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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Onzekerheidsrelatie van Heisenberg</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="Ga naar een artikel in een andere taal. Beschikbaar in 69 talen" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-69" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">69 talen</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Onsekerheidsbeginsel" title="Onsekerheidsbeginsel – Afrikaans" lang="af" hreflang="af" data-title="Onsekerheidsbeginsel" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%A8%D8%AF%D8%A3_%D8%A7%D9%84%D8%B1%D9%8A%D8%A8%D8%A9" title="مبدأ الريبة – Arabisch" lang="ar" hreflang="ar" data-title="مبدأ الريبة" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Rellaci%C3%B3n_d%27indeterminaci%C3%B3n_de_Heisenberg" title="Rellación d'indeterminación de Heisenberg – Asturisch" lang="ast" hreflang="ast" data-title="Rellación d'indeterminación de Heisenberg" data-language-autonym="Asturianu" data-language-local-name="Asturisch" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Qeyri-m%C3%BC%C9%99yy%C9%99nlik_prinsipi" title="Qeyri-müəyyənlik prinsipi – Azerbeidzjaans" lang="az" hreflang="az" data-title="Qeyri-müəyyənlik prinsipi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbeidzjaans" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D1%80%D1%8B%D0%BD%D1%86%D1%8B%D0%BF_%D0%BD%D1%8F%D0%B2%D1%8B%D0%B7%D0%BD%D0%B0%D1%87%D0%B0%D0%BD%D0%B0%D1%81%D1%86%D1%96_%D0%93%D0%B5%D0%B9%D0%B7%D0%B5%D0%BD%D0%B1%D0%B5%D1%80%D0%B3%D0%B0" title="Прынцып нявызначанасці Гейзенберга – Belarussisch" lang="be" hreflang="be" data-title="Прынцып нявызначанасці Гейзенберга" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" 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%A1%D1%8A%D0%BE%D1%82%D0%BD%D0%BE%D1%88%D0%B5%D0%BD%D0%B8%D0%B5_%D0%BD%D0%B0_%D0%BD%D0%B5%D0%BE%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%BE%D1%81%D1%82_%D0%BD%D0%B0_%D0%A5%D0%B0%D0%B9%D0%B7%D0%B5%D0%BD%D0%B1%D0%B5%D1%80%D0%B3" title="Съотношение на неопределеност на Хайзенберг – Bulgaars" lang="bg" hreflang="bg" data-title="Съотношение на неопределеност на Хайзенберг" data-language-autonym="Български" data-language-local-name="Bulgaars" 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%85%E0%A6%A8%E0%A6%BF%E0%A6%B6%E0%A7%8D%E0%A6%9A%E0%A6%AF%E0%A6%BC%E0%A6%A4%E0%A6%BE_%E0%A6%A8%E0%A7%80%E0%A6%A4%E0%A6%BF" title="অনিশ্চয়তা নীতি – Bengaals" lang="bn" hreflang="bn" data-title="অনিশ্চয়তা নীতি" data-language-autonym="বাংলা" data-language-local-name="Bengaals" 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/Princip_neodre%C4%91enosti" title="Princip neodređenosti – Bosnisch" lang="bs" hreflang="bs" data-title="Princip neodređenosti" data-language-autonym="Bosanski" data-language-local-name="Bosnisch" 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/Principi_d%27incertesa_de_Heisenberg" title="Principi d'incertesa de Heisenberg – Catalaans" lang="ca" hreflang="ca" data-title="Principi d'incertesa de Heisenberg" data-language-autonym="Català" data-language-local-name="Catalaans" 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/Princip_neur%C4%8Ditosti" title="Princip neurčitosti – Tsjechisch" lang="cs" hreflang="cs" data-title="Princip neurčitosti" data-language-autonym="Čeština" data-language-local-name="Tsjechisch" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A3%C3%A7%C4%83%D0%BC%D1%81%C4%83%D1%80%D0%BB%C4%83%D1%85_%D0%BF%D1%80%D0%B8%D0%BD%D1%86%D0%B8%D0%BF%C4%95" title="Уçăмсăрлăх принципĕ – Tsjoevasjisch" lang="cv" hreflang="cv" data-title="Уçăмсăрлăх принципĕ" data-language-autonym="Чӑвашла" data-language-local-name="Tsjoevasjisch" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Heisenbergs_ubestemthedsrelation" title="Heisenbergs ubestemthedsrelation – Deens" lang="da" hreflang="da" data-title="Heisenbergs ubestemthedsrelation" data-language-autonym="Dansk" data-language-local-name="Deens" 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/Heisenbergsche_Unsch%C3%A4rferelation" title="Heisenbergsche Unschärferelation – Duits" lang="de" hreflang="de" data-title="Heisenbergsche Unschärferelation" data-language-autonym="Deutsch" data-language-local-name="Duits" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%81%CF%87%CE%AE_%CF%84%CE%B7%CF%82_%CE%B1%CF%80%CF%81%CE%BF%CF%83%CE%B4%CE%B9%CE%BF%CF%81%CE%B9%CF%83%CF%84%CE%AF%CE%B1%CF%82" title="Αρχή της απροσδιοριστίας – Grieks" lang="el" hreflang="el" data-title="Αρχή της απροσδιοριστίας" data-language-autonym="Ελληνικά" data-language-local-name="Grieks" 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/Uncertainty_principle" title="Uncertainty principle – Engels" lang="en" hreflang="en" data-title="Uncertainty principle" data-language-autonym="English" data-language-local-name="Engels" 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/Necerteca_principo_de_Heisenberg" title="Necerteca principo de Heisenberg – Esperanto" lang="eo" hreflang="eo" data-title="Necerteca principo de Heisenberg" 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/Relaci%C3%B3n_de_indeterminaci%C3%B3n_de_Heisenberg" title="Relación de indeterminación de Heisenberg – Spaans" lang="es" hreflang="es" data-title="Relación de indeterminación de Heisenberg" data-language-autonym="Español" data-language-local-name="Spaans" 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/M%C3%A4%C3%A4ramatuse_printsiip" title="Määramatuse printsiip – Estisch" lang="et" hreflang="et" data-title="Määramatuse printsiip" data-language-autonym="Eesti" data-language-local-name="Estisch" 