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<span>Mechanisms</span> </div> </a> <button aria-controls="toc-Mechanisms-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>Toggle Mechanisms subsection</span> </button> <ul id="toc-Mechanisms-sublist" class="vector-toc-list"> <li id="toc-Real-valued_wave_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Real-valued_wave_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Real-valued wave functions</span> </div> </a> <ul id="toc-Real-valued_wave_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Between_two_plane_waves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Between_two_plane_waves"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Between two plane waves</span> </div> </a> <ul id="toc-Between_two_plane_waves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Between_two_spherical_waves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Between_two_spherical_waves"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Between two spherical waves</span> </div> </a> <ul id="toc-Between_two_spherical_waves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiple_beams" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiple_beams"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Multiple beams</span> </div> </a> <ul id="toc-Multiple_beams-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Complex_valued_wave_functions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Complex_valued_wave_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Complex valued wave functions</span> </div> </a> <button aria-controls="toc-Complex_valued_wave_functions-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>Toggle Complex valued wave functions subsection</span> </button> <ul id="toc-Complex_valued_wave_functions-sublist" class="vector-toc-list"> <li id="toc-Optical_wave_interference" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Optical_wave_interference"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Optical wave interference</span> </div> </a> <ul id="toc-Optical_wave_interference-sublist" class="vector-toc-list"> <li id="toc-Light_source_requirements" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Light_source_requirements"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Light source requirements</span> </div> </a> <ul id="toc-Light_source_requirements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Optical_arrangements" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Optical_arrangements"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2</span> <span>Optical arrangements</span> </div> </a> <ul id="toc-Optical_arrangements-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Quantum_interference" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_interference"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Quantum interference</span> </div> </a> <ul id="toc-Quantum_interference-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-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>Toggle Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Beat" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Beat"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Beat</span> </div> </a> <ul id="toc-Beat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Optical_interferometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Optical_interferometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Optical interferometry</span> </div> </a> <ul id="toc-Optical_interferometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Radio_interferometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Radio_interferometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Radio interferometry</span> </div> </a> <ul id="toc-Radio_interferometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Acoustic_interferometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Acoustic_interferometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Acoustic interferometry</span> </div> </a> <ul id="toc-Acoustic_interferometry-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" 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="Toggle the table of contents" > <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">Toggle the table of contents</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">Wave interference</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="Go to an article in another language. Available in 71 languages" > <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-71" 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">71 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%A8%E1%89%A5%E1%88%AD%E1%88%83%E1%8A%95_%E1%88%98%E1%8C%A0%E1%88%8B%E1%88%88%E1%8D%8D" title="የብርሃን መጠላለፍ – Amharic" lang="am" hreflang="am" data-title="የብርሃን መጠላለፍ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AF%D8%A7%D8%AE%D9%84_%D8%A7%D9%84%D9%85%D9%88%D8%AC%D8%A7%D8%AA" title="تداخل الموجات – Arabic" lang="ar" hreflang="ar" data-title="تداخل الموجات" data-language-autonym="العربية" data-language-local-name="Arabic" 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/Interferencia" title="Interferencia – Asturian" lang="ast" hreflang="ast" data-title="Interferencia" data-language-autonym="Asturianu" data-language-local-name="Asturian" 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/Dal%C4%9Falar%C4%B1n_interferensiyas%C4%B1" title="Dalğaların interferensiyası – Azerbaijani" lang="az" hreflang="az" data-title="Dalğaların interferensiyası" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A4%E0%A6%B0%E0%A6%99%E0%A7%8D%E0%A6%97_%E0%A6%AC%E0%A7%8D%E0%A6%AF%E0%A6%A4%E0%A6%BF%E0%A6%9A%E0%A6%BE%E0%A6%B0" title="তরঙ্গ ব্যতিচার – Bangla" lang="bn" hreflang="bn" data-title="তরঙ্গ ব্যতিচার" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Kan-sia%CC%8Dp" title="Kan-sia̍p – Minnan" lang="nan" hreflang="nan" data-title="Kan-sia̍p" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%86%D0%BD%D1%82%D1%8D%D1%80%D1%84%D0%B5%D1%80%D1%8D%D0%BD%D1%86%D1%8B%D1%8F_%D1%85%D0%B2%D0%B0%D0%BB%D1%8C" title="Інтэрферэнцыя хваль – Belarusian" lang="be" hreflang="be" data-title="Інтэрферэнцыя хваль" data-language-autonym="Беларуская" data-language-local-name="Belarusian" 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%98%D0%BD%D1%82%D0%B5%D1%80%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D1%8F_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Интерференция (физика) – Bulgarian" lang="bg" hreflang="bg" data-title="Интерференция (физика)" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Interferencija_talasa" title="Interferencija talasa – Bosnian" lang="bs" hreflang="bs" data-title="Interferencija talasa" data-language-autonym="Bosanski" data-language-local-name="Bosnian" 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/Interfer%C3%A8ncia_(propagaci%C3%B3_d%27ones)#Interferència_òptica" title="Interferència (propagació d'ones) – Catalan" lang="ca" hreflang="ca" data-title="Interferència (propagació d'ones)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A5%D1%83%D0%BC%D1%81%D0%B5%D0%BD_%D0%B8%D0%BD%D1%82%D0%B5%D1%80%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B9%C4%95" title="Хумсен интерференцийĕ – Chuvash" lang="cv" hreflang="cv" data-title="Хумсен интерференцийĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Interference" title="Interference – Czech" lang="cs" hreflang="cs" data-title="Interference" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Ymyriant" title="Ymyriant – Welsh" lang="cy" hreflang="cy" data-title="Ymyriant" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Interferens" title="Interferens – Danish" lang="da" hreflang="da" data-title="Interferens" data-language-autonym="Dansk" data-language-local-name="Danish" 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/Interferenz_(Physik)" title="Interