Efeutu túnel - Wikipedia

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</a> <ul id="toc-Na_cultura_popular-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ver_tamién" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ver_tamién"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Ver tamién</span> </div> </a> <ul id="toc-Ver_tamién-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_y_referencies" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_y_referencies"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes y referencies</span> </div> </a> <button aria-controls="toc-Notes_y_referencies-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>Alternar subsección Notes y referencies</span> </button> <ul id="toc-Notes_y_referencies-sublist" class="vector-toc-list"> <li id="toc-Llectures_y_publicaciones" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Llectures_y_publicaciones"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Llectures y publicaciones</span> </div> </a> <ul id="toc-Llectures_y_publicaciones-sublist" class="vector-toc-list"> </ul> </li> </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="Conteníu" 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="Cambiar a la tabla de contenidos" > <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">Cambiar a la tabla de contenidos</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">Efeutu túnel</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="Ir a un artículo en otro idioma. Disponible en 57 idiomas" > <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-57" 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">57 llingües</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D9%81%D9%82_%D9%85%D9%8A%D9%83%D8%A7%D9%86%D9%8A%D9%83%D8%A7_%D8%A7%D9%84%D9%83%D9%85" title="نفق ميكانيكا الكم – árabe" lang="ar" hreflang="ar" data-title="نفق ميكانيكا الكم" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D1%83%D0%BD%D1%8D%D0%BB%D1%8C%D0%BD%D1%8B_%D1%8D%D1%84%D0%B5%D0%BA%D1%82" title="Тунэльны эфект – bielorrusu" lang="be" hreflang="be" data-title="Тунэльны эфект" data-language-autonym="Беларуская" data-language-local-name="bielorrusu" 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%A2%D1%83%D0%BD%D0%B5%D0%BB%D0%B5%D0%BD_%D0%BF%D1%80%D0%B5%D1%85%D0%BE%D0%B4" title="Тунелен преход – búlgaru" lang="bg" hreflang="bg" data-title="Тунелен преход" data-language-autonym="Български" data-language-local-name="búlgaru" 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%95%E0%A7%8B%E0%A6%AF%E0%A6%BC%E0%A6%BE%E0%A6%A8%E0%A7%8D%E0%A6%9F%E0%A6%BE%E0%A6%AE_%E0%A6%9F%E0%A6%BE%E0%A6%A8%E0%A7%87%E0%A6%B2%E0%A6%BF%E0%A6%82" title="কোয়ান্টাম টানেলিং – bengalín" lang="bn" hreflang="bn" data-title="কোয়ান্টাম টানেলিং" data-language-autonym="বাংলা" data-language-local-name="bengalín" 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/Tunelski_efekt" title="Tunelski efekt – bosniu" lang="bs" hreflang="bs" data-title="Tunelski efekt" data-language-autonym="Bosanski" data-language-local-name="bosniu" 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/Efecte_t%C3%BAnel" title="Efecte túnel – catalán" lang="ca" hreflang="ca" data-title="Efecte túnel" data-language-autonym="Català" data-language-local-name="catalán" 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/Tunelov%C3%BD_jev" title="Tunelový jev – checu" lang="cs" hreflang="cs" data-title="Tunelový jev" data-language-autonym="Čeština" data-language-local-name="checu" 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/Kvantemekanisk_tunnelering" title="Kvantemekanisk tunnelering – danés" lang="da" hreflang="da" data-title="Kvantemekanisk tunnelering" data-language-autonym="Dansk" data-language-local-name="danés" 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/Tunneleffekt" title="Tunneleffekt – alemán" lang="de" hreflang="de" data-title="Tunneleffekt" data-language-autonym="Deutsch" data-language-local-name="alemán" 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%A6%CE%B1%CE%B9%CE%BD%CF%8C%CE%BC%CE%B5%CE%BD%CE%BF_%CF%83%CE%AE%CF%81%CE%B1%CE%B3%CE%B3%CE%B1%CF%82" title="Φαινόμενο σήραγγας – griegu" lang="el" hreflang="el" data-title="Φαινόμενο σήραγγας" data-language-autonym="Ελληνικά" data-language-local-name="griegu" 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/Quantum_tunnelling" title="Quantum tunnelling – inglés" lang="en" hreflang="en" data-title="Quantum tunnelling" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Tunela_efiko" title="Tunela efiko – esperanto" lang="eo" hreflang="eo" data-title="Tunela efiko" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Efecto_t%C3%BAnel" title="Efecto túnel – español" lang="es" hreflang="es" data-title="Efecto túnel" data-language-autonym="Español" data-language-local-name="español" 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/Tunneliefekt" title="Tunneliefekt – estoniu" lang="et" hreflang="et" data-title="Tunneliefekt" data-language-autonym="Eesti" data-language-local-name="estoniu" 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/Tunel-efektua" title="Tunel-efektua – vascu" lang="eu" hreflang="eu" data-title="Tunel-efektua" data-language-autonym="Euskara" data-language-local-name="vascu" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%88%D9%86%D9%84%E2%80%8C%D8%B2%D9%86%DB%8C_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85%DB%8C" title="تونل‌زنی کوانتومی – persa" lang="fa" hreflang="fa" data-title="تونل‌زنی کوانتومی" data-language-autonym="فارسی" data-language-local-name="persa" 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/Tunneloituminen" title="Tunneloituminen – finlandés" lang="fi" hreflang="fi" data-title="Tunneloituminen" data-language-autonym="Suomi" data-language-local-name="finlandés" 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/Effet_tunnel" title="Effet tunnel – francés" lang="fr" hreflang="fr" data-title="Effet tunnel" data-language-autonym="Français" data-language-local-name="francés" 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/Toll%C3%A1n%C3%BA_candamach" title="Tollánú candamach – irlandés" lang="ga" hreflang="ga" data-title="Tollánú candamach" data-language-autonym="Gaeilge" data-language-local-name="irlandés" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A0%D7%94%D7%95%D7%A8_%D7%A7%D7%95%D7%95%D7%A0%D7%98%D7%99" title="מנהור קוונטי – hebréu" lang="he" hreflang="he" data-title="מנהור קוונטי" data-language-autonym="עברית" data-language-local-name="hebréu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Tuneliranje" title="Tuneliranje – croata" lang="hr" hreflang="hr" data-title="Tuneliranje" data-language-autonym="Hrvatski" data-language-local-name="croata" 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/Alag%C3%BAthat%C3%A1s" title="Alagúthatás – húngaru" lang="hu" hreflang="hu" data-title="Alagúthatás" data-language-autonym="Magyar" data-language-local-name="húngaru" 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%B9%D5%B8%D6%82%D5%B6%D5%A5%D5%AC%D5%A1%D5%B5%D5%AB%D5%B6_%D5%A5%D6%80%D6%87%D5%B8%D6%82%D5%B5%D5%A9" title="Թունելային երևույթ – armeniu" lang="hy" hreflang="hy" data-title="Թունելային երևույթ" data-language-autonym="Հայերեն" data-language-local-name="armeniu" 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/Penerowongan_kuantum" title="Penerowongan kuantum – indonesiu" lang="id" hreflang="id" data-title="Penerowongan kuantum" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiu" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Skammtasmug" title="Skammtasmug – islandés" lang="is" hreflang="is" data-title="Skammtasmug" data-language-autonym="Íslenska" data-language-local-name="islandés" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Effetto_tunnel" title="Effetto tunnel – italianu" lang="it" hreflang="it" data-title="Effetto tunnel" data-language-autonym="Italiano" data-language-local-name="italianu" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%88%E3%83%B3%E3%83%8D%E3%83%AB%E5%8A%B9%E6%9E%9C" title="トンネル効果 – xaponés" lang="ja" hreflang="ja" data-title="トンネル効果" data-language-autonym="日本語" data-language-local-name="xaponés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D1%83%D0%BD%D0%BD%D0%B5%D0%BB%D1%8C%D0%B4%D1%96%D0%BA_%D1%8D%D1%84%D1%84%D0%B5%D0%BA%D1%82" title="Туннельдік эффект – kazaquistanín" lang="kk" hreflang="kk" data-title="Туннельдік эффект" data-language-autonym="Қазақша" data-language-local-name="kazaquistanín" 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/%ED%84%B0%EB%84%90_%ED%9A%A8%EA%B3%BC" title="터널 효과 – coreanu" lang="ko" hreflang="ko" data-title="터널 효과" data-language-autonym="한국어" data-language-local-name="coreanu" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Tune%C4%BCefekts" title="Tuneļefekts – letón" lang="lv" hreflang="lv" data-title="Tuneļefekts" data-language-autonym="Latviešu" data-language-local-name="letón" 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%A2%D1%83%D0%BD%D0%B5%D0%BB%D1%81%D0%BA%D0%B8_%D0%B5%D1%84%D0%B5%D0%BA%D1%82" title="Тунелски ефект – macedoniu" lang="mk" hreflang="mk" data-title="Тунелски ефект" data-language-autonym="Македонски" data-language-local-name="macedoniu" 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%9F%E0%B4%A3%E0%B4%B2%E0%B4%BF%E0%B4%99%E0%B5%8D_(%E0%B4%87%E0%B4%B2%E0%B4%95%E0%B5%8D%E0%B4%9F%E0%B5%8D%E0%B4%B0%E0%B5%8B%E0%B4%A3%E0%B4%BF%E0%B4%95%E0%B4%82)" title="ടണലിങ് (ഇലക്ട്രോണികം) – malayalam" lang="ml" hreflang="ml" data-title="ടണലിങ് (ഇലക്ട്രോണികം)" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Tunneleffect" title="Tunneleffect – neerlandés" lang="nl" hreflang="nl" data-title="Tunneleffect" data-language-autonym="Nederlands" data-language-local-name="neerlandés" 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/Kvantemekanisk_tunnelering" title="Kvantemekanisk tunnelering – noruegu Nynorsk" lang="nn" hreflang="nn" data-title="Kvantemekanisk tunnelering" data-language-autonym="Norsk nynorsk" data-language-local-name="noruegu 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/Kvantetunnelering" title="Kvantetunnelering – noruegu Bokmål" lang="nb" hreflang="nb" data-title="Kvantetunnelering" data-language-autonym="Norsk bokmål" data-language-local-name="noruegu Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Efiech_tun%C3%A8u" title="Efiech tunèu – occitanu" lang="oc" hreflang="oc" data-title="Efiech tunèu" data-language-autonym="Occitan" data-language-local-name="occitanu" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A9%81%E0%A8%86%E0%A8%82%E0%A8%9F%E0%A8%AE_%E0%A8%9F%E0%A9%B1%E0%A8%A8%E0%A8%B2%E0%A8%BF%E0%A9%B0%E0%A8%97" title="ਕੁਆਂਟਮ ਟੱਨਲਿੰਗ – punyabí" lang="pa" hreflang="pa" data-title="ਕੁਆਂਟਮ ਟੱਨਲਿੰਗ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punyabí" 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/Zjawisko_tunelowe" title="Zjawisko tunelowe – polacu" lang="pl" hreflang="pl" data-title="Zjawisko tunelowe" data-language-autonym="Polski" data-language-local-name="polacu" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Tunelamento_qu%C3%A2ntico" title="Tunelamento quântico – portugués" lang="pt" hreflang="pt" data-title="Tunelamento quântico" data-language-autonym="Português" data-language-local-name="portugués" 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/Efectul_tunel" title="Efectul tunel – rumanu" lang="ro" hreflang="ro" data-title="Efectul tunel" data-language-autonym="Română" data-language-local-name="rumanu" 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%A2%D1%83%D0%BD%D0%BD%D0%B5%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D1%8D%D1%84%D1%84%D0%B5%D0%BA%D1%82" title="Туннельный эффект – rusu" lang="ru" hreflang="ru" data-title="Туннельный эффект" data-language-autonym="Русский" data-language-local-name="rusu" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Quantum_tunnellin" title="Quantum tunnellin – scots" lang="sco" hreflang="sco" data-title="Quantum tunnellin" data-language-autonym="Scots" data-language-local-name="scots" 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/Tuneliranje" title="Tuneliranje – serbo-croata" lang="sh" hreflang="sh" data-title="Tuneliranje" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croata" 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/Quantum_tunnelling" title="Quantum tunnelling – Simple English" lang="en-simple" hreflang="en-simple" data-title="Quantum tunnelling" 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/Tunelov%C3%BD_jav" title="Tunelový jav – eslovacu" lang="sk" hreflang="sk" data-title="Tunelový jav" data-language-autonym="Slovenčina" data-language-local-name="eslovacu" 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/Tunelski_pojav" title="Tunelski pojav – eslovenu" lang="sl" hreflang="sl" data-title="Tunelski pojav" data-language-autonym="Slovenščina" data-language-local-name="eslovenu" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Efekti_tunel" title="Efekti tunel – albanu" lang="sq" hreflang="sq" data-title="Efekti tunel" data-language-autonym="Shqip" data-language-local-name="albanu" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a 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hreflang="ta" data-title="புரை ஊடுருவு மின்னோட்டம்" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kuantum_t%C3%BCnelleme" title="Kuantum tünelleme – turcu" lang="tr" hreflang="tr" data-title="Kuantum tünelleme" data-language-autonym="Türkçe" data-language-local-name="turcu" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A2%D1%83%D0%BD%D0%BD%D0%B5%D0%BB%D1%8C_%D1%8D%D1%84%D1%84%D0%B5%D0%BA%D1%82%D1%8B" title="Туннель эффекты – tártaru" lang="tt" hreflang="tt" data-title="Туннель эффекты" data-language-autonym="Татарча / tatarça" data-language-local-name="tártaru" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a 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esiste)">visual</a> del conductor, <a href="/w/index.php?title=Distorsi%C3%B3n&action=edit&redlink=1" class="new" title="Distorsión (la páxina nun esiste)">Distorsión</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheru:EffetTunnel.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/50/EffetTunnel.gif" decoding="async" width="200" height="200" class="mw-file-element" data-file-width="200" data-file-height="200" /></a><figcaption>Reflexón y "tunelado" d'un <a href="/wiki/Electr%C3%B3n" title="Electrón">electrón</a> empobináu escontra una <a href="/w/index.php?title=Barrera_de_potencial&action=edit&redlink=1" class="new" title="Barrera de potencial (la páxina nun esiste)">barrera de potencial</a>. El puntu resplandorientu moviéndose de derecha a izquierda ye la seición reflexada del <a href="/w/index.php?title=Quantum&action=edit&redlink=1" class="new" title="Quantum (la páxina nun esiste)">paquete d'onda</a>. Un acolumbre pue reparase a la derecha de la barrera. Esta pequeña fracción del paquete d'onda traviesa'l túnel d'una forma imposible pa los sistemes clásicos. Tamién ye notable la interferencia de les contornes ente les ondes d'emisión y de reflexón.</figcaption></figure> <p>En <a href="/wiki/Mec%C3%A1nica_cu%C3%A1ntica" title="Mecánica cuántica">mecánica cuántica</a>, el <b>efeutu túnel</b> ye un fenómenu cuánticu pol qu'una partícula viola los principios de la <a href="/wiki/Mec%C3%A1nica_cl%C3%A1sica" title="Mecánica clásica">mecánica clásica</a> enfusando una <a href="/w/index.php?title=Barrera_de_potencial&action=edit&redlink=1" class="new" title="Barrera de potencial (la páxina nun esiste)">barrera de potencial</a> o <a href="/wiki/Impedancia" title="Impedancia">impedancia</a> mayor que la <a href="/wiki/Enerx%C3%ADa_cin%C3%A9tica" class="mw-redirect" title="Enerxía cinética">enerxía cinética</a> de la mesma partícula. Una barrera, en términos cuánticos aplicaos al efeuto túnel, tratar d'una cualidá del <a href="/w/index.php?title=Est%C3%A1u_enerx%C3%A9ticu&action=edit&redlink=1" class="new" title="Estáu enerxéticu (la páxina nun esiste)">estáu enerxéticu</a> de la materia análogo a una "llomba" o pendiente clásica, compuesta por crestes y lladrales alternos, que suxer que'l camín más curtiu d'un móvil ente dos o más lladrales tien de travesar la so correspondiente cresta entemedia. Si l'oxetu nun dispón d'<a href="/w/index.php?title=Enerx%C3%ADa_mec%C3%A1nica&action=edit&redlink=1" class="new" title="Enerxía mecánica (la páxina nun esiste)">enerxía mecánica</a> abonda como pa travesar la barrera, la mecánica clásica afirma que nunca va poder apaecer nun estáu perteneciente al otru llau de la barrera. </p><p>A escala cuántica, los oxetos exhiben un <a href="/wiki/Dualid%C3%A1_onda_corp%C3%BAsculu" title="Dualidá onda corpúsculu">comportamientu ondular</a>; na teoría cuántica, un <a href="/w/index.php?title=Cuanto&action=edit&redlink=1" class="new" title="Cuanto (la páxina nun esiste)">cuanto</a> moviéndose en direición a una "llomba" potencialmente enerxética pue ser descritu pola so <a href="/wiki/Funci%C3%B3n_d%27onda" title="Función d'onda">función d'onda</a>, que representa l'amplitú probable que tien la partícula de ser atopada na posición allende la estructura de la <a href="/wiki/Curva" title="Curva">curva</a>. Si esta función describe la posición de la partícula perteneciente al lladral axacente al que supunxo'l so puntu de partida, esiste cierta probabilidá de que se moviera "al traviés" de la estructura, en cuenta de superala pela ruta convencional que traviesa'l visu enerxéticu relativa. A esto conozse como efeutu túnel. </p><p>L'efeutu túnel xuega un papel esencial en munchos fenómenos físicos como, por casu, na <a href="/wiki/Fusi%C3%B3n_nuclear" title="Fusión nuclear">fusión nuclear</a> qu'asocede na <a href="/wiki/Secuencia_principal" title="Secuencia principal">secuencia principal</a> d'estrelles como'l <a href="/wiki/Sol" title="Sol">Sol</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> La <a href="/wiki/Enerx%C3%ADa_cin%C3%A9tica" class="mw-redirect" title="Enerxía cinética">enerxía</a> que tienen los <a href="/w/index.php?title=Hidr%C3%B3n&action=edit&redlink=1" class="new" title="Hidrón (la páxina nun esiste)">hidrones</a> (<a href="/w/index.php?title=Cati%C3%B3n&action=edit&redlink=1" class="new" title="Catión (la páxina nun esiste)">catión</a> hidróxenu) nel nucleu del Sol nun ye abondu pa superar la barrera de potencial que produz la <a href="/w/index.php?title=Barrera_de_Coulomb&action=edit&redlink=1" class="new" title="Barrera de Coulomb (la páxina nun esiste)">repulsión electromagnética</a> ente ellos. Gracies al efeuto túnel, esiste una pequeña probabilidá de que dellos hidrones devasar, produciendo la fusión de los mesmos y lliberando enerxía en forma de <a href="/wiki/Radiaci%C3%B3n_electromagn%C3%A9tico" title="Radiación electromagnético">radiación electromagnético</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Anque la probabilidá de que se produza esti efeutu túnel ye bien pequeña, la inmensa cantidá de partícules que componen el Sol fai qu'esti efeutu prodúzase constantemente. Esto esplica por qué cuanto más masiva ye una estrella (como una <a href="/wiki/Superxigante_azul" title="Superxigante azul">superxigante azul</a>), más curtia ye la so secuencia principal, una y bones la enerxía cinética de los hidrones ye mayor y, arriendes d'ello, la probabilidá del efeutu túnel tamién.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historia">Historia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Efeutu_t%C3%BAnel&veaction=edit&section=1" title="Editar seición: Historia" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Efeutu_t%C3%BAnel&action=edit&section=1" title="Editar el código fuente de la sección: Historia"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Aproximao, en 1928, <a href="/w/index.php?title=George_Gamow&action=edit&redlink=1" class="new" title="George Gamow (la páxina nun esiste)">George Gamow</a> resolvió la teoría de la <a href="/w/index.php?title=Desintegraci%C3%B3n_Alfa&action=edit&redlink=1" class="new" title="Desintegración Alfa (la páxina nun esiste)">desintegración Alfa</a> de los <a href="/wiki/Nucleu_at%C3%B3micu" title="Nucleu atómicu">nucleos atómicos</a> al traviés de les propiedaes del efeutu túnel. Clásicamente, la <a href="/wiki/Part%C3%ADcula_subat%C3%B3mica" title="Partícula subatómica">partícula</a> atópase confinada al nucleu por cuenta de la ingente cantidá d'enerxía riquida pa escapar al so potencial. Análogamente, ye necesariu un apurra enorme d'enerxía pa esgazar les mesmes del nucleu. Na mecánica cuántica, sicasí, esiste una probabilidá razonable de que la partícula traviese'l potencial enérxicu descritu pol nucleu y llogre escapar de la influencia del mesmu. Gamow resolvió un modelu potencial pa los nucleos atómicos y derivó una rellación ente la vida media de la partícula y la enerxía d'emisión. </p><p>La descomposición alpha tamién foi resuelta coles mesmes por <a href="/w/index.php?title=Ronald_Gurney&action=edit&redlink=1" class="new" title="Ronald Gurney (la páxina nun esiste)">Ronald Gurney</a> y <a href="/w/index.php?title=Edward_Condon&action=edit&redlink=1" class="new" title="Edward Condon (la páxina nun esiste)">Edward Condon</a>. A partir d'entós, consideróse que les partícules pueden introducise nun túnel enerxéticu qu'inclusive traviese'l mesmu nucleu atómicu, dotando de validez completa al modelu enerxéticu pa cualquier aplicación del "efeutu túnel". </p><p>Dempués de l'asistencia de <a href="/wiki/Max_Born" title="Max Born">Max Born</a> al seminariu de Gamow, el primeru reconoció les xeneralidaes o básiques de la mecánica del efeutu. Diose cuenta de que'l "efeutu túnel" nun s'acutaba namái a la <a href="/wiki/F%C3%ADsica_nuclear" title="Física nuclear">física nuclear</a>, sinón qu'aprovía un resultáu xeneral que s'aplica a un conxuntu bien heteroxéneu de sistemes que se rixen poles lleis de la mecánica cuántica. Anguaño, la teoría de los túneles enerxéticos o "efeutu túnel" ta siendo aplicada a la física de la <a href="/wiki/Cosmolox%C3%ADa" title="Cosmoloxía">cosmoloxía</a> del universu. Los sos usos tán, coles mesmes, derivándose a otres árees del progresu teunolóxicu, como la tresmisión en fríu d'electrones, y quiciabes, de forma más importante y reconocida a la física de <a href="/wiki/Semiconductor" title="Semiconductor">semiconductores</a> y <a href="/w/index.php?title=Superconductor&action=edit&redlink=1" class="new" title="Superconductor (la páxina nun esiste)">superconductores</a>. Fenómenos como la emisión de campu, vital pa les <a href="/w/index.php?title=Memoria_flash&action=edit&redlink=1" class="new" title="Memoria flash (la páxina nun esiste)">memories flash</a> son resueltos cuánticamente al traviés de les consecuencies del efeutu túnel. Esti efeutu tamién ye un recursu p'ampliar l'escape na electrónica d'Integración a Bien Altes Escales o <a href="/w/index.php?title=VLSI&action=edit&redlink=1" class="new" title="VLSI (la páxina nun esiste)">VLSI</a> y resulta nel substancial poder de drenado y efeutu de calentamientu que mina la <a href="/w/index.php?title=Teunolox%C3%ADa_m%C3%B3vil&action=edit&redlink=1" class="new" title="Teunoloxía móvil (la páxina nun esiste)">teunoloxía móvil</a> d'alta velocidá. </p><p><br /> Otres aplicaciones son el <a href="/w/index.php?title=Microscopiu_d%27efeutu_t%C3%BAnel&action=edit&redlink=1" class="new" title="Microscopiu d'efeutu túnel (la páxina nun esiste)">microscopiu d'efeutu túnel</a> (electrón-túnel) que puede presentar y resolver oxetos que son bien pequeños pa ser visualizaos al traviés de microscopios convencionales. Estos artificios superen les llimitaciones de los microscopios ópticos; <a href="/w/index.php?title=Aberraci%C3%B3n_en_sistemes_%C3%B3pticos&action=edit&redlink=1" class="new" title="Aberración en sistemes ópticos (la páxina nun esiste)">aberración visual</a>, llendes de <a href="/wiki/Llonxit%C3%BA_d%27onda" title="Llonxitú d'onda">llonxitú d'onda</a> realizando un barríu de superficie sobre l'oxetu con electrones "tuneladores". </p><p>Ye notable comprobar que demostró tratase d'un efeutu que se lleva a cabu naturalmente poles <a href="/wiki/Enzimes" class="mw-redirect" title="Enzimes">enzimes</a> para <a href="/wiki/Cat%C3%A1lisis" title="Catálisis">catalizar</a> reacciones químiques, y demostróse que, estes mesmes, son avezadas al so usu a la de tresferir dos talos electrones y átomos nucleares, como'l <a href="/wiki/Hidr%C3%B3xenu" title="Hidróxenu">hidróxenu</a> y el <a href="/wiki/Deuteriu" title="Deuteriu">deuteriu</a>. Tamién se reparó na enzima (GOx) (EC 1.1.3.4) que los nucleos d'osíxenu pueden travesar túneles enerxéticos baxu condicionantes fisiolóxicos. </p> <div class="mw-heading mw-heading2"><h2 id="Esplicación_simplificada"><span id="Esplicaci.C3.B3n_simplificada"></span>Esplicación simplificada</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Efeutu_t%C3%BAnel&veaction=edit&section=2" title="Editar seición: Esplicación simplificada" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Efeutu_t%C3%BAnel&action=edit&section=2" title="Editar el código fuente de la sección: Esplicación simplificada"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheru:QuantumTunnel.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/QuantumTunnel.jpg/300px-QuantumTunnel.jpg" decoding="async" width="300" height="253" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/QuantumTunnel.jpg/450px-QuantumTunnel.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/QuantumTunnel.jpg/600px-QuantumTunnel.jpg 2x" data-file-width="681" data-file-height="575" /></a><figcaption>Fig.2</figcaption></figure> <p>La diagrama (Fig.2) compara l'efeutu de túnel col movimientu clásicu d'un oxetu. Por analoxía cola <a href="/wiki/Graved%C3%A1" title="Gravedá">gravedá</a>, l'oxetu tiende a movese en direición al centru de la tierra. Clásicamente, p'algamar l'estáu mínimu, tien d'aprovise con enerxía adicional. So la llei de la mecánica cuántica, sicasí, l'oxetu puede dacuando "travesar" l'estáu enerxéticu representáu poles dos pendientes y la cresta hasta llograr un estáu de mínimu de potencial enerxéticu. Considerando un móvil que circula a lo llargo de la trayeutoria que describe una <a href="/w/index.php?title=Vaguada_(xeomorfolox%C3%ADa)&action=edit&redlink=1" class="new" title="Vaguada (xeomorfoloxía) (la páxina nun esiste)">vaguada</a> (Pa los propósitos de la dilucidación, discriminar fuercies adicionales a la gravedá). Dizse que'l mesmu, atopar a 500 metros sobre'l nivel del mar, el visu del monte, simbolizada por una cresta enerxética, algama los 1000 metros, y el planu más allá de la mesma, atopar al altor del mar. Toa instancia o entidá material que conocemos tiende a la so <a href="/w/index.php?title=Est%C3%A1u_fundamental_(f%C3%ADsica)&action=edit&redlink=1" class="new" title="Estáu fundamental (física) (la páxina nun esiste)">nivel mínimu enerxéticu</a> (esto ye, mayor <a href="/wiki/Entrop%C3%ADa" title="Entropía">entropía</a>, polo que l'oxetu va tratar de baxar tanto como seya posible). Na mecánica clásica, mientres una posición del planu seya energéticamente menor qu'aquella qu'ocupa'l móvil, ensin compromisu ulterior coles fuercies añadíes al sistema, esti nun va tener la capacidá de por sí p'algamar esa posición. Sicasí si esistiera un túnel comunicante ente dambos lladrales del monte, el móvil moveríase al traviés d'ella ensin la necesidá d'una enerxía suplementaria a la mesma gravedá. N'aplicación a una partícula que se rixe so los preceptos de la mecánica clásica, esto ye consideráu "tunelado cuánticu".<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Nótese que se trata d'un efeutu válidu n'escales <a href="/w/index.php?title=Fenomenolox%C3%ADa_(ciencia)&action=edit&redlink=1" class="new" title="Fenomenoloxía (ciencia) (la páxina nun esiste)">fenomenológicas</a> desaxeradamente mínimes, xeneralmente, solo puede ser reparáu cuando esiste un intercambiu enerxéticu ente partícules de tamañu atómicu o más amenorgaes, nes cualos el potencial del intercambiu o tresvase coles fuercies qu'ello arreya, tresformar nun fenómenu notablemente más complexu, y nel que nun esisten vasos comunicantes ente túneles de percorríu creciente. Esti fenómenu, como s'espunxo antes, solo dexa graduar la enerxía del espaciu que percuerre la partícula de forma decreciente y acordies cola <a href="/w/index.php?title=Segunda_llei_de_la_termodin%C3%A1mica&action=edit&redlink=1" class="new" title="Segunda llei de la termodinámica (la páxina nun esiste)">segunda llei de la termodinámica</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Cálculu_en_sistemes_semiclásicos"><span id="C.C3.A1lculu_en_sistemes_semicl.C3.A1sicos"></span>Cálculu en sistemes semiclásicos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Efeutu_t%C3%BAnel&veaction=edit&section=3" title="Editar seición: Cálculu en sistemes semiclásicos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Efeutu_t%C3%BAnel&action=edit&section=3" title="Editar el código fuente de la sección: Cálculu en sistemes semiclásicos"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consideremos la forma atemporal de la <a href="/wiki/Ecuaci%C3%B3n_de_Schr%C3%B6dinger" title="Ecuación de Schrödinger">ecuación de Schrödinger</a> pa una partícula unidimensional, so la influencia d'una "llomba" potencial. <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(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ab3e825c2bf9c80d11d12e070a4626d48e03c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.926ex; height:2.843ex;" alt="{\displaystyle V(x)}"></span>. </p> <style data-mw-deduplicate="TemplateStyles:r4219090">.mw-parser-output .ecuacion{padding:5px 10px;background-color:var(--background-color-base);color:var(--color-base);margin-left:30px;margin-bottom:0.8em;margin-top:0.5em;min-width:50%}.mw-parser-output .ecuacion .referencia{float:right;width:10%;text-align:end}.mw-parser-output .ecuacion cite{font-style:normal}</style><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\Psi (x)+V(x)\Psi (x)=Y\Psi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Y</mi> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\Psi (x)+V(x)\Psi (x)=Y\Psi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4921c5cdaf4bfacbd49c34aca13300e42ed7a3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:37.763ex; height:6.009ex;" alt="{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\Psi (x)+V(x)\Psi (x)=Y\Psi (x)}"></span> </p> </blockquote> <p>Agora, recuperemos la función 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 \Psi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8411ce878bfee5fba76012cdf74e18f8a0eba447" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.947ex; height:2.843ex;" alt="{\displaystyle \Psi (x)}"></span> como esponencial d'una función. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (x)=y^{\Phi (x)},\ \mathrm {con} \quad \Phi ''(x)+\Phi '(x)^{2}={\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> </mrow> <mspace width="1em" /> <msup> <mi mathvariant="normal">Φ</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi mathvariant="normal">Φ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Y</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (x)=y^{\Phi (x)},\ \mathrm {con} \quad \Phi ''(x)+\Phi '(x)^{2}={\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afb0e8d3101568721e07aadc944be028895c969f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:55.527ex; height:5.509ex;" alt="{\displaystyle \Psi (x)=y^{\Phi (x)},\ \mathrm {con} \quad \Phi ''(x)+\Phi '(x)^{2}={\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right).}"></span> </p> </blockquote> <p>Dixebramos <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 \Phi '(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Φ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi '(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6608bb3c4e9e8aae7399b4f57fe36df25e35f2c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.502ex; height:3.009ex;" alt="{\displaystyle \Phi '(x)}"></span> en dos tales partes reales ya imaxinaries, emplegando pa ello les funciones de variable real A y B. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><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 {\begin{cases}\Phi '(x)=A(x)+iB(x)\\A'(x)+A(x)^{2}-B(x)^{2}={\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mi mathvariant="normal">Φ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>B</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Y</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\Phi '(x)=A(x)+iB(x)\\A'(x)+A(x)^{2}-B(x)^{2}={\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8babfbedc4186b9d45a355859bd3e2ae8ca3e43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.701ex; height:7.509ex;" alt="{\displaystyle {\begin{cases}\Phi '(x)=A(x)+iB(x)\\A'(x)+A(x)^{2}-B(x)^{2}={\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)\end{cases}}}"></span>, </p> </blockquote> <p>porque la parte imaxinaria pura sume por cuenta de la evaluación real del segundu miembru: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\left(B'(x)-2A(x)B(x)\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\left(B'(x)-2A(x)B(x)\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a22568ad4a294053960b171562cc6e327f6bc4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.635ex; height:3.009ex;" alt="{\displaystyle i\left(B'(x)-2A(x)B(x)\right)=0}"></span> </p> </blockquote> <p>Lo siguiente ye tomar l'aproximamientu semiclásica pa resolver la ecuación. Esto significa que vamos haber d'espandir cada función como una superserie en <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>. De les ecuaciones, inferimos que les superseries tienen d'empezar, cuandoquier un orde de <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 ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2f8d3667641a33c62d4ecd729d8bf01fe32c57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.655ex; height:2.676ex;" alt="{\displaystyle \hbar ^{-1}}"></span> pa satisfaer la parte real de les mesmes. Pero, cuando'l cálculu rique d'una llende clásica razonablemente más precisu, tamién vamos precisar empezar con un orde de magnitú cimera a la constante de Planck como seya posible. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><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(x)={\frac {1}{\hbar }}\sum _{k=0}^{\infty }\hbar ^{k}A_{k}(x),\qquad B(x)={\frac {1}{\hbar }}\sum _{k=0}^{\infty }\hbar ^{k}B_{k}(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em" /> <mi>B</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(x)={\frac {1}{\hbar }}\sum _{k=0}^{\infty }\hbar ^{k}A_{k}(x),\qquad B(x)={\frac {1}{\hbar }}\sum _{k=0}^{\infty }\hbar ^{k}B_{k}(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/662ca0e0936be9c53ddf25097f4022bd42163cd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:51.635ex; height:7.009ex;" alt="{\displaystyle A(x)={\frac {1}{\hbar }}\sum _{k=0}^{\infty }\hbar ^{k}A_{k}(x),\qquad B(x)={\frac {1}{\hbar }}\sum _{k=0}^{\infty }\hbar ^{k}B_{k}(x).}"></span> </p> </blockquote> <p>Les llimitaciones nos términos de mínimu orde queden: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><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}(x)^{2}-B_{0}(x)^{2}=2m\left(V(x)-Y\right),\ \mathrm {con} \ A_{0}(x)B_{0}(x)=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> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Y</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> </mrow> <mtext> </mtext> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{0}(x)^{2}-B_{0}(x)^{2}=2m\left(V(x)-Y\right),\ \mathrm {con} \ A_{0}(x)B_{0}(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6bc88d46e2fabec44c418f8c1a56bcf13ab2abc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.104ex; height:3.176ex;" alt="{\displaystyle A_{0}(x)^{2}-B_{0}(x)^{2}=2m\left(V(x)-Y\right),\ \mathrm {con} \ A_{0}(x)B_{0}(x)=0}"></span> </p> </blockquote> <p>Si l'amplitú varia amodo en comparanza cola fase, especificamos <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}(x)=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> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{0}(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b527cadbac41ac17ee670e03b32615f766630d92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.197ex; height:2.843ex;" alt="{\displaystyle A_{0}(x)=0}"></span> y llogramos: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><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 B_{0}(x)=\pm {\sqrt {2m\left(Y-V(x)\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mi>Y</mi> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{0}(x)=\pm {\sqrt {2m\left(Y-V(x)\right)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/943446033910dedab14cd3fe3da111caece5da89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.127ex; height:4.843ex;" alt="{\displaystyle B_{0}(x)=\pm {\sqrt {2m\left(Y-V(x)\right)}}}"></span> </p> </blockquote> <p>que ye namái válida cuando se dispón de más enerxía que potencial - movimientu clásicu. Dempués aplícase'l mesmu procedimientos nel siguiente orde de la espansión y llogramos: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (x)\approx C{\frac {y^{i\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(Y-V(x)\right)}}+\theta }}{\sqrt[{4}]{{\frac {2m}{\hbar ^{2}}}\left(Y-V(x)\right)}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∫</mo> <mi>d</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>Y</mi> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> <mo>+</mo> <mi>θ</mi> </mrow> </msup> <mroot> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>Y</mi> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (x)\approx C{\frac {y^{i\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(Y-V(x)\right)}}+\theta }}{\sqrt[{4}]{{\frac {2m}{\hbar ^{2}}}\left(Y-V(x)\right)}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe95177c99599c29ec9ee85938723f775fccf5c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:29.848ex; height:10.843ex;" alt="{\displaystyle \Psi (x)\approx C{\frac {y^{i\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(Y-V(x)\right)}}+\theta }}{\sqrt[{4}]{{\frac {2m}{\hbar ^{2}}}\left(Y-V(x)\right)}}}}"></span> </p> </blockquote> <p>Per otra parte, si la fase varia amodo en comparanza cola amplitú, podemos afaer <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 B_{0}(x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{0}(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38399ff8720cd56d9eb7bed45ff312a3e50874fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.218ex; height:2.843ex;" alt="{\displaystyle B_{0}(x)=0}"></span> y llograr: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><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}(x)=\pm {\sqrt {2m\left(V(x)-Y\right)}}}"> <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 stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Y</mi> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{0}(x)=\pm {\sqrt {2m\left(V(x)-Y\right)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2cb5aaf096e5eef75d9a534b2266a261174789c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.106ex; height:4.843ex;" alt="{\displaystyle A_{0}(x)=\pm {\sqrt {2m\left(V(x)-Y\right)}}}"></span> </p> </blockquote> <p>que ye válidu solu si tien mayor potencia qu'energía - movimiento tunelado. Resolviendo la siguiente espansión con un orde cimeru, llogramos: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (x)\approx {\frac {C_{+}y^{+\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)}}}+C_{-}y^{-\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)}}}}{\sqrt[{4}]{{\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mo>∫</mo> <mi>d</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Y</mi> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mrow> </msup> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>∫</mo> <mi>d</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Y</mi> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mrow> </msup> </mrow> <mroot> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (x)\approx {\frac {C_{+}y^{+\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)}}}+C_{-}y^{-\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)}}}}{\sqrt[{4}]{{\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b67b4517f58c8bf4877fdf81e4b39cfc704917b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:53.79ex; height:10.843ex;" alt="{\displaystyle \Psi (x)\approx {\frac {C_{+}y^{+\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)}}}+C_{-}y^{-\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)}}}}{\sqrt[{4}]{{\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)}}}}"></span> </p> </blockquote> <p>Ye aparente pol denominador, que dambes soluciones averaes alloñar del puntu de combadura clásicu <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 Y=V(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=V(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5038f303e977282e6555f0edc634e55deefa1f4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.798ex; height:2.843ex;" alt="{\displaystyle Y=V(x)}"></span>. Lo que tenemos son les soluciones averaes más allá del potencial de la "llomba" y debaxo de la mesma. Más allá d'esta, la partícula pórtase como una onda llibre - la fase ye trémbole. Debaxo, la partícula sufre cambeos esponenciales na amplitú. </p><p> Nun problema específicu del "efeutu túnel", tendríamos d'abarruntar que l'amplitú de la transición ye proporcional a <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"></p><blockquote class="ecuacion" style="text-align:left"> <p><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 y^{-\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>∫</mo> <mi>d</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Y</mi> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{-\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cda63b1362e80ebe5c5a843a9110d81bebe1a9a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.861ex; height:5.176ex;" alt="{\displaystyle y^{-\int dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)}}}}"></span> </p> </blockquote><p>, polo que, d'esta manera, l'efeutu ta exponencialmente complicáu por llargues esviaciones provenientes de la permisividad motriz clásica. Pero pa completalo, tenemos d'atopar les soluciones averaes ayures y rellacionar los coeficientes pa llograr un aproximamientu global al problema. Emplegamos pa ello les soluciones que s'averen con fundamentu a aquelles topaes antes de los puntos de combadura clásicos <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 Y=V(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=V(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5038f303e977282e6555f0edc634e55deefa1f4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.798ex; height:2.843ex;" alt="{\displaystyle Y=V(x)}"></span>. </p><p>Si <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 x_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"></span> designa a un puntu de cruvatura, y por cuenta de que asítiense próximos <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 Y=V(x_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mi>V</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=V(x_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba23ec53d80e8ba8ed77d063f130b0300eed6b89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.852ex; height:2.843ex;" alt="{\displaystyle Y=V(x_{1})}"></span>, puede espandise <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 2m/\hbar ^{2}\left(V(x)-Y\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Y</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2m/\hbar ^{2}\left(V(x)-Y\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb32fce521d67fd6681af3359df2430a679517b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.478ex; height:3.176ex;" alt="{\displaystyle 2m/\hbar ^{2}\left(V(x)-Y\right)}"></span> nuna serie de Taylor: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><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 {2m}{\hbar ^{2}}}\left(V(x)-Y\right)=v_{1}(x-x_{1})+v_{2}(x-x_{1})^{2}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)=v_{1}(x-x_{1})+v_{2}(x-x_{1})^{2}+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26644fe24ee85ee350a26299f03e6f8dada96f01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:49.422ex; height:5.509ex;" alt="{\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)=v_{1}(x-x_{1})+v_{2}(x-x_{1})^{2}+\cdots }"></span> </p> </blockquote> <p>Aproximémonos namái al orde llinial <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 {2m}{\hbar ^{2}}}\left(V(x)-Y\right)=v_{1}(x-x_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)=v_{1}(x-x_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58f1f45d14a2358db9f9f93b5e6cdb838194bc9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:29.419ex; height:5.509ex;" alt="{\displaystyle {\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)=v_{1}(x-x_{1})}"></span> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><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 {d^{2}}{dx^{2}}}\Psi (x)=v_{1}(x-x_{1})\Psi (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)=v_{1}(x-x_{1})\Psi (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ce9f669a3efde36cabdcc0130dded83764e51ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:27.974ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)=v_{1}(x-x_{1})\Psi (x)}"></span> </p> </blockquote> <p>Esta ecuación diferencial paez sospechosa y engañosamente simple. Les sos soluciones son <a href="/w/index.php?title=Funci%C3%B3n_de_Airy&action=edit&redlink=1" class="new" title="Función de Airy (la páxina nun esiste)">funciones de Airy</a>: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (x)=C_{A}Ai\left({\sqrt[{3}]{v_{1}}}(x-x_{1})\right)+C_{B}Bi\left({\sqrt[{3}]{v_{1}}}(x-x_{1})\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mi>A</mi> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>B</mi> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (x)=C_{A}Ai\left({\sqrt[{3}]{v_{1}}}(x-x_{1})\right)+C_{B}Bi\left({\sqrt[{3}]{v_{1}}}(x-x_{1})\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/735a1e9ea0d738f9153e0ceab4ef3171c08a6215" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:52.262ex; height:3.343ex;" alt="{\displaystyle \Psi (x)=C_{A}Ai\left({\sqrt[{3}]{v_{1}}}(x-x_{1})\right)+C_{B}Bi\left({\sqrt[{3}]{v_{1}}}(x-x_{1})\right)}"></span> </p> </blockquote> <p>Supuestamente, esta solución tendría de coneutar les soluciones topaes pa los puntos del espaciu allende les crestes y debaxo del sistema. Daos dos coeficientes nun llau del puntu de combadura, tendríamos de poder determinar otros dos coeficientes, al otru llau de la mesma emplegando esta solución local que los conecte. Por esta ende, agora atopemos una rellación ente <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C,\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>,</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C,\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1649a72d1df41b0d37da2d829df4e88268f47e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.891ex; height:2.509ex;" alt="{\displaystyle C,\theta }"></span> y <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{+},C_{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{+},C_{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a1efa6708759eb46829c9536ff715e9a89baba0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.379ex; height:2.509ex;" alt="{\displaystyle C_{+},C_{-}}"></span>. </p><p>Afortunadamente, les funciones de Airy son asíntóticas pa los senos, cosenos y funciones esponenciales, dientro de los mesmos llendes que les definen. La rellación pos, determinar como siguen estes llinies: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <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 C_{+}={\frac {1}{2}}C\cos {\left(\theta -{\frac {\pi }{4}}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>C</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{+}={\frac {1}{2}}C\cos {\left(\theta -{\frac {\pi }{4}}\right)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/748f40b8fd68ca0f00949910a16b48cef170adb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.796ex; height:5.176ex;" alt="{\displaystyle C_{+}={\frac {1}{2}}C\cos {\left(\theta -{\frac {\pi }{4}}\right)}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{-}=-C\sin {\left(\theta -{\frac {\pi }{4}}\right)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>C</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>θ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{-}=-C\sin {\left(\theta -{\frac {\pi }{4}}\right)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba6277d16147999e0517614e3409c4ed6caf3cbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.35ex; height:4.843ex;" alt="{\displaystyle C_{-}=-C\sin {\left(\theta -{\frac {\pi }{4}}\right)}}"></span></dd></dl> </blockquote><p> Agora, podemos construyir soluciones globales y resolver problemes de "tunelación". El coeficiente de tresmisión, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"></p><blockquote class="ecuacion" style="text-align:left"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|{\frac {C_{\mbox{tresmitida}}}{C_{\mbox{reflexada}}}}\right|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>tresmitida</mtext> </mstyle> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>reflexada</mtext> </mstyle> </mrow> </msub> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\frac {C_{\mbox{tresmitida}}}{C_{\mbox{reflexada}}}}\right|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd7eda927fd65b868d98c369995034f0e843c778" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.432ex; height:7.343ex;" alt="{\displaystyle \left|{\frac {C_{\mbox{tresmitida}}}{C_{\mbox{reflexada}}}}\right|^{2}}"></span> </p> </blockquote> <p>pa una partícula "tuneladora" al traviés d'un potencial enérxicu o barrera, llogramos que ten de ser: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4219090"><blockquote class="ecuacion" style="text-align:left"> <p><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^{2}={\frac {y^{-2\int _{x_{1}}^{x_{2}}dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)}}}}{\left(1+{\frac {1}{4}}y^{-2\int _{x_{1}}^{x_{2}}dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)}}}\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>d</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Y</mi> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>d</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>Y</mi> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{2}={\frac {y^{-2\int _{x_{1}}^{x_{2}}dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)}}}}{\left(1+{\frac {1}{4}}y^{-2\int _{x_{1}}^{x_{2}}dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)}}}\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78f7b5a7262b1a2a3126d40588eb33053facf9ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:37.585ex; height:14.176ex;" alt="{\displaystyle T^{2}={\frac {y^{-2\int _{x_{1}}^{x_{2}}dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)}}}}{\left(1+{\frac {1}{4}}y^{-2\int _{x_{1}}^{x_{2}}dx{\sqrt {{\frac {2m}{\hbar ^{2}}}\left(V(x)-Y\right)}}}\right)^{2}}}}"></span> </p> </blockquote> <p>Onde, <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 x_{1},x_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},x_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/188263943645114e27e316cc1be787861e5b67be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.802ex; height:2.009ex;" alt="{\displaystyle x_{1},x_{2}}"></span> nun son sinón los dos puntos de la curva clásicos definíos pola barrera de potencial. Si tomamos la llende clásica de tolos demás parámetros mayores que la constante de Planck, embrivíos como <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 \rightarrow 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar \rightarrow 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf00388b93a754b255edf3d102b97121f59c65de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.083ex; height:2.176ex;" alt="{\displaystyle \hbar \rightarrow 0}"></span>, podemos reparar que'l coeficiente de tresmisión tiende a cero. Esta llende clásica puede fallar virtualmente, pero ye más bono de resolver, como ye'l casu del potencial cuadrático. </p> <div class="mw-heading mw-heading2"><h2 id="Na_cultura_popular">Na cultura popular</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Efeutu_t%C3%BAnel&veaction=edit&section=4" title="Editar seición: Na cultura popular" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Efeutu_t%C3%BAnel&action=edit&section=4" title="Editar el código fuente de la sección: Na cultura popular"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Nel episodiu Drama-Futuru (Future-Drama) de "<a href="/wiki/The_Simpsons" title="The Simpsons">The Simpsons</a>", Homer y Bart traviesen un monte na que puede lleese "Quantum tunnel" (Túnel cuánticu)</li> <li>Nel show de ciencia ficción "<a href="/w/index.php?title=Sliders&action=edit&redlink=1" class="new" title="Sliders (la páxina nun esiste)">Sliders</a>", los personaxes principales viaxen a universos paralelos emplegando'l "efeutu túnel al traviés d'una puerta d'<a href="/w/index.php?title=Paradoxa_d%27Einstein-Podolsky-Rosen&action=edit&redlink=1" class="new" title="Paradoxa d'Einstein-Podolsky-Rosen (la páxina nun esiste)">Einstein-Rosen-Podolsky</a>".</li></ul> <ul><li>Na serie de ciencia ficción "<a href="/w/index.php?title=Zeta_Disconnect&action=edit&redlink=1" class="new" title="Zeta Disconnect (la páxina nun esiste)">Zeta Disconnect</a>", el portón que los personaxes principales utilicen pa viaxar al traviés del tiempu ye referíu como un túnel cuánticu.</li> <li>Nel videoxuegu "<a href="/w/index.php?title=Supreme_Commander&action=edit&redlink=1" class="new" title="Supreme Commander (la páxina nun esiste)">Supreme Commander</a>", los humanos empleguen l'efeutu túnel como mediu de teletransporte, mediu pol cual pueden colonizar árees bien distantes.</li></ul> <ul><li>Na novela <a href="/w/index.php?title=Rescate_nel_tiempu_(novela)&action=edit&redlink=1" class="new" title="Rescate nel tiempu (novela) (la páxina nun esiste)">Rescate nel Tiempu</a> de <a href="/wiki/Michael_Crichton" title="Michael Crichton">Michael Crichton</a>, los personaxes faen usu del efeutu túnel como mediu experimiental pa los viaxes nel tiempu.</li></ul> <ul><li><a href="/w/index.php?title=Kitty_Pryde&action=edit&redlink=1" class="new" title="Kitty Pryde (la páxina nun esiste)">Kitty Pryde</a>, un personaxe de los cómics <a href="/wiki/Marvel" class="mw-redirect" title="Marvel">Marvel</a>, usar pa travesar les parés.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Ver_tamién"><span id="Ver_tami.C3.A9n"></span>Ver tamién</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Efeutu_t%C3%BAnel&veaction=edit&section=5" title="Editar seición: Ver tamién" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Efeutu_t%C3%BAnel&action=edit&section=5" title="Editar el código fuente de la sección: Ver tamién"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Efeutu_Josephson&action=edit&redlink=1" class="new" title="Efeutu Josephson (la páxina nun esiste)">Efeutu Josephson</a></li> <li><a href="/w/index.php?title=SQUID&action=edit&redlink=1" class="new" title="SQUID (la páxina nun esiste)">SQUID</a></li> <li><a href="/w/index.php?title=Diodu_t%C3%BAnel&action=edit&redlink=1" class="new" title="Diodu túnel (la páxina nun esiste)">Diodu túnel</a></li> <li><a href="/w/index.php?title=Aproximamientu_WKB&action=edit&redlink=1" class="new" title="Aproximamientu WKB (la páxina nun esiste)">Aproximamientu WKB</a></li> <li><a href="/w/index.php?title=Microscopiu_d%27efeutu_t%C3%BAnel&action=edit&redlink=1" class="new" title="Microscopiu d'efeutu túnel (la páxina nun esiste)">Microscopiu d'efeutu túnel</a></li> <li><a href="/w/index.php?title=Barrera_de_potencial&action=edit&redlink=1" class="new" title="Barrera de potencial (la páxina nun esiste)">Barrera de potencial</a></li> <li><a