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/Heisenbergen_ziurgabetasunaren_printzipioa" title="Heisenbergen ziurgabetasunaren printzipioa – Baskisch" lang="eu" hreflang="eu" data-title="Heisenbergen ziurgabetasunaren printzipioa" data-language-autonym="Euskara" data-language-local-name="Baskisch" 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%A7%D8%B5%D9%84_%D8%B9%D8%AF%D9%85_%D9%82%D8%B7%D8%B9%DB%8C%D8%AA" title="اصل عدم قطعیت – Perzisch" lang="fa" hreflang="fa" data-title="اصل عدم قطعیت" data-language-autonym="فارسی" data-language-local-name="Perzisch" 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/Heisenbergin_ep%C3%A4tarkkuusperiaate" title="Heisenbergin epätarkkuusperiaate – Fins" lang="fi" hreflang="fi" data-title="Heisenbergin epätarkkuusperiaate" data-language-autonym="Suomi" data-language-local-name="Fins" 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/Principe_d%27incertitude" title="Principe d'incertitude – Frans" lang="fr" hreflang="fr" data-title="Principe d'incertitude" data-language-autonym="Français" data-language-local-name="Frans" 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/Prionsabal_%C3%A9iginnteachta_Heisenberg" title="Prionsabal éiginnteachta Heisenberg – Iers" lang="ga" hreflang="ga" data-title="Prionsabal éiginnteachta Heisenberg" data-language-autonym="Gaeilge" data-language-local-name="Iers" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Principio_de_indeterminaci%C3%B3n_de_Heisenberg" title="Principio de indeterminación de Heisenberg – Galicisch" lang="gl" hreflang="gl" data-title="Principio de indeterminación de Heisenberg" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A2%D7%A7%D7%A8%D7%95%D7%9F_%D7%94%D7%90%D7%99-%D7%95%D7%93%D7%90%D7%95%D7%AA" title="עקרון האי-ודאות – Hebreeuws" lang="he" hreflang="he" data-title="עקרון האי-ודאות" data-language-autonym="עברית" data-language-local-name="Hebreeuws" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%A8%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%9A%E0%A4%BF%E0%A4%A4%E0%A4%A4%E0%A4%BE_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4" title="अनिश्चितता सिद्धान्त – Hindi" lang="hi" hreflang="hi" data-title="अनिश्चितता सिद्धान्त" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Heisenbergovo_na%C4%8Delo_neodre%C4%91enosti" title="Heisenbergovo načelo neodređenosti – Kroatisch" lang="hr" hreflang="hr" data-title="Heisenbergovo načelo neodređenosti" data-language-autonym="Hrvatski" data-language-local-name="Kroatisch" 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/Hat%C3%A1rozatlans%C3%A1gi_rel%C3%A1ci%C3%B3" title="Határozatlansági reláció – Hongaars" lang="hu" hreflang="hu" data-title="Határozatlansági reláció" data-language-autonym="Magyar" data-language-local-name="Hongaars" 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%B1%D5%B6%D5%B8%D6%80%D5%B8%D5%B7%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6%D5%B6%D5%A5%D6%80%D5%AB_%D5%BD%D5%AF%D5%A6%D5%A2%D5%B8%D6%82%D5%B6%D6%84" title="Անորոշությունների սկզբունք – Armeens" lang="hy" hreflang="hy" data-title="Անորոշությունների սկզբունք" data-language-autonym="Հայերեն" data-language-local-name="Armeens" 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/Prinsip_ketidakpastian_Heisenberg" title="Prinsip ketidakpastian Heisenberg – Indonesisch" lang="id" hreflang="id" data-title="Prinsip ketidakpastian Heisenberg" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Principio_di_indeterminazione_di_Heisenberg" title="Principio di indeterminazione di Heisenberg – Italiaans" lang="it" hreflang="it" data-title="Principio di indeterminazione di Heisenberg" data-language-autonym="Italiano" data-language-local-name="Italiaans" 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/%E4%B8%8D%E7%A2%BA%E5%AE%9A%E6%80%A7%E5%8E%9F%E7%90%86" title="不確定性原理 – Japans" lang="ja" hreflang="ja" data-title="不確定性原理" data-language-autonym="日本語" data-language-local-name="Japans" 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%90%D0%BD%D1%8B%D2%9B%D1%82%D0%B0%D0%BB%D0%BC%D0%B0%D2%93%D0%B0%D0%BD%D0%B4%D1%8B%D2%9B_%D2%9B%D0%B0%D1%82%D1%8B%D0%BD%D0%B0%D1%81%D1%8B" title="Анықталмағандық қатынасы – Kazachs" lang="kk" hreflang="kk" data-title="Анықталмағандық қатынасы" data-language-autonym="Қазақша" data-language-local-name="Kazachs" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%85%E0%B2%A8%E0%B2%BF%E0%B2%B0%E0%B3%8D%E0%B2%A6%E0%B2%BF%E0%B2%B7%E0%B3%8D%E0%B2%9F%E0%B2%A4%E0%B3%86%E0%B2%AF_%E0%B2%A4%E0%B2%A4%E0%B3%8D%E0%B2%A4%E0%B3%8D%E0%B2%B5" title="ಅನಿರ್ದಿಷ್ಟತೆಯ ತತ್ತ್ವ – Kannada" lang="kn" hreflang="kn" data-title="ಅನಿರ್ದಿಷ್ಟತೆಯ ತತ್ತ್ವ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B6%88%ED%99%95%EC%A0%95%EC%84%B1_%EC%9B%90%EB%A6%AC" title="불확정성 원리 – Koreaans" lang="ko" hreflang="ko" data-title="불확정성 원리" data-language-autonym="한국어" data-language-local-name="Koreaans" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Heizenbergo_neapibr%C4%97%C5%BEtumo_s%C4%85ry%C5%A1is" title="Heizenbergo neapibrėžtumo sąryšis – Litouws" lang="lt" hreflang="lt" data-title="Heizenbergo neapibrėžtumo sąryšis" data-language-autonym="Lietuvių" data-language-local-name="Litouws" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Heizenberga_nenoteikt%C4%ABbas_princips" title="Heizenberga nenoteiktības princips – Lets" lang="lv" hreflang="lv" data-title="Heizenberga nenoteiktības princips" data-language-autonym="Latviešu" data-language-local-name="Lets" 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%9D%D0%B0%D1%87%D0%B5%D0%BB%D0%BE_%D0%BD%D0%B0_%D0%BD%D0%B5%D0%BE%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%BE%D1%81%D1%82" title="Начело на неопределеност – Macedonisch" lang="mk" hreflang="mk" data-title="Начело на