fd2270ec0fa693" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.555ex; height:5.676ex;" alt="{\displaystyle U=-{\frac {kZe^{2}}{r}}}"></span></dd></dl> <p>Quindi: </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=T+U={\frac {1}{2}}mv^{2}-{\frac {kZe^{2}}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>T</mi> <mo>+</mo> <mi>U</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>Z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=T+U={\frac {1}{2}}mv^{2}-{\frac {kZe^{2}}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aee3989fd84404a784659d3cee254e87e3fdeb8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.159ex; height:5.676ex;" alt="{\displaystyle E=T+U={\frac {1}{2}}mv^{2}-{\frac {kZe^{2}}{r}}}"></span></dd></dl> <p>Sostituendo l'espressione per la velocità <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> </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 v={\sqrt {\frac {kZe^{2}}{mr}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>k</mi> <mi>Z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>m</mi> <mi>r</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\sqrt {\frac {kZe^{2}}{mr}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7097d55a72a17e09781019efcc3d5d3a05ed078" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.415ex; height:6.176ex;" alt="{\displaystyle v={\sqrt {\frac {kZe^{2}}{mr}}}}"></span></dd></dl> <p>nella formula per l'energia cinetica <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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>, si ricava che l'energia cinetica risulta essere pari alla metà del <a href="/wiki/Valore_assoluto" title="Valore assoluto">valore assoluto</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 |U|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |U|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62d03a31f61b08711b143345ea90d7f4b1a6d2e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.076ex; height:2.843ex;" alt="{\displaystyle |U|}"></span> dell'energia potenziale: </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 T={\frac {1}{2}}\,{\frac {kZe^{2}}{r}}={\frac {1}{2}}\,|U|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>Z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {1}{2}}\,{\frac {kZe^{2}}{r}}={\frac {1}{2}}\,|U|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b65d269ea05f576c127e9f3b9bb0e47386d2c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.546ex; height:5.676ex;" alt="{\displaystyle T={\frac {1}{2}}\,{\frac {kZe^{2}}{r}}={\frac {1}{2}}\,|U|}"></span></dd></dl> <p>L'energia totale risulta quindi essere pari a: </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=T+U={\frac {1}{2}}|U|-|U|=-{\frac {1}{2}}|U|=-{\frac {1}{2}}\,{\frac {kZe^{2}}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>T</mi> <mo>+</mo> <mi>U</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>Z</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=T+U={\frac {1}{2}}|U|-|U|=-{\frac {1}{2}}|U|=-{\frac {1}{2}}\,{\frac {kZe^{2}}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a479b335ba551e737bbbf1cddac35d5da541bef7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:48.363ex; height:5.676ex;" alt="{\displaystyle E=T+U={\frac {1}{2}}|U|-|U|=-{\frac {1}{2}}|U|=-{\frac {1}{2}}\,{\frac {kZe^{2}}{r}}}"></span></dd></dl> <p>Sostituendo il raggio quantizzato <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57784cdf7f49f2c46baad4eb7b6a7d5c14eb5fa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.267ex; height:2.009ex;" alt="{\displaystyle r_{n}}"></span> nella formula precedente, si ricavano le energie quantizzate di Bohr: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}=-{\frac {Z^{2}}{n^{2}}}\,{\frac {k^{2}me^{4}}{2\hbar ^{2}}}=-{\frac {Z^{2}}{n^{2}}}\,{\frac {me^{4}}{8\varepsilon _{0}^{2}h^{2}}}=-{\frac {Z^{2}}{n^{2}}}\,E_{0}}"> <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> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi class="MJX-variant">ℏ</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>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <msubsup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>h</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>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{n}=-{\frac {Z^{2}}{n^{2}}}\,{\frac {k^{2}me^{4}}{2\hbar ^{2}}}=-{\frac {Z^{2}}{n^{2}}}\,{\frac {me^{4}}{8\varepsilon _{0}^{2}h^{2}}}=-{\frac {Z^{2}}{n^{2}}}\,E_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adddcdf43724587514a540777819f9f7339def53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.19ex; height:6.843ex;" alt="{\displaystyle E_{n}=-{\frac {Z^{2}}{n^{2}}}\,{\frac {k^{2}me^{4}}{2\hbar ^{2}}}=-{\frac {Z^{2}}{n^{2}}}\,{\frac {me^{4}}{8\varepsilon _{0}^{2}h^{2}}}=-{\frac {Z^{2}}{n^{2}}}\,E_{0}}"></span></dd></dl> <p>Si noti che i possibili valori dell'energia dell'elettrone sono tutti negativi (vedi Figura). Ciò discende dal fatto che l'elettrone si trova in uno stato legato e l'<a href="/wiki/Energia_di_legame" title="Energia di legame">energia di legame</a> è sempre negativa. Fissato il nucleo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84eb0c0feacfe730c6e1b53a4892259ccabd8b88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.49ex; height:2.843ex;" alt="{\displaystyle (Z)}"></span> a cui si fa riferimento, le 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_{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> delle orbite elettroniche quantizzate dipendono solo dal valore del <a href="/wiki/Numero_quantico_principale" title="Numero quantico principale">numero quantico principale</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 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/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>. La formula derivata da Bohr per i livelli energetici degli atomi risulta accurata solo per l'<a href="/wiki/Idrogeno" title="Idrogeno">idrogeno</a>, e in buon accordo coi dati sperimentali per gli <a href="/wiki/Atomo_idrogenoide" title="Atomo idrogenoide">atomi idrogenoidi</a> (quelli con un solo elettrone nel guscio<sup id="cite_ref-Bohr_1922_11-1" class="reference"><a href="#cite_note-Bohr_1922-11"><span class="cite-bracket">[</span>Nota 3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-guscio_12-1" class="reference"><a href="#cite_note-guscio-12"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> più esterno). </p><p>Si definisce <i>Rydberg</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 \mathrm {Ry} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">y</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Ry} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1122ccf4c6e7f1a72eef95e562aec27d41ba7f03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.938ex; height:2.509ex;" alt="{\displaystyle \mathrm {Ry} }"></span>) l'unità di misura delle energie atomiche, che coincide con <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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/411d268de7b1cf300d7481e3fe59f3b20887e0d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{0}}"></span> e vale<sup id="cite_ref-codata_13-0" class="reference"><a href="#cite_note-codata-13"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </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_{0}={\frac {me^{4}}{8\varepsilon _{0}^{2}h^{2}}}=1\,\mathrm {Ry} =13,605\,693\,122\,994(26)\,\mathrm {eV} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <msubsup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">y</mi> </mrow> <mo>=</mo> <mn>13</mn> <mo>,</mo> <mn>605</mn> <mspace width="thinmathspace" /> <mn>693</mn> <mspace width="thinmathspace" /> <mn>122</mn> <mspace width="thinmathspace" /> <mn>994</mn> <mo stretchy="false">(</mo> <mn>26</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">V</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0}={\frac {me^{4}}{8\varepsilon _{0}^{2}h^{2}}}=1\,\mathrm {Ry} =13,605\,693\,122\,994(26)\,\mathrm {eV} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f66ac4ba4ea0f7b01d8d63a7728b81071c9c9b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.848ex; height:6.843ex;" alt="{\displaystyle E_{0}={\frac {me^{4}}{8\varepsilon _{0}^{2}h^{2}}}=1\,\mathrm {Ry} =13,605\,693\,122\,994(26)\,\mathrm {eV} }"></span></dd></dl> <p>dove <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 {eV} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">V</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {eV} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a77208febc077e5d6033f6e6e3135537959f4830" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \mathrm {eV} }"></span> è il simbolo dell'<a href="/wiki/Elettronvolt" title="Elettronvolt">elettronvolt</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/411d268de7b1cf300d7481e3fe59f3b20887e0d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{0}}"></span> è l'<a href="/wiki/Energia_di_ionizzazione" title="Energia di ionizzazione">energia di ionizzazione</a> per un elettrone dell'atomo d'<a href="/wiki/Idrogeno" title="Idrogeno">idrogeno</a> che si trovi nello stato fondamentale. </p> <div class="mw-heading mw-heading3"><h3 id="Derivazione_della_formula_di_Rydberg">Derivazione della formula di Rydberg</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_atomico_di_Bohr&veaction=edit&section=8" title="Modifica la sezione Derivazione della formula di Rydberg" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_atomico_di_Bohr&action=edit&section=8" title="Edit section's source code: Derivazione della formula di Rydberg"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Lyman_Series.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Lyman_Series.svg/290px-Lyman_Series.svg.png" decoding="async" width="290" height="321" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Lyman_Series.svg/435px-Lyman_Series.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/07/Lyman_Series.svg/580px-Lyman_Series.svg.png 2x" data-file-width="114" data-file-height="126" /></a><figcaption>Prime righe della serie di Lyman per l'atomo d'idrogeno.</figcaption></figure> <p>La transizione <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_{m}\to E_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{m}\to E_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4186e88da8f1f0bbf373fd107c3dc5dfbb69f4ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.938ex; height:2.509ex;" alt="{\displaystyle E_{m}\to E_{n}}"></span> tra due livelli energetici <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> ed <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/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> (con <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>n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>></mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m>n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/637039c4a193f33fee72ebfeb6cb003593696160" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.843ex;" alt="{\displaystyle m>n}"></span>) dell'<a href="/wiki/Idrogeno" title="Idrogeno">idrogeno</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 (Z=1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>Z</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Z=1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c5f9384a91d760d3f6e62dac0b032447773c3b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.751ex; height:2.843ex;" alt="{\displaystyle (Z=1)}"></span> può essere scritta come </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_{m}-E_{n}=E_{0}\left({\frac {1}{n^{2}}}-{\frac {1}{m^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{m}-E_{n}=E_{0}\left({\frac {1}{n^{2}}}-{\frac {1}{m^{2}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/088c560e3ba9f209d8526260fde40833c518f61c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.897ex; height:6.176ex;" alt="{\displaystyle E_{m}-E_{n}=E_{0}\left({\frac {1}{n^{2}}}-{\frac {1}{m^{2}}}\right)}"></span></dd></dl> <p>Il <a href="/wiki/Numero_d%27onda" title="Numero d'onda">numero d'onda</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 1/\lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/\lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1add931eabb9aec89275acb2f50d50428010b0fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.68ex; height:2.843ex;" alt="{\displaystyle 1/\lambda }"></span> è correlabile alla <a href="/wiki/Frequenza" title="Frequenza">frequenza</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 \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> e all'energia della transizione <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_{m}\to E_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{m}\to E_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4186e88da8f1f0bbf373fd107c3dc5dfbb69f4ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.938ex; height:2.509ex;" alt="{\displaystyle E_{m}\to E_{n}}"></span> tra i due livelli mediante la <a href="/wiki/Condizione_della_frequenza_di_Bohr" title="Condizione