href="/w/index.php?title=Barrera_de_potencial_Delta&action=edit&redlink=1" class="new" title="Barrera de potencial Delta (la páxina nun esiste)">Barrera de potencial Delta</a></li> <li><a href="/w/index.php?title=M%C3%A9todu_de_Holstein%E2%80%93Herring&action=edit&redlink=1" class="new" title="Métodu de Holstein–Herring (la páxina nun esiste)">Métodu de Holstein–Herring</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes_y_referencies">Notes y referencies</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Efeutu_t%C3%BAnel&veaction=edit&section=6" title="Editar seición: Notes y referencies" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Efeutu_t%C3%BAnel&action=edit&section=6" title="Editar el código fuente de la sección: Notes y referencies"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r3503771">@media only screen and (max-width:600px){.mw-parser-output .llistaref{column-count:1!important}}</style><div class="llistaref" style="-moz-column-count:2; -webkit-column-count:2; column-count:2; list-style-type: decimal;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><cite style="font-style:normal">Serway;  Vuille (2008). <i>College Physics</i> <b>2</b>. <a href="/wiki/Especial:FuentesDeLibros/978-0-495-55475-2" title="Especial:FuentesDeLibros/978-0-495-55475-2">ISBN 978-0-495-55475-2</a>.</cite></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><span class="citation cita-Journal">«<a rel="nofollow" class="external text" href="http://www.astro.princeton.edu/~gk/A403/fusion.pdf">The physics of fusion in stars</a>». <i>Department of Astrophisical Science</i>. 2011<span class="printonly">. <a rel="nofollow" class="external free" href="http://www.astro.princeton.edu/~gk/A403/fusion.pdf">http://www.astro.princeton.edu/~gk/A403/fusion.pdf</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+physics+of+fusion+in+stars&rft.jtitle=Department+of+Astrophisical+Science&rft.date=2011&rft_id=http%3A%2F%2Fwww.astro.princeton.edu%2F%7Egk%2FA403%2Ffusion.pdf&rfr_id=info:sid/ast.wikipedia.org:Efeutu_t%C3%BAnel"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><cite style="font-style:normal">Hale Bradt (2010). <i><a rel="nofollow" class="external text" href="https://books.google.es/books?id=wnQ2sm04sVsC&pg=PA66&lpg=PA66&dq=kinetic+energy+proton+sun&source=bl&ots=bx8bX7vvIy&sig=zGnhX41xV3MdvDKnKA3PZZx64dM&hl=es&sa=X&ved=0ahUKEwjDiNasm5DLAhWEthoKHamvALY4ChDoAQggMAE#v=onepage&q=kinetic%20energy%20proton%20sun&f=false">Astrophysics Processes: The Physics of Astronomical Phenomena</a></i>. <a href="/wiki/Especial:FuentesDeLibros/9781107677241" title="Especial:FuentesDeLibros/9781107677241">ISBN 9781107677241</a>.</cite></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><span class="citation cita-Journal" id="CITAREFDavies2005">Davies, P. C. W. (2005). «<a rel="nofollow" class="external text" href="http://www.quantum3000.narod.ru/papers/edu/quantum_tunelling.pdf">Quantum tunneling time</a>». <i>American Journal of Physics</i> <b>73</b>:  p. 23. <small><a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<span class="neverexpand"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1119%2F1.1810153">10.1119/1.1810153</a></span></small><span class="printonly">. <a rel="nofollow" class="external free" href="http://www.quantum3000.narod.ru/papers/edu/quantum_tunelling.pdf">http://www.quantum3000.narod.ru/papers/edu/quantum_tunelling.pdf</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Quantum+tunneling+time&rft.jtitle=American+Journal+of+Physics&rft.aulast=Davies&rft.aufirst=P.+C.+W.&rft.au=Davies%2C%26%2332%3BP.+C.+W.&rft.date=2005&rft.volume=73&rft.pages=%26nbsp%3Bp.%26nbsp%3B23&rft_id=info:doi/10.1119%2F1.1810153&rft_id=http%3A%2F%2Fwww.quantum3000.narod.ru%2Fpapers%2Fedu%2Fquantum_tunelling.pdf&rfr_id=info:sid/ast.wikipedia.org:Efeutu_t%C3%BAnel"><span style="display: none;"> </span></span></span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Llectures_y_publicaciones">Llectures y publicaciones</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Efeutu_t%C3%BAnel&veaction=edit&section=7" title="Editar seición: Llectures y publicaciones" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Efeutu_t%C3%BAnel&action=edit&section=7" title="Editar el código fuente de la sección: Llectures y publicaciones"><span>editar la fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite style="font-style:normal">Razavy, Mohsen (2003). <i>Quantum Theory of Tunneling</i>. World Scientific. <a href="/wiki/Especial:FuentesDeLibros/9812380191" class="internal mw-magiclink-isbn">ISBN 981-238-019-1</a>.</cite></li> <li><cite style="font-style:normal">Griffiths, David J. (2004). <i>Introduction to Quantum Mechanics (2nd ed.)</i>. Prentice Hall. <a href="/wiki/Especial:FuentesDeLibros/013805326X" class="internal mw-magiclink-isbn">ISBN 0-13-805326-X</a>.</cite></li> <li><cite style="font-style:normal"><a href="/w/index.php?title=Liboff,_Richard_L.&action=edit&redlink=1" class="new" title="Liboff, Richard L. (la páxina nun esiste)">Liboff, Richard L.</a> (2002). <i>Introductory Quantum Mechanics</i>. Addison-Wesley. <a href="/wiki/Especial:FuentesDeLibros/0805387145" class="internal mw-magiclink-isbn">ISBN 0-8053-8714-5</a>.</cite></li> <li><span class="citation cita-Journal" id="CITAREFVilenkin2003">Vilenkin, Alexander (2003). «<a rel="nofollow" class="external text" href="http://arxiv.org/abs/gr-qc/0210034">Particle creation in a tunneling universe</a>». <i>Phys.Rev. D</i> <b>68</b>:  páxs. 023520. <small><a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<span class="neverexpand"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1103%2FPhysRevD.68.023520">10.1103/PhysRevD.68.023520</a></span></small><span class="printonly">. <a rel="nofollow" class="external free" href="http://arxiv.org/abs/gr-qc/0210034">http://arxiv.org/abs/gr-qc/0210034</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Particle+creation+in+a+tunneling+universe&rft.jtitle=Phys.Rev.+D&rft.aulast=Vilenkin&rft.aufirst=Alexander&rft.au=Vilenkin%2C%26%2332%3BAlexander&rft.date=2003&rft.volume=68&rft.pages=%26nbsp%3Bp%C3%A1xs.%26nbsp%3B023520&rft_id=info:doi/10.1103%2FPhysRevD.68.023520&rft_id=http%3A%2F%2Farxiv.org%2Fabs%2Fgr-qc%2F0210034&rfr_id=info:sid/ast.wikipedia.org:Efeutu_t%C3%BAnel"><span style="display: 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/></a></span> Multimedia:</span> <span class="uid"><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Quantum_tunneling">Quantum tunneling</a></span></span></li></ul> <hr /> <ul><li><b>Identificadores</b></li> <li><span style="white-space:nowrap;"><a href="/wiki/Integrated_Authority_File" class="mw-redirect" title="Integrated Authority File">GND</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4136216-0">4136216-0</a></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Library_of_Congress_Control_Number" class="mw-redirect" title="Library of Congress Control Number">LCCN</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85138672">sh85138672</a></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/National_Diet_Library" class="mw-redirect" title="National Diet Library">NDL</a>:</span> <span class="uid"><a 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href="https://commons.wikimedia.org/wiki/Category:Quantum_tunneling">Quantum tunneling</a></span></span></li></ul> </div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Sacáu de «<a dir="ltr" href="https://ast.wikipedia.org/w/index.php?title=Efeutu_túnel&oldid=4273455">https://ast.wikipedia.org/w/index.php?title=Efeutu_túnel&oldid=4273455</a>»</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categor%C3%ADas" title="Especial:Categorías">Categoríes</a>: <ul><li><a href="/wiki/Categor%C3%ADa:Wikipedia:Revisar_traducci%C3%B3n" title="Categoría:Wikipedia:Revisar traducción">Wikipedia:Revisar traducción</a></li><li><a href="/wiki/Categor%C3%ADa:Mec%C3%A1nica_cu%C3%A1ntica" title="Categoría:Mecánica 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