неопределеност" data-language-autonym="Македонски" data-language-local-name="Macedonisch" 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%85%E0%B4%A8%E0%B4%BF%E0%B4%B6%E0%B5%8D%E0%B4%9A%E0%B4%BF%E0%B4%A4%E0%B4%A4%E0%B5%8D%E0%B4%B5_%E0%B4%A4%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B5%8D%E0%B4%B5%E0%B4%82" title="അനിശ്ചിതത്വ തത്ത്വം – Malayalam" lang="ml" hreflang="ml" data-title="അനിശ്ചിതത്വ തത്ത്വം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%85%E0%A4%A8%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%9A%E0%A4%BF%E0%A4%A4%E0%A4%A4%E0%A5%87%E0%A4%9A%E0%A5%87_%E0%A4%A4%E0%A4%A4%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B5" title="अनिश्चिततेचे तत्त्व – Marathi" lang="mr" hreflang="mr" data-title="अनिश्चिततेचे तत्त्व" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-mzn mw-list-item"><a href="https://mzn.wikipedia.org/wiki/%D8%B9%D8%AF%D9%85_%D9%82%D8%B7%D8%B9%DB%8C%D8%AA_%D8%A7%D8%B5%D9%84" title="عدم قطعیت اصل – Mazanderani" lang="mzn" hreflang="mzn" data-title="عدم قطعیت اصل" data-language-autonym="مازِرونی" data-language-local-name="Mazanderani" class="interlanguage-link-target"><span>مازِرونی</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Heisenbergs_uskarpleiksrelasjon" title="Heisenbergs uskarpleiksrelasjon – Noors - Nynorsk" lang="nn" hreflang="nn" data-title="Heisenbergs uskarpleiksrelasjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Noors - 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/Heisenbergs_uskarphetsrelasjon" title="Heisenbergs uskarphetsrelasjon – Noors - Bokmål" lang="nb" hreflang="nb" data-title="Heisenbergs uskarphetsrelasjon" data-language-autonym="Norsk bokmål" data-language-local-name="Noors - Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%85%E0%A8%A8%E0%A8%B8%E0%A8%B0%E0%A8%9F%E0%A8%A8%E0%A8%9F%E0%A9%80_%E0%A8%AA%E0%A9%8D%E0%A8%B0%E0%A8%BF%E0%A9%B0%E0%A8%B8%E0%A9%80%E0%A8%AA%E0%A8%B2" title="ਅਨਸਰਟਨਟੀ ਪ੍ਰਿੰਸੀਪਲ – Punjabi" lang="pa" hreflang="pa" data-title="ਅਨਸਰਟਨਟੀ ਪ੍ਰਿੰਸੀਪਲ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Zasada_nieoznaczono%C5%9Bci" title="Zasada nieoznaczoności – Pools" lang="pl" hreflang="pl" data-title="Zasada nieoznaczoności" data-language-autonym="Polski" data-language-local-name="Pools" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AF_%D9%82%D8%B7%D8%B9%DB%8C%D8%AA_%D8%AF_%D9%86%D8%B4%D8%AA%D9%88%D8%A7%D9%84%D9%8A_%D8%A7%D8%B5%D9%84" title="د قطعیت د نشتوالي اصل – Pasjtoe" lang="ps" hreflang="ps" data-title="د قطعیت د نشتوالي اصل" data-language-autonym="پښتو" data-language-local-name="Pasjtoe" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Princ%C3%ADpio_da_incerteza_de_Heisenberg" title="Princípio da incerteza de Heisenberg – Portugees" lang="pt" hreflang="pt" data-title="Princípio da incerteza de Heisenberg" data-language-autonym="Português" data-language-local-name="Portugees" 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/Principiul_incertitudinii" title="Principiul incertitudinii – Roemeens" lang="ro" hreflang="ro" data-title="Principiul incertitudinii" data-language-autonym="Română" data-language-local-name="Roemeens" 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%9F%D1%80%D0%B8%D0%BD%D1%86%D0%B8%D0%BF_%D0%BD%D0%B5%D0%BE%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D1%91%D0%BD%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Принцип неопределённости – Russisch" lang="ru" hreflang="ru" data-title="Принцип неопределённости" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Na%C4%8Delo_neodre%C4%91enosti" title="Načelo neodređenosti – Servo-Kroatisch" lang="sh" hreflang="sh" data-title="Načelo neodređenosti" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Servo-Kroatisch" 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/Uncertainty_principle" title="Uncertainty principle – Simple English" lang="en-simple" hreflang="en-simple" data-title="Uncertainty principle" 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/Heisenbergov_princ%C3%ADp_neur%C4%8Ditosti" title="Heisenbergov princíp neurčitosti – Slowaaks" lang="sk" hreflang="sk" data-title="Heisenbergov princíp neurčitosti" data-language-autonym="Slovenčina" data-language-local-name="Slowaaks" 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/Na%C4%8Delo_nedolo%C4%8Denosti" title="Načelo nedoločenosti – Sloveens" lang="sl" hreflang="sl" data-title="Načelo nedoločenosti" data-language-autonym="Slovenščina" data-language-local-name="Sloveens" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A0%D0%B5%D0%BB%D0%B0%D1%86%D0%B8%D1%98%D0%B5_%D0%BD%D0%B5%D0%BE%D0%B4%D1%80%D0%B5%D1%92%D0%B5%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Релације неодређености – Servisch" lang="sr" hreflang="sr" data-title="Релације неодређености" data-language-autonym="Српски / srpski" data-language-local-name="Servisch" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Os%C3%A4kerhetsprincipen" title="Osäkerhetsprincipen – Zweeds" lang="sv" hreflang="sv" data-title="Osäkerhetsprincipen" data-language-autonym="Svenska" data-language-local-name="Zweeds" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Kanuni_ya_Heisenberg_ya_Utovu_wa_Hakika" title="Kanuni ya Heisenberg ya Utovu wa Hakika – Swahili" lang="sw" hreflang="sw" data-title="Kanuni ya Heisenberg ya Utovu wa Hakika" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%85%E0%AE%B1%E0%AF%81%E0%AE%A4%E0%AE%BF%E0%AE%AF%E0%AE%BF%E0%AE%A9%E0%AF%8D%E0%AE%AE%E0%AF%88%E0%AE%95%E0%AF%8D_%E0%AE%95%E0%AF%8A%E0%AE%B3%E0%AF%8D%E0%AE%95%E0%AF%88" title="அறுதியின்மைக் கொள்கை – Tamil" lang="ta" hreflang="ta" data-title="அறுதியின்மைக் கொள்கை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AB%E0%B8%A5%E0%B8%B1%E0%B8%81%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%84%E0%B8%A1%E0%B9%88%E0%B9%81%E0%B8%99%E0%B9%88%E0%B8%99%E0%B8%AD%E0%B8%99" title="หลักความไม่แน่นอน – Thai" lang="th" hreflang="th" data-title="หลักความไม่แน่นอน" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Prinsipyong_walang_katiyakan" title="Prinsipyong walang katiyakan – Tagalog" lang="tl" hreflang="tl" data-title="Prinsipyong walang katiyakan" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Belirsizlik_ilkesi" title="Belirsizlik ilkesi – Turks" lang="tr" hreflang="tr" data-title="Belirsizlik ilkesi" data-language-autonym="Türkçe" data-language-local-name="Turks" 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%91%D0%B8%D0%BB%D0%B3%D0%B5%D1%81%D0%B5%D0%B7%D0%BB%D0%B5%D0%BA_%D0%BF%D1%80%D0%B8%D0%BD%D1%86%D0%B8%D0%B1%D1%8B" title="Билгесезлек принцибы – Tataars" lang="tt" hreflang="tt" data-title="Билгесезлек принцибы" data-language-autonym="Татарча / tatarça" data-language-local-name="Tataars" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D0%BD%D1%86%D0%B8%D0%BF_%D0%BD%D0%B5%D0%B2%D0%B8%D0%B7%D0%BD%D0%B0%D1%87%D0%B5%D0%BD%D0%BE%D1%81%D1%82%D1%96" title="Принцип невизначеності – Oekraïens" lang="uk" hreflang="uk" data-title="Принцип невизначеності" data-language-autonym="Українська" data-language-local-name="Oekraïens" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Noaniqlik_prinsipi" title="Noaniqlik prinsipi – Oezbeeks" lang="uz" hreflang="uz" data-title="Noaniqlik prinsipi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Oezbeeks" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Nguy%C3%AAn_l%C3%BD_b%E1%BA%A5t_%C4%91%E1%BB%8Bnh" title="Nguyên lý bất định – Vietnamees" lang="vi" hreflang="vi" data-title="Nguyên lý bất định" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamees" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%BC%97%E7%A1%AE%E5%AE%9A%E6%80%A7%E5%8E%9F%E7%90%86" title="弗确定性原理 – Wuyu" lang="wuu" hreflang="wuu" data-title="弗确定性原理" data-language-autonym="吴语" data-language-local-name="Wuyu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%90%D7%95%D7%9E%D7%96%D7%99%D7%9B%D7%A2%D7%A8%D7%A7%D7%99%D7%99%D7%98_%D7%A4%D7%A8%D7%99%D7%A0%D7%A6%D7%99%D7%A4" title="אומזיכערקייט פרינציפ – Jiddisch" lang="yi" hreflang="yi" data-title="אומזיכערקייט פרינציפ" data-language-autonym="ייִדיש" data-language-local-name="Jiddisch" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh badge-Q17437798 badge-goodarticle mw-list-item" title="goed artikel"><a href="https://zh.wikipedia.org/wiki/%E4%B8%8D%E7%A1%AE%E5%AE%9A%E6%80%A7%E5%8E%9F%E7%90%86" title="不确定性原理 – Chinees" lang="zh" hreflang="zh" data-title="不确定性原理" data-language-autonym="中文" data-language-local-name="Chinees" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%B8%AC%E4%B8%8D%E6%BA%96%E5%8E%9F%E7%90%86" title="測不準原理 – Klassiek Chinees" lang="lzh" hreflang="lzh" data-title="測不準原理" data-language-autonym="文言" data-language-local-name="Klassiek Chinees" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%94%94%E7%A2%BA%E5%AE%9A%E5%8E%9F%E7%90%86" title="唔確定原理 – Kantonees" lang="yue" hreflang="yue" data-title="唔確定原理" data-language-autonym="粵語" data-language-local-name="Kantonees" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q44746#sitelinks-wikipedia" title="Taalkoppelingen bewerken" class="wbc-editpage">Koppelingen bewerken</a></span></div> </div> 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id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="nl" dir="ltr"><table class="infobox" cellpadding="1" cellspacing="1" style="width:270px;"> <tbody><tr><th colspan="2" style="text-align:center; background-color:#DCF0FF;"><big> <a href="/wiki/Kwantummechanica" title="Kwantummechanica">Kwantummechanica</a> </big></th></tr><tr><td colspan="2" style="text-align:center; padding-top:7px;padding-bottom:4px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Delta x}\,{\Delta p}\geq {\frac {\hbar }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </mrow> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Delta x}\,{\Delta p}\geq {\frac {\hbar }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f95982f3955dee58787e7d3fa091216cbb51bb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.999ex; height:5.343ex;" alt="{\displaystyle {\Delta x}\,{\Delta p}\geq {\frac {\hbar }{2}}}"></span> </td></tr><tr><td colspan="2" style="text-align:center; padding-top:0px;line-height:150%;"><a class="mw-selflink selflink">Onzekerheidsrelatie</a> <br /> </td></tr><tr><td colspan="2" style="text-align:center; line-height:150%;"><a href="https://en.wikibooks.org/wiki/nl:In_mensentaal/Kwantummechanica" class="extiw" title="wikibooks:nl:In mensentaal/Kwantummechanica">Algemene inleiding...</a> <br /> </td></tr> <tr> <td colspan="2"><div class="mw-collapsible mw-collapsed" style="border:none; font-size:100%;"> <div style="padding:0; text-align:left; background-color:#DCF0FF;"><b>Achtergrond</b></div> <div class="mw-collapsible-content" style="text-align:center;"><a href="/wiki/Klassieke_mechanica" title="Klassieke mechanica">Klassieke mechanica</a> <br /> <a href="/wiki/Interferentie_(natuurkunde)" title="Interferentie (natuurkunde)">Interferentie</a> <br /> <a href="/wiki/Hamiltonformalisme" title="Hamiltonformalisme">Hamiltonformalisme</a></div> </div>