della frequenza di Bohr">condizione della frequenza di Bohr</a>: </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}{\lambda }}={\frac {\nu }{c}}={\frac {E_{m}-E_{n}}{ch}}={\frac {E_{0}}{ch}}\left({\frac {1}{n^{2}}}-{\frac {1}{m^{2}}}\right)=R_{\infty }\left({\frac {1}{n^{2}}}-{\frac {1}{m^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ν</mi> <mi>c</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mi>c</mi> <mi>h</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mi>c</mi> <mi>h</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\lambda }}={\frac {\nu }{c}}={\frac {E_{m}-E_{n}}{ch}}={\frac {E_{0}}{ch}}\left({\frac {1}{n^{2}}}-{\frac {1}{m^{2}}}\right)=R_{\infty }\left({\frac {1}{n^{2}}}-{\frac {1}{m^{2}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4999dd43f8c0ed57014f8dcda4d1096e3c9460f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:61.628ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{\lambda }}={\frac {\nu }{c}}={\frac {E_{m}-E_{n}}{ch}}={\frac {E_{0}}{ch}}\left({\frac {1}{n^{2}}}-{\frac {1}{m^{2}}}\right)=R_{\infty }\left({\frac {1}{n^{2}}}-{\frac {1}{m^{2}}}\right)}"></span></dd></dl> <p>La possibilità di spiegare, a partire da principi primi, la <a href="/wiki/Equazione_di_Rydberg" class="mw-redirect" title="Equazione di Rydberg">formula di Rydberg</a> segnò la definitiva affermazione del modello atomico di Bohr. </p><p>È quindi possibile calcolare, mediante la quantizzazione delle energie, il valore della <a href="/wiki/Costante_di_Rydberg" title="Costante di Rydberg">costante di Rydberg</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 R_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/639146705058e9d335aed47b55a9f802bc290b5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.639ex; height:2.509ex;" alt="{\displaystyle R_{\infty }}"></span> per un nucleo di massa infinita (<a href="/wiki/CODATA" class="mw-redirect" title="CODATA">CODATA</a>, 2014).<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Si trova che </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 R_{\infty }={\frac {E_{0}}{ch}}={\frac {me^{4}}{8\varepsilon _{0}^{2}ch^{3}}}=1.097\,373\,156\,850\,8(65)\times 10^{7}\,\mathrm {m} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mi>c</mi> <mi>h</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <msubsup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>c</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1.097</mn> <mspace width="thinmathspace" /> <mn>373</mn> <mspace width="thinmathspace" /> <mn>156</mn> <mspace width="thinmathspace" /> <mn>850</mn> <mspace width="thinmathspace" /> <mn>8</mn> <mo stretchy="false">(</mo> <mn>65</mn> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\infty }={\frac {E_{0}}{ch}}={\frac {me^{4}}{8\varepsilon _{0}^{2}ch^{3}}}=1.097\,373\,156\,850\,8(65)\times 10^{7}\,\mathrm {m} ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa2234446fb605c6ad4fe1ea161a514a2f88ae3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:57.556ex; height:6.843ex;" alt="{\displaystyle R_{\infty }={\frac {E_{0}}{ch}}={\frac {me^{4}}{8\varepsilon _{0}^{2}ch^{3}}}=1.097\,373\,156\,850\,8(65)\times 10^{7}\,\mathrm {m} ^{-1}}"></span></dd></dl> <p>Tenendo conto del fatto che la massa del nucleo non è infinita e che quindi il nucleo stesso ruota intorno al centro di massa dell'atomo, si introduce una lieve dipendenza della <a href="/wiki/Costante_di_Rydberg" title="Costante di Rydberg">costante di Rydberg</a> dalla massa del nucleo, migliorando così l'accordo con i dati sperimentali. Per l'atomo d'idrogeno </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 R_{H}={\frac {R_{\infty }}{1+m/m_{p}}}={\frac {R_{\infty }}{1+1/1836}}=0,999\,455\,634\,R_{\infty }=1.096\,775\,784\times 10^{7}\,\mathrm {m} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mrow> <mn>1</mn> <mo>+</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mrow> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>1836</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>999</mn> <mspace width="thinmathspace" /> <mn>455</mn> <mspace width="thinmathspace" /> <mn>634</mn> <mspace width="thinmathspace" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>=</mo> <mn>1.096</mn> <mspace width="thinmathspace" /> <mn>775</mn> <mspace width="thinmathspace" /> <mn>784</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{H}={\frac {R_{\infty }}{1+m/m_{p}}}={\frac {R_{\infty }}{1+1/1836}}=0,999\,455\,634\,R_{\infty }=1.096\,775\,784\times 10^{7}\,\mathrm {m} ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3cb5e90393f80c56792ed093891d3c8bffbad3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:80.185ex; height:6.009ex;" alt="{\displaystyle R_{H}={\frac {R_{\infty }}{1+m/m_{p}}}={\frac {R_{\infty }}{1+1/1836}}=0,999\,455\,634\,R_{\infty }=1.096\,775\,784\times 10^{7}\,\mathrm {m} ^{-1}}"></span></dd></dl> <p>con <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_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7d93b292637f64f747f3fe0ea4bad9f2bd65637" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.1ex; height:2.343ex;" alt="{\displaystyle m_{p}}"></span> <a href="/wiki/Massa_(fisica)" title="Massa (fisica)">massa</a> del <a href="/wiki/Protone" title="Protone">protone</a>. Il valore teorico di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{H}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{H}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e233f422302cd1a45d6e0b06cb123b716e63dba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.455ex; height:2.509ex;" alt="{\displaystyle R_{H}}"></span> è in ottimo accordo con il valore numerico ottenuto sperimentalmente in <a href="/wiki/Spettroscopia" title="Spettroscopia">spettroscopia</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Correzione_per_la_massa_finita_del_nucleo">Correzione per la massa finita del nucleo</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_atomico_di_Bohr&veaction=edit&section=9" title="Modifica la sezione Correzione per la massa finita del nucleo" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_atomico_di_Bohr&action=edit&section=9" title="Edit section's source code: Correzione per la massa finita del nucleo"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nel modello di Bohr si considera la massa del nucleo infinitamente grande rispetto alla massa dell'elettrone, cosicché il nucleo rimane fisso nello spazio; questa è un'approssimazione ragionevole in quanto la massa del protone è circa 1.836 volte quella dell'elettrone. Tuttavia, l'accuratezza raggiunta dalle misurazioni spettroscopiche richiede di tenere conto del fatto che la massa del nucleo è finita e che il nucleo e l'elettrone si muovono attorno al comune centro di massa. </p><p>Facendo uso del concetto di <a href="/wiki/Massa_ridotta" title="Massa ridotta">massa ridotta</a> si può mostrare che, per tenere conto del fatto che la massa del nucleo è finita, è sufficiente sostituire, nelle equazioni del moto, la massa dell'elettrone con la massa ridotta <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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> del sistema nucleo-elettrone.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </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 \mu ={\frac {m\,M}{m+M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mspace width="thinmathspace" /> <mi>M</mi> </mrow> <mrow> <mi>m</mi> <mo>+</mo> <mi>M</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu ={\frac {m\,M}{m+M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0335bc06ba4dc12fa0a002f697a5f008a7d163d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.659ex; height:5.343ex;" alt="{\displaystyle \mu ={\frac {m\,M}{m+M}}}"></span></dd></dl> <p>dove <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> è la massa dell'elettrone e <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/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> la massa del nucleo. In questo modo si introduce un fattore correttivo <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 \mu /m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu /m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c042bc381a60fcb487a26c85325a37fa7757cdc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.605ex; height:2.843ex;" alt="{\displaystyle \mu /m}"></span> alle energie e <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/\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m/\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9339be8bc91f2697f939d13d7e05419956eb9d65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.605ex; height:2.843ex;" alt="{\displaystyle m/\mu }"></span> ai raggi atomici. </p><p>Per tutti gli <a href="/wiki/Atomo_idrogenoide" title="Atomo idrogenoide">atomi idrogenoidi</a> (quelli con un solo elettrone nel guscio<sup id="cite_ref-Bohr_1922_11-2" class="reference"><a href="#cite_note-Bohr_1922-11"><span class="cite-bracket">[</span>Nota 3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-guscio_12-2" class="reference"><a href="#cite_note-guscio-12"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> più esterno), la <a href="/wiki/Costante_di_Rydberg" title="Costante di Rydberg">costante di Rydberg</a> effettiva <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a600516077fb78afb09d1aa981513e4d60dc16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.723ex; height:2.509ex;" alt="{\displaystyle R_{M}}"></span> può essere derivata dalla costante di Rydberg "all'infinito" (per un nucleo infinitamente pesante), da </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 R_{M}={\frac {R_{\infty }}{1+m/M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mrow> <mn>1</mn> <mo>+</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>M</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{M}={\frac {R_{\infty }}{1+m/M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e06eca78521bb41c42343cc5159ef1be9643a0a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.306ex; height:6.009ex;" alt="{\displaystyle R_{M}={\frac {R_{\infty }}{1+m/M}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Limiti_del_modello_di_Bohr">Limiti del modello di Bohr</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_atomico_di_Bohr&veaction=edit&section=10" title="Modifica la sezione Limiti del modello di Bohr" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_atomico_di_Bohr&action=edit&section=10" title="Edit section's source code: Limiti del modello di Bohr"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Stark_splitting-en.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Stark_splitting-en.svg/310px-Stark_splitting-en.svg.png" decoding="async" width="310" height="232" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Stark_splitting-en.svg/465px-Stark_splitting-en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Stark_splitting-en.svg/620px-Stark_splitting-en.svg.png 2x" data-file-width="2186" data-file-height="1633" /></a><figcaption>Spostamenti di Stark-Lo Surdo del secondo ordine, relativi all'<a href="/wiki/Idrogeno" title="Idrogeno">idrogeno</a>. Ogni <a href="/wiki/Numero_quantico_principale" title="Numero quantico principale">livello energetico n</a> si separa in (n - 1) <a href="/wiki/Livelli_di_energia_degeneri" title="Livelli di energia degeneri">sottolivelli degeneri</a>.</figcaption></figure> <p>Il modello atomico di Bohr ha difficoltà a spiegare, oppure non spiega del tutto: </p> <ul><li>Il valore del momento angolare orbitale per lo stato fondamentale <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"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> </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/863277921bb6a619f684b3dee34605e3dc132fcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.465ex; height:2.843ex;" alt="{\displaystyle (n=1)}"></span>. Bohr prevede che valga <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{1}=\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{1}=\hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1eddcb937a658119385996308d79a4ab45d4c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.042ex; height:2.509ex;" alt="{\displaystyle L_{1}=\hbar }"></span>, mentre è noto sperimentalmente che il momento angolare dello stato fondamentale è nullo in tutti gli atomi.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>Nota 4<span class="cite-bracket">]</span></a></sup></li></ul> <ul><li>Essendo un modello semi-<a href="/wiki/Fisica_classica" title="Fisica classica">classico</a>, quello di Bohr viola la <a href="/wiki/Principio_di_indeterminazione_di_Heisenberg" title="Principio di indeterminazione di Heisenberg">relazione d'indeterminazione posizione/momento lineare</a> di <a href="/wiki/Werner_Heisenberg" class="mw-redirect" title="Werner Heisenberg">Heisenberg</a>, in quanto assegna agli elettroni orbite precise, con posizione e quantità di moto determinati in ogni punto.