 </td></tr> <tr> <td colspan="2"><div class="mw-collapsible mw-collapsed" style="border:none; font-size:100%;"> <div style="padding:0; text-align:left; background-color:#DCF0FF;"><b>Fundamentele begrippen</b></div> <div class="mw-collapsible-content" style="text-align:center;"><a href="/wiki/Kwantumtoestand" title="Kwantumtoestand">Kwantumtoestand</a> · <a href="/wiki/Golffunctie" title="Golffunctie">Golffunctie</a> · <a href="/wiki/Postulaten_van_de_kwantummechanica" title="Postulaten van de kwantummechanica">Postulaten</a><br /><a href="/wiki/Superpositie_(natuurkunde)" title="Superpositie (natuurkunde)">Superpositie</a> · <a class="mw-selflink selflink">Onzekerheidsprincipe</a> <br /><a href="/wiki/Schr%C3%B6dingervergelijking" title="Schrödingervergelijking">Schrödingervergelijking</a> · <a href="/wiki/Tunneleffect" title="Tunneleffect">Tunneleffect</a> <br /> <a href="/wiki/Uitsluitingsprincipe_van_Pauli" title="Uitsluitingsprincipe van Pauli">Uitsluitingsprincipe</a> <br /> <a href="/wiki/Diracnotatie" title="Diracnotatie">Diracnotatie</a></div> </div>
 </td></tr> <tr> <td colspan="2"><div class="mw-collapsible mw-collapsed" style="border:none; font-size:100%;"> <div style="padding:0; text-align:left; background-color:#DCF0FF;"><b>Gevorderde onderwerpen</b></div> <div class="mw-collapsible-content" style="text-align:center;"><a href="/wiki/Interpretatie_van_de_kwantummechanica" title="Interpretatie van de kwantummechanica">Interpretatie</a> <br /> <a href="/wiki/Klein-Gordonvergelijking" title="Klein-Gordonvergelijking">Klein-Gordonvergelijking</a><br /><a href="/wiki/Diracvergelijking" title="Diracvergelijking">Diracvergelijking</a><br /><a href="/wiki/Kwantumveldentheorie" title="Kwantumveldentheorie">Kwantumveldentheorie</a> <br /> <a href="/wiki/Kwantumgravitatie" title="Kwantumgravitatie">Kwantumgravitatie</a></div> </div>
 </td></tr> <tr> <td colspan="2"><div class="mw-collapsible mw-collapsed" style="border:none; font-size:100%;"> <div style="padding:0; text-align:left; background-color:#DCF0FF;"><b>Experimenten</b></div> <div class="mw-collapsible-content" style="text-align:center;"><a href="/wiki/Schr%C3%B6dingers_kat" title="Schrödingers kat">Schrödingers kat</a> <br /> <a href="/wiki/Tweespletenexperiment" title="Tweespletenexperiment">Tweespletenexperiment</a> <br /> <a href="/wiki/Tunneleffect" title="Tunneleffect">Tunneleffect</a> <br /> <a href="/wiki/Stern-Gerlach-experiment" title="Stern-Gerlach-experiment">Stern-Gerlach-experiment</a></div> </div>
 </td></tr> <tr> <td colspan="2"><div class="mw-collapsible mw-collapsed" style="border:none; font-size:100%;"> <div style="padding:0; text-align:left; background-color:#DCF0FF;"><b>Wetenschappers</b></div> <div class="mw-collapsible-content" style="text-align:center;"><a href="/wiki/Max_Planck" title="Max Planck">Planck</a> · <a href="/wiki/Albert_Einstein_(hoofdbetekenis)" class="mw-redirect" title="Albert Einstein (hoofdbetekenis)">Einstein</a> · <a href="/wiki/Niels_Bohr" title="Niels Bohr">Bohr</a> · <a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Sommerfeld</a> · <a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Bose</a> · <a href="/wiki/Hendrik_Kramers" title="Hendrik Kramers">Kramers</a> · <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a> · <a href="/wiki/Max_Born" title="Max Born">Born</a> · <a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a> · <a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a> · <a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a> · <a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">de Broglie</a> · <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödinger</a> · <a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a> · <a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner</a> · <a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a> · <a href="/wiki/David_Bohm" title="David Bohm">Bohm</a> · <a href="/wiki/Hugh_Everett" title="Hugh Everett">Everett</a> · <a href="/wiki/John_Bell_(natuurkundige)" title="John Bell (natuurkundige)">Bell</a></div> </div> </td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Bestand:Werner_Heisenberg_-_Canonical_commutation_rule_for_position_and_momentum_variables_of_a_particle_-_Uncertainty_principle,_1927.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Werner_Heisenberg_-_Canonical_commutation_rule_for_position_and_momentum_variables_of_a_particle_-_Uncertainty_principle%2C_1927.jpg/260px-Werner_Heisenberg_-_Canonical_commutation_rule_for_position_and_momentum_variables_of_a_particle_-_Uncertainty_principle%2C_1927.jpg" decoding="async" width="260" height="109" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Werner_Heisenberg_-_Canonical_commutation_rule_for_position_and_momentum_variables_of_a_particle_-_Uncertainty_principle%2C_1927.jpg/390px-Werner_Heisenberg_-_Canonical_commutation_rule_for_position_and_momentum_variables_of_a_particle_-_Uncertainty_principle%2C_1927.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/1/10/Werner_Heisenberg_-_Canonical_commutation_rule_for_position_and_momentum_variables_of_a_particle_-_Uncertainty_principle%2C_1927.jpg 2x" data-file-width="437" data-file-height="183" /></a><figcaption>Commutatieregel voor impuls p en plaats q:<br /><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 \quad pq-qp=h/2\pi i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <mi>p</mi> <mi>q</mi> <mo>−</mo> <mi>q</mi> <mi>p</mi> <mo>=</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad pq-qp=h/2\pi i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3656c0b72f215e3181290288561f278c50ca14a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.538ex; height:2.843ex;" alt="{\displaystyle \quad pq-qp=h/2\pi i}"></span><br />Artikel van Werner Heisenberg, 1927.</figcaption></figure> <p>De <b>onzekerheidsrelatie van Heisenberg</b>, ook het <b>onzekerheidsprincipe van Heisenberg</b>, door <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a> in <a href="/wiki/1927" title="1927">1927</a> gepubliceerd, is een van de belangrijkste resultaten van de <a href="/wiki/Kwantummechanica" title="Kwantummechanica">kwantummechanica</a>. De relatie drukt uit dat er zogenaamde <a href="/wiki/Commensurabiliteit" title="Commensurabiliteit">incommensurabele</a> paren van grootheden bestaan, waarvoor geldt dat niet van beide grootheden de waarden tegelijkertijd exact vastgelegd kunnen worden of met een willekeurige mate van nauwkeurigheid bepaald kunnen worden. Een voorbeeld van een dergelijk paar is plaats en <a href="/wiki/Impuls_(natuurkunde)" title="Impuls (natuurkunde)">impuls</a>, een ander voorbeeld is <a href="/wiki/Energie" title="Energie">energie</a> en <a href="/wiki/Tijd" title="Tijd">tijd</a>. </p><p>Elke theorie die een kwantummechanisch systeem beschrijft moet deze relatie tussen een of meer paren <a href="/wiki/Geconjugeerde_operator" class="mw-redirect" title="Geconjugeerde operator">geconjugeerde</a> <a href="/wiki/Observabele" title="Observabele">observabelen</a> bevatten. Sommige fysici spreken liever over een <b>onbepaaldheidsrelatie</b> of over het <b>onbepaaldheidsprincipe</b> dan over een <i>onzekerheidsrelatie</i> als ze over deze fundamentele relatie spreken. </p><p>In <a href="/wiki/Golffunctie" title="Golffunctie">golffuncties</a> die oplossingen zijn van <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödingers</a> <a href="/wiki/Golfvergelijking" title="Golfvergelijking">golfvergelijking</a> hebben de daaruit af te leiden plaats en impuls geen scherp gedefinieerde waarden, maar zijn zij <a href="/wiki/Stochastische_variabele" title="Stochastische variabele">stochastische variabelen</a> met <a href="/wiki/Kansverdeling" title="Kansverdeling">kansverdelingen</a> die impliciet in de golfvergelijking besloten liggen. De kansverdelingen van die twee grootheden hangen met elkaar samen: als de <a href="/wiki/Standaardafwijking" title="Standaardafwijking">standaardafwijking</a> van de ene kleiner wordt, wordt die van de andere automatisch groter. Heisenberg formuleerde de ondergrens voor het product van standaarddeviaties voor de kansverdelingen van plaats en impuls als volgt: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x\Delta p\geq {\frac {h}{4\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mi mathvariant="normal">Δ</mi> <mi>p</mi> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x\Delta p\geq {\frac {h}{4\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d590fae49786ef71895573276f343cee6129eae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.8ex; height:5.343ex;" alt="{\displaystyle \Delta x\Delta p\geq {\frac {h}{4\pi }}}"></span></dd></dl> <p>waarin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3890eb866b6258d7a304fc34c70ee3fb3a81a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.176ex;" alt="{\displaystyle \Delta x}"></span> de onzekerheid (<a href="/wiki/Standaardafwijking" title="Standaardafwijking">standaardafwijking</a>) in de plaats is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0758c326125ad3d8b96e515c7fd69164ec587b81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.105ex; height:2.509ex;" alt="{\displaystyle \Delta p}"></span> de onzekerheid in de impuls 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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> de <a href="/wiki/Constante_van_Planck" title="Constante van Planck">constante van Planck</a>. De waarde van deze constante is gedefinieerd als exact 6,626 070 15 × 10<sup>−34</sup> <a href="/wiki/Joule" title="Joule">J</a><a href="/wiki/Seconde" title="Seconde">s</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> . </p><p>Tegenwoordig schrijft men deze uitdrukking vaak met een aangepaste versie van de constante <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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>, de zogenaamde <a href="/wiki/Constante_van_Dirac" title="Constante van Dirac">constante van Dirac</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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>, <i>h</i>-streep geheten: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar ={\frac {h}{2\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar ={\frac {h}{2\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df05180652a87fe1af83af4bba3402117bd18466" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.736ex; height:5.343ex;" alt="{\displaystyle \hbar ={\frac {h}{2\pi }}}"></span>.</dd></dl> <p>De onzekerheidsrelatie van Heisenberg heeft belangrijke gevolgen in veel takken van de <a href="/wiki/Natuurkunde" title="Natuurkunde">natuurkunde</a>, op subatomaire schaal (<a href="/wiki/Kwantumfysica" title="Kwantumfysica">kwantumfysica</a>). Voor macroscopische voorwerpen, zoals stoelen, huizen of stuifmeelkorrels, geldt de relatie uiteraard ook, maar is de onzekerheid verwaarloosbaar doordat de constante van Planck zo klein is. Een belangrijk gevolg van de onzekerheidsrelatie is dat metingen altijd invloed hebben op het systeem. Wordt bijvoorbeeld zeer exact de plaats van een deeltje gemeten, dan zal hierdoor de impuls, en dus de snelheid, zeer onzeker worden. Een inperking van de plaats heeft hetzelfde effect. Een goed voorbeeld hiervan is een stroom elektronen die op een plaat met een klein gaatje valt. De elektronen die door het gaatje vliegen, hebben een kort moment een zeer exact bepaalde positie in het vlak van de plaat. Hieruit volgt dat hun impuls parallel aan de plaat zeer onzeker is; de stroom elektronen zal uiteen waaieren achter de plaat en vertoont dus <a href="/wiki/Diffractie" title="Diffractie">diffractie</a>. Een elektron heeft dus ook het karakter van een <a href="/wiki/Golf_(natuurkunde)" title="Golf (natuurkunde)">golf</a>. </p><p>Ook voor andere