</li></ul> <ul><li>Gli atomi multi elettronici non hanno i livelli energetici previsti dal modello. Non funziona, ad esempio per l'<a href="/wiki/Elio" title="Elio">elio</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 (Z=2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>Z</mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Z=2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f40a5691702cbd109ea4963726d5b576cffabf16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.751ex; height:2.843ex;" alt="{\displaystyle (Z=2)}"></span>.</li></ul> <ul><li>Gli spettri di emissione per atomi con un singolo elettrone nel guscio<sup id="cite_ref-Bohr_1922_11-3" class="reference"><a href="#cite_note-Bohr_1922-11"><span class="cite-bracket">[</span>Nota 3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-guscio_12-3" class="reference"><a href="#cite_note-guscio-12"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> esterno (<a href="/wiki/Atomo_idrogenoide" title="Atomo idrogenoide">atomi idrogenoidi</a>) possono essere previsti approssimativamente. Per gli altri spettri di atomi multi-elettronici il modello di Bohr fa, nella migliore delle ipotesi, previsioni sull'emissione di raggi X delle righe <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\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/741e4e6b24d5d8025694573bdac057e497f8fb7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.554ex; height:2.176ex;" alt="{\displaystyle K\alpha }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n=1,\,m=2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mi>m</mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n=1,\,m=2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f2de3e6e831e9d5ae4ab6005a91544430bf9079" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.187ex; height:2.843ex;" alt="{\displaystyle (n=1,\,m=2)}"></span> e, in alcuni casi, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8461661976ac11b877cfd99f9c73d3df50575d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.07ex; height:2.176ex;" alt="{\displaystyle L\alpha }"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n=2,\,m=3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mi>m</mi> <mo>=</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n=2,\,m=3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79a8601413a92c92907d0ef2b860167b1ab6f96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.187ex; height:2.843ex;" alt="{\displaystyle (n=2,\,m=3)}"></span> dello <a href="/wiki/Spettro_atomico" title="Spettro atomico">spettro atomico</a>, se vengono fatte ipotesi aggiuntive ad hoc. Altre linee spettrali possono essere dedotte tramite il <a href="/wiki/Principio_di_combinazione_di_Ritz" title="Principio di combinazione di Ritz">principio di combinazione di Ritz</a>.</li></ul> <ul><li>Le intensità (larghezze) relative delle <a href="/wiki/Linea_spettrale" title="Linea spettrale">righe spettrali</a>. La formula di Bohr per i livelli energetici o sue modifiche sono state in grado di fornire stime ragionevoli in alcuni casi semplici, come ad esempio per l'<a href="/wiki/Effetto_Stark-Lo_Surdo" title="Effetto Stark-Lo Surdo">effetto Stark-Lo Surdo</a>.</li></ul> <ul><li>Doppietti<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>Nota 5<span class="cite-bracket">]</span></a></sup> e tripletti appaiono negli spettri di alcuni atomi come linee molto vicine. Il modello di Bohr non spiega perché alcuni livelli energetici siano così vicini tra loro.</li></ul> <ul><li>L'esistenza di una <a href="/wiki/Struttura_fine" title="Struttura fine">struttura fine</a> e di una <a href="/wiki/Struttura_iperfine" title="Struttura iperfine">struttura iperfine</a> nelle <a href="/wiki/Linea_spettrale" title="Linea spettrale">righe spettrali</a>, dovute a effetti <a href="/wiki/Relativit%C3%A0_speciale" class="mw-redirect" title="Relatività speciale">relativistici</a> nonché allo <a href="/wiki/Spin" title="Spin">spin</a> degli elettroni, che non vengono considerati dal modello atomico di Bohr.</li></ul> <ul><li>L'<a href="/wiki/Effetto_Zeeman" title="Effetto Zeeman">effetto Zeeman</a> (cambiamenti nelle righe spettrali dovuti a <a href="/wiki/Campo_magnetico" title="Campo magnetico">campi magnetici</a> esterni). In questo caso intervengono lo <a href="/wiki/Spin" title="Spin">spin</a> degli elettroni e i <a href="/wiki/Campo_magnetico" title="Campo magnetico">campi magnetici</a> orbitali, di cui si riesce a tener conto solo in trattazioni <a href="/wiki/Meccanica_quantistica" title="Meccanica quantistica">quantistiche</a> più sofisticate.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Modello_di_Bohr-Sommerfeld">Modello di Bohr-Sommerfeld</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_atomico_di_Bohr&veaction=edit&section=11" title="Modifica la sezione Modello di Bohr-Sommerfeld" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_atomico_di_Bohr&action=edit&section=11" title="Edit section's source code: Modello di Bohr-Sommerfeld"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sommerfeld_ellipses.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Sommerfeld_ellipses.svg/310px-Sommerfeld_ellipses.svg.png" decoding="async" width="310" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Sommerfeld_ellipses.svg/465px-Sommerfeld_ellipses.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/75/Sommerfeld_ellipses.svg/620px-Sommerfeld_ellipses.svg.png 2x" data-file-width="650" data-file-height="460" /></a><figcaption>Orbite ellittiche di Sommerfeld con la stessa energia e momento angolare quantizzato (<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 \ell =0,1,2,3,4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell =0,1,2,3,4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b98a086dce524405a746bf4e09f253036135d53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.016ex; height:2.509ex;" alt="{\displaystyle \ell =0,1,2,3,4}"></span>).