grootheden dan plaats en impuls geldt een vergelijkbaar verband. Van belang is de onzekerheidsrelatie voor <a href="/wiki/Tijd" title="Tijd">tijd</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> en <a href="/wiki/Energie" title="Energie">energie</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta E\Delta t\geq {\frac {h}{4\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>E</mi> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta E\Delta t\geq {\frac {h}{4\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fcd8c041ecc845cc2290ae9c6793520e596401f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.916ex; height:5.343ex;" alt="{\displaystyle \Delta E\Delta t\geq {\frac {h}{4\pi }}}"></span></dd></dl> <p>Dit betekent dat de hoeveelheid energie in een systeem des te onzekerder is naarmate de tijdschaal waarop het systeem varieert kleiner is. Hierdoor kan er ook als het ware energie 'geleend' worden, wat onder meer aanleiding geeft tot het bestaan van <a href="/wiki/Virtueel_deeltje" title="Virtueel deeltje">virtuele deeltjes</a> en het <a href="/wiki/Tunneleffect" title="Tunneleffect">tunneleffect</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Afleiding">Afleiding</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Onzekerheidsrelatie_van_Heisenberg&veaction=edit&section=1" title="Bewerk dit kopje: Afleiding" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Onzekerheidsrelatie_van_Heisenberg&action=edit&section=1" title="De broncode bewerken van de sectie: Afleiding"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In de kwantummechanica zijn plaats <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> en impuls <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> geconjugeerde, <a href="/wiki/Hermitisch" title="Hermitisch">Hermitische</a> <a href="/wiki/Operator_(wiskunde)" title="Operator (wiskunde)">operatoren</a>. Volgens de <a href="/wiki/Ongelijkheid_van_Schwarz" class="mw-redirect" title="Ongelijkheid van Schwarz">ongelijkheid van Schwarz</a> geldt algemeen voor de varianties <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 \sigma ^{2}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/563e81ed7c5ede55b3d8844ea9bc881ec7ad0aa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.524ex; height:3.176ex;" alt="{\displaystyle \sigma ^{2}(x)}"></span> van <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> 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 \sigma ^{2}(p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}(p)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49ef48e80d6280805b461adf23a7c20b2b724b5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.363ex; height:3.176ex;" alt="{\displaystyle \sigma ^{2}(p)}"></span> van <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}(x)\sigma ^{2}(p)\geq |\operatorname {cov} (x,p)|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}(x)\sigma ^{2}(p)\geq |\operatorname {cov} (x,p)|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab753eb3629f4df81d5e2ea34152fd4f00a3f57a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.485ex; height:3.343ex;" alt="{\displaystyle \sigma ^{2}(x)\sigma ^{2}(p)\geq |\operatorname {cov} (x,p)|^{2}}"></span></dd></dl> <p>Zonder de algemeenheid te schaden kan men aannemen dat de verwachtingswaarden gelijk zijn aan 0: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73c5e00878f88ce373a5a1932654d2bcb4dc1177" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.561ex; height:2.176ex;" alt="{\displaystyle \operatorname {E} x=0}"></span> 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 \operatorname {E} p=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mi>p</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} p=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec758aaf38bb65658c87678c4c5dddef860bb78c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.4ex; height:2.509ex;" alt="{\displaystyle \operatorname {E} p=0}"></span></dd></dl> <p>Nu is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2|\operatorname {cov} (x,p)|\geq 2|\mathrm {Im} \ \operatorname {cov} (x,p)|=|\operatorname {cov} (x,p)-\operatorname {cov} (p,x)|=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">m</mi> </mrow> <mtext> </mtext> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2|\operatorname {cov} (x,p)|\geq 2|\mathrm {Im} \ \operatorname {cov} (x,p)|=|\operatorname {cov} (x,p)-\operatorname {cov} (p,x)|=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b01e4a4b48fbb1a43e9b78062758930ec7e79e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.273ex; height:2.843ex;" alt="{\displaystyle 2|\operatorname {cov} (x,p)|\geq 2|\mathrm {Im} \ \operatorname {cov} (x,p)|=|\operatorname {cov} (x,p)-\operatorname {cov} (p,x)|=}"></span> <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 =|\operatorname {E} (xp)-\operatorname {E} (px)|=|\operatorname {E} (xp-px)|=|i\hbar |=\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>p</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>p</mi> <mo>−</mo> <mi>p</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =|\operatorname {E} (xp)-\operatorname {E} (px)|=|\operatorname {E} (xp-px)|=|i\hbar |=\hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cf75d86aba3e6746b02b9d253083ac8797b2501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.673ex; height:2.843ex;" alt="{\displaystyle =|\operatorname {E} (xp)-\operatorname {E} (px)|=|\operatorname {E} (xp-px)|=|i\hbar |=\hbar }"></span>,</dd></dl></dd></dl> <p>zodat volgt: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (x)\sigma (p)\geq {\frac {\hbar }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (x)\sigma (p)\geq {\frac {\hbar }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a49c5d993c92be038c59e7f5fe2f8b18491950b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.018ex; height:5.343ex;" alt="{\displaystyle \sigma (x)\sigma (p)\geq {\frac {\hbar }{2}}}"></span></dd></dl> <p>Daarbij is gebruikgemaakt van het feit dat de operatoren <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> 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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> niet <a href="/wiki/Commutator_(wiskunde)" title="Commutator (wiskunde)">commuteren</a>, wat inhoudt dat hun volgorde belangrijk is. In de kwantummechanica wordt gewerkt met het volgende <a href="/wiki/Axioma" title="Axioma">postulaat</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x,p]=xp-px=i\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>x</mi> <mi>p</mi> <mo>−</mo> <mi>p</mi> <mi>x</mi> <mo>=</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x,p]=xp-px=i\hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a7321d8f2f154308d3592dfd96c45821e437bc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.971ex; height:2.843ex;" alt="{\displaystyle [x,p]=xp-px=i\hbar }"></span></dd></dl> <p>Dit is ook te zien aan de Schrödinger-operatoren in de plaats-ruimte: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (xp-px)\psi =x{\frac {\hbar }{i}}{\frac {\partial }{\partial x}}\psi -{\frac {\hbar }{i}}{\frac {\partial }{\partial x}}(x\psi )=x{\frac {\hbar }{i}}{\frac {\partial }{\partial x}}\psi -{\frac {\hbar }{i}}\psi -x{\frac {\hbar }{i}}{\frac {\partial }{\partial x}}\psi =i\hbar \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mi>p</mi> <mo>−</mo> <mi>p</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mi>i</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mi>i</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mi>i</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mi>i</mi> </mfrac> </mrow> <mi>ψ</mi> <mo>−</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mi>i</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>ψ</mi> <mo>=</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (xp-px)\psi =x{\frac {\hbar }{i}}{\frac {\partial }{\partial x}}\psi -{\frac {\hbar }{i}}{\frac {\partial }{\partial x}}(x\psi )=x{\frac {\hbar }{i}}{\frac {\partial }{\partial x}}\psi -{\frac {\hbar }{i}}\psi -x{\frac {\hbar }{i}}{\frac {\partial }{\partial x}}\psi =i\hbar \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dce4b8c31b0b83e945efa0c02543b231de7ae43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:71.942ex; height:5.509ex;" alt="{\displaystyle (xp-px)\psi =x{\frac {\hbar }{i}}{\frac {\partial }{\partial x}}\psi -{\frac {\hbar }{i}}{\frac {\partial }{\partial x}}(x\psi )=x{\frac {\hbar }{i}}{\frac {\partial }{\partial x}}\psi -{\frac {\hbar }{i}}\psi -x{\frac {\hbar }{i}}{\frac {\partial }{\partial x}}\psi =i\hbar \psi }"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Zie_ook">Zie ook</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Onzekerheidsrelatie_van_Heisenberg&veaction=edit&section=2" title="Bewerk dit kopje: Zie ook" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Onzekerheidsrelatie_van_Heisenberg&action=edit&section=2" title="De broncode bewerken van de sectie: Zie ook"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Principe_van_lokaliteit" title="Principe van lokaliteit">Principe van lokaliteit</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Externe_links">Externe links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Onzekerheidsrelatie_van_Heisenberg&veaction=edit&section=3" title="Bewerk dit kopje: Externe links" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Onzekerheidsrelatie_van_Heisenberg&action=edit&section=3" title="De broncode bewerken van de sectie: Externe links"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r67679320">.mw-parser-output .taalaanduiding{font-family:sans-serif;font-size:85%;cursor:help;color:var(--color-subtle,#555)}.mw-parser-output .taalaanduiding span{border-bottom:1px dotted var(--color-subtle,#555)}</style><span class="taalaanduiding" title="Taal: Engels">(<span>en</span>) </span> <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/qt-uncertainty/">Uitgebreide uitleg</a></li></ul> <div class="toccolours appendix" role="presentation" style="font-size:90%; margin:1em 0 -0.5em; clear:both;"> <div><span style="font-weight:bold">Bronnen, noten en/of referenties</span></div> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external free" href="https://web.archive.org/web/20180429025229/https://www.bipm.org/utils/en/pdf/CGPM/Draft-Resolution-A-EN.pdf">https://web.archive.org/web/20180429025229/https://www.bipm.org/utils/en/pdf/CGPM/Draft-Resolution-A-EN.pdf</a></span> </li> </ol></div></div> </div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Overgenomen van "<a dir="ltr" href="https://nl.wikipedia.org/w/index.php?title=Onzekerheidsrelatie_van_Heisenberg&oldid=60442534">https://nl.wikipedia.org/w/index.php?title=Onzekerheidsrelatie_van_Heisenberg&oldid=60442534</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Categorie:Alles" title="Categorie:Alles">Categorie</a>: <ul><li><a href="/wiki/Categorie:Kwantummechanica" title="Categorie:Kwantummechanica">Kwantummechanica</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Deze pagina is voor het laatst bewerkt op 30 nov 2021 om 13:05.</li> <li id="footer-info-copyright">De tekst is beschikbaar onder de licentie <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.nl">Creative Commons Naamsvermelding/Gelijk delen</a>, er kunnen aanvullende voorwaarden van toepassing zijn. 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