</figcaption></figure> <p><a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Arnold Sommerfeld</a> migliorò nel 1916 il modello di Bohr. Innanzitutto, vi fu un ritorno alle orbite ellittiche di tipo planetario, che Bohr aveva inizialmente introdotto (<i>prima condizione di Bohr</i>), ma poi sostituito con l'ipotesi semplificativa di un'orbita circolare, caso particolare di <a href="/wiki/Ellisse" title="Ellisse">ellisse</a> con <a href="/wiki/Eccentricit%C3%A0_(matematica)" title="Eccentricità (matematica)">eccentricità</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c2f5932668126c63c844dc00ca187bc58a29e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle e=1}"></span>. </p><p>Inoltre questo modello sostituì la condizione del modello di Bohr sul momento angolare quantizzato <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (L_{n}=n\hbar )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mi class="MJX-variant">ℏ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (L_{n}=n\hbar )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7067b5a8c6f8d11d8b735cf3a937318df30bc040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.41ex; height:2.843ex;" alt="{\displaystyle (L_{n}=n\hbar )}"></span> con una condizione di <a href="/wiki/Quantizzazione_(fisica)" title="Quantizzazione (fisica)">quantizzazione</a> suggerita dal <a href="/wiki/Principio_di_corrispondenza" title="Principio di corrispondenza">principio di corrispondenza</a> e nota come <a href="/wiki/Formula_di_Wilson-Sommerfeld" title="Formula di Wilson-Sommerfeld">formula di Wilson-Sommerfeld</a>:<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> l'<a href="/wiki/Azione_ridotta" class="mw-redirect" title="Azione ridotta">azione ridotta</a> sull'orbita deve valere </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 A_{n}=\oint {\mathbf {p} \cdot \operatorname {d} \!\mathbf {q} }=nh}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">d</mi> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mrow> <mo>=</mo> <mi>n</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}=\oint {\mathbf {p} \cdot \operatorname {d} \!\mathbf {q} }=nh}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7cd419791b5b75dc4726b5d1b52f027bcedd84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.346ex; height:5.676ex;" alt="{\displaystyle A_{n}=\oint {\mathbf {p} \cdot \operatorname {d} \!\mathbf {q} }=nh}"></span></dd></dl> <p>dove <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> è il momento e <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 dq}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dq}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/212a4169ef66be3955464b7b64e185e2b78ef365" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle dq}"></span> rappresenta il <a href="/wiki/Differenziale_(matematica)" title="Differenziale (matematica)">differenziale</a> della generica funzione coordinata <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1f8079f76d0e8a89cf19db8fc43f34ec569d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.718ex; height:2.843ex;" alt="{\displaystyle q(t)}"></span>. </p><p>I calcoli basati sul modello di Bohr-Sommerfeld sono stati in grado di spiegare accuratamente una serie di effetti spettrali atomici complessi. Ad esempio, fino alle perturbazioni del primo ordine, il modello di Bohr e la meccanica quantistica fanno le stesse previsioni per la divisione di una riga spettrale dovuta all’<a href="/wiki/Effetto_Stark-Lo_Surdo" title="Effetto Stark-Lo Surdo">effetto Stark-Lo Surdo</a>. Nel caso di perturbazioni di ordine superiore, tuttavia, il modello di Bohr e la meccanica quantistica differiscono, e le misurazioni dell’effetto Stark con intensità di <a href="/wiki/Campo_elettrico" title="Campo elettrico">campo elettrico</a> elevate hanno contribuito a confermare la correttezza della meccanica quantistica rispetto al modello di Bohr. La spiegazione di questa differenza risiede nella forma degli orbitali degli elettroni, che variano a seconda dello stato energetico dell'elettrone. Il modello di Bohr-Sommerfeld, a differenza di quello di Bohr, riesce a darne conto. </p><p>Il modello di Bohr-Sommerfeld, come il precedente di Bohr, aveva limiti e difetti: </p> <ul><li>L'approccio di Sommerfeld non si estendeva ai moti non integrabili; quindi molti sistemi non potevano essere trattati, nemmeno in linea di principio.</li> <li>La <a href="/wiki/Quantizzazione_(fisica)" title="Quantizzazione (fisica)">quantizzazione</a> di Sommerfeld può essere eseguita in diverse coordinate canoniche e talvolta fornisce risposte diverse.</li> <li>Anche il modello di Bohr-Sommerfeld non è corretto per piccoli <a href="/wiki/Numero_quantico" title="Numero quantico">numeri quantici</a>, perché mescola concetti classici e quantistici.</li> <li>Anche il modello di Bohr-Sommerfeld, essendo semi-<a href="/wiki/Fisica_classica" title="Fisica classica">classico</a>, viola la <a href="/wiki/Principio_di_indeterminazione_di_Heisenberg" title="Principio di indeterminazione di Heisenberg">relazione d'indeterminazione posizione/momento lineare</a> di <a href="/wiki/Werner_Heisenberg" class="mw-redirect" title="Werner Heisenberg">Heisenberg</a>, in quanto assegna agli elettroni orbite precise, con posizione e quantità di moto determinati in ogni punto.</li> <li>Il <a href="/wiki/Numero_quantico_magnetico" title="Numero quantico magnetico">numero quantico magnetico</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> misurava l'inclinazione del piano orbitale rispetto al piano XY e poteva assumere solo pochi valori discreti. Ciò ovviamente contraddiceva il fatto che un atomo può essere ruotato senza restrizioni.</li> <li>L'inclusione delle correzioni radiative nel modello è stata difficile, perché richiedeva di trovare le coordinate dell'angolo di azione per un sistema combinato radiazione/atomo, cosa difficile quando la radiazione viene rilasciata.</li></ul> <p>Il modello fu sostituito dalla moderna trattazione <a href="/wiki/Meccanica_quantistica" title="Meccanica quantistica">quantomeccanica</a> dell'atomo d'idrogeno, proposta da <a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Wolfgang Pauli</a> nel 1925, utilizzando la <a href="/wiki/Meccanica_delle_matrici" title="Meccanica delle matrici">meccanica delle matrici</a>. L'immagine attuale dell'atomo d'idrogeno si basa sugli <a href="/wiki/Orbitale_atomico" title="Orbitale atomico">orbitali atomici</a> della <a href="/wiki/Meccanica_ondulatoria" title="Meccanica ondulatoria">meccanica ondulatoria</a>, che <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a> sviluppò nel 1926. </p> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_atomico_di_Bohr&veaction=edit&section=12" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_atomico_di_Bohr&action=edit&section=12" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>Approfondimenti</dt></dl> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text">Essa fu introdotta in <a href="/wiki/Fisica" title="Fisica">Fisica</a> da <a href="/wiki/Max_Planck" title="Max Planck">Max Planck</a> nel 1900 per la spiegazione teorica della <a href="/wiki/Radiazione_del_corpo_nero" class="mw-redirect" title="Radiazione del corpo nero">radiazione del corpo nero</a> e quindi utilizzata da <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> nel 1905 per la comprensione dell'<a href="/wiki/Effetto_fotoelettrico" title="Effetto fotoelettrico">effetto fotoelettrico</a>, dando avvio alla <a href="/wiki/Teoria_dei_quanti" title="Teoria dei quanti">teoria dei quanti</a> che precedette la formulazione della <a href="/wiki/Meccanica_quantistica" title="Meccanica quantistica">meccanica quantistica</a>.</span> </li> <li id="cite_note-7"><a href="#cite_ref-7"><b>^</b></a> <span class="reference-text">Oggi si considera l'interazione tra l'elettrone ed il campo elettromagnetico come dovuta allo scambio di un <a href="/wiki/Fotone" title="Fotone">fotone</a> o quanto di luce. Tuttavia Bohr nei suoi articoli non chiama in causa i quanti di luce di Einstein, dei quali sarà un deciso oppositore fino al 1924.</span> </li> <li id="cite_note-Bohr_1922-11"><span class="mw-cite-backlink"><b>^</b> <sup><i><a href="#cite_ref-Bohr_1922_11-0">a</a></i></sup> <sup><i><a href="#cite_ref-Bohr_1922_11-1">b</a></i></sup> <sup><i><a href="#cite_ref-Bohr_1922_11-2">c</a></i></sup> <sup><i><a href="#cite_ref-Bohr_1922_11-3">d</a></i></sup></span> <span class="reference-text">Bohr aggiornò il suo modello atomico nel 1922, assumendo che a certi numeri di elettroni (2, 8 e 18, ad esempio) corrispondano "gusci chiusi" energeticamente stabili.</span> </li> <li id="cite_note-17"><a href="#cite_ref-17"><b>^</b></a> <span class="reference-text">Si può immaginare, sebbene le immagini mentali falliscano a livelli di scala microscopici, che un elettrone nell'<a href="/wiki/Orbitale_atomico" title="Orbitale atomico">orbitale atomico</a> fondamentale, privo di <a href="/wiki/Momento_angolare" title="Momento angolare">momento angolare</a>, non ruoti affatto "intorno" al nucleo come un pianeta attorno al Sole, ma sia localizzato in una regione spaziale a simmetria sferica, centrata sul nucleo. Secondo la descrizione <a href="/wiki/Meccanica_quantistica" title="Meccanica quantistica">quantomeccanica</a> moderna, l'elettrone del primo orbitale è rappresentato da una nuvola sferica di probabilità di presenza, che diventa più densa man mano che ci si avvicina al suo centro, il nucleo.</span> </li> <li id="cite_note-18"><a href="#cite_ref-18"><b>^</b></a> <span class="reference-text">In <a href="/wiki/Spettroscopia" title="Spettroscopia">spettroscopia</a>, un doppietto è una coppia di <a href="/wiki/Linea_spettrale" title="Linea spettrale">righe spettrali</a> aventi <a href="/wiki/Lunghezza_d%27onda" title="Lunghezza d'onda">lunghezza d'onda</a> molto prossima, discriminabili solo mediante tecniche ad alta risoluzione.</span> </li> </ol></div> <dl><dt>Fonti</dt></dl> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-2"><a href="#cite_ref-2"><b>^</b></a> <span class="reference-text"><cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.unirc.it/documentazione/materiale_didattico/597_2009_221_6520.pdf"><span style="font-style:italic;">Luce, occhio, visione</span></a> (<span style="font-weight: bolder; font-size:80%"><abbr title="documento in formato PDF">PDF</abbr></span>), su <span style="font-style:italic;">unirc.it</span>. <small>URL consultato il 4 novembre 2023</small>.</cite></span> </li> <li id="cite_note-3"><a href="#cite_ref-3"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFParodi">Parodi</a></cite> p.450.</span> </li> <li id="cite_note-a-4"><span class="mw-cite-backlink"><b>^</b> <sup><i><a href="#cite_ref-a_4-0">a</a></i></sup> <sup><i><a href="#cite_ref-a_4-1">b</a></i></sup></span> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFParodi">Parodi</a></cite> p.451.</span> </li> <li id="cite_note-5"><a href="#cite_ref-5"><b>^</b></a> <span class="reference-text"> <cite class="citation pubblicazione" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) Ernest Rutherford, <a rel="nofollow" class="external text" href="http://www.chemteam.info/Chem-History/Rutherford-1911/Rutherford-1911.html"><span style="font-style:italic;">The Scattering of α and β Particles by Matter and the Structure of the Atom</span></a>, in <span style="font-style:italic;">Philosophical Magazine</span>, 6, vol. 21, n. 125, 1911, pp. 669–688, <a href="/wiki/Digital_object_identifier" title="Digital object identifier">DOI</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1080%2F14786440508637080">10.1080/14786440508637080</a>.</cite></span> </li> <li id="cite_note-6"><a href="#cite_ref-6"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFParodi">Parodi</a></cite> pp.453-455.</span> </li> <li id="cite_note-8"><a href="#cite_ref-8"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFParodi">Parodi</a></cite> pp.455-456.</span> </li> <li id="cite_note-9"><a href="#cite_ref-9"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFCatalano">Catalano</a></cite> p.377.</span> </li> <li id="cite_note-10"><a href="#cite_ref-10"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> Paolo Mazzoldi, Massimo Nigro, Cesare Voci, <span style="font-style:italic;">Fisica (Volume II)</span>, EdiSES Editore, 2001, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-7959-152-5" title="Speciale:RicercaISBN/88-7959-152-5">88-7959-152-5</a>.</cite> p.44</span> </li> <li id="cite_note-guscio-12"><span class="mw-cite-backlink"><b>^</b> <sup><i><a href="#cite_ref-guscio_12-0">a</a></i></sup> <sup><i><a href="#cite_ref-guscio_12-1">b</a></i></sup> <sup><i><a href="#cite_ref-guscio_12-2">c</a></i></sup> <sup><i><a href="#cite_ref-guscio_12-3">d</a></i></sup></span> <span class="reference-text"><cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://online.scuola.zanichelli.it/chimicarippa/files/2016/09/02-Modello_elettronico_a_gusci.pdf"><span style="font-style:italic;">Il modello elettronico a gusci</span></a> (<span style="font-weight: bolder; font-size:80%"><abbr title="documento in formato PDF">PDF</abbr></span>), su <span style="font-style:italic;">online.scuola.zanichelli.it</span>. <small>URL consultato il 20 ottobre 2023</small>.</cite></span> </li> <li id="cite_note-codata-13"><a href="#cite_ref-codata_13-0"><b>^</b></a> <span class="reference-text">Eite Tiesinga, Peter J. Mohr, David B. Newell, and Barry N. Taylor (2019), "The 2018 CODATA Recommended Values of the Fundamental Physical Constants" (Web Version 8.0). Database developed by J. Baker, M. Douma, and S. Kotochigova. Available at <a rel="nofollow" class="external free" href="http://physics.nist.gov/constants">http://physics.nist.gov/constants</a>, National Institute of Standards and Technology, Gaithersburg, MD 20899. <a rel="nofollow" class="external text" href="http://physics.nist.gov/cgi-bin/cuu/Value?ryd">Link to R<sub>∞</sub></a>, <a rel="nofollow" class="external text" href="http://physics.nist.gov/cgi-bin/cuu/Value?rydhcev">Link to hcR<sub>∞</sub></a></span> </li> <li id="cite_note-14"><a href="#cite_ref-14"><b>^</b></a> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFMazzoldi,_Nigro,_Voci">Mazzoldi, Nigro, Voci</a></cite> p.709.</span> </li> <li id="cite_note-15"><a href="#cite_ref-15"><b>^</b></a> <span class="reference-text"><cite class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://physics.nist.gov/cgi-bin/cuu/Value?ryd"><span style="font-style:italic;">Costante di Rydberg all'infinito</span></a>, su <span style="font-style:italic;">physics.nist.gov</span>. <small>URL consultato il 12 maggio 2019</small>.</cite></span> </li> <li id="cite_note-16"><a href="#cite_ref-16"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> Robert Martin Eisberg e Robert Resnick, <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/10779839"><span style="font-style:italic;">4-7 Correction for finite nuclear mas</span></a>, in <span style="font-style:italic;">Quantum physics of atoms, molecules, solids, nuclei, and particles</span>, Wiley, 1985, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-0-471-87373-0" title="Speciale:RicercaISBN/978-0-471-87373-0">978-0-471-87373-0</a>, <a href="/wiki/Online_Computer_Library_Center" title="Online Computer Library Center">OCLC</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/oclc/10779839">10779839</a>.</cite></span> </li> <li id="cite_note-19"><a href="#cite_ref-19"><b>^</b></a> <span class="reference-text"><cite class="citation pubblicazione" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="tedesco">DE</abbr></span>) Arnold Sommerfeld, <a rel="nofollow" class="external text" href="https://zenodo.org/record/1424309"><span style="font-style:italic;">Zur Quantentheorie der Spektrallinien</span></a>, in <span style="font-style:italic;"><a href="/wiki/Annalen_der_Physik" title="Annalen der Physik">Annalen der Physik</a></span>, vol. 51, n. 17, 1916, pp. 1–94, <a href="/wiki/Bibcode" title="Bibcode">Bibcode</a>:<a rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/1916AnP...356....1S">1916AnP...356....1S</a>, <a href="/wiki/Digital_object_identifier" title="Digital object identifier">DOI</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1002%2Fandp.19163561702">10.1002/andp.19163561702</a>.</cite></span> </li> <li id="cite_note-20"><a href="#cite_ref-20"><b>^</b></a> <span class="reference-text"><cite class="citation pubblicazione" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) William Wilson, <a rel="nofollow" class="external text" href="https://zenodo.org/record/1430790"><span style="font-style:italic;">The quantum theory of radiation and line spectra</span></a>, in <span style="font-style:italic;"><a href="/wiki/Philosophical_Magazine" title="Philosophical Magazine">Philosophical Magazine</a></span>, vol. 29, n. 174, 1915, pp. 795–802, <a href="/wiki/Digital_object_identifier" title="Digital object identifier">DOI</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1080%2F14786440608635362">10.1080/14786440608635362</a>.</cite></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_atomico_di_Bohr&veaction=edit&section=13" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_atomico_di_Bohr&action=edit&section=13" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="CITEREFMazzoldi,_Nigro,_Voci" class="citation libro" style="font-style:normal"> Paolo Mazzoldi, Massimo Nigro, Cesare Voci, <span style="font-style:italic;">Fisica (Volume II)</span>, EdiSES Editore, 2001, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-7959-152-5" title="Speciale:RicercaISBN/88-7959-152-5">88-7959-152-5</a>.</cite></li> <li><cite id="CITEREFParodi" class="citation libro" style="font-style:normal"> Gianpaolo Parodi, Marco Ostili, Guglielmo Mochi Onori, <span style="font-style:italic;">L'evoluzione della Fisica (Volume 3)</span>, Paravia, 2006, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-395-1611-5" title="Speciale:RicercaISBN/88-395-1611-5">88-395-1611-5</a>.</cite></li> <li><cite id="CITEREFCatalano" class="citation libro" style="font-style:normal"> Ferdinando Catalano, <span style="font-style:italic;">Elementi di Ottica Generale</span>, Bologna, Zanichelli, 2014, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-88-08-09786-6" title="Speciale:RicercaISBN/978-88-08-09786-6">978-88-08-09786-6</a>.</cite></li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modello_atomico_di_Bohr&veaction=edit&section=14" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Modello_atomico_di_Bohr&action=edit&section=14" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Atomo_di_idrogeno" title="Atomo di idrogeno">Atomo di idrogeno</a></li> <li><a href="/wiki/Formula_di_Wilson-Sommerfeld" title="Formula di Wilson-Sommerfeld">Formula di Wilson-Sommerfeld</a></li> <li><a href="/wiki/Modello_atomico_di_Rutherford" title="Modello atomico di Rutherford">Modello atomico di Rutherford</a></li> <li><a href="/wiki/Modello_atomico_di_Nagaoka" title="Modello atomico di Nagaoka">Modello atomico di Nagaoka</a></li> <li><a href="/wiki/Modello_atomico_di_Thomson" title="Modello atomico di Thomson">Modello atomico di Thomson</a></li></ul> <div class="mw-heading mw-heading2"><h2 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Bohr">Wikizionario</a></li> <li class="" title=""><span class="plainlinks" title="commons:Category:Bohr model"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Bohr_model?uselang=it">Wikimedia Commons</a></span></li></ul></div> <ul><li><span typeof="mw:File"><a href="https://it.wiktionary.org/wiki/" title="Collabora a Wikizionario"><img alt="Collabora a Wikizionario" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/18px-Wiktionary_small.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/27px-Wiktionary_small.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/36px-Wiktionary_small.svg.png 2x" data-file-width="350" data-file-height="350" /></a></span> <a href="https://it.wiktionary.org/wiki/" class="extiw" title="wikt:">Wikizionario</a> contiene il lemma di dizionario «<b><a 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Modello atomico di Bohr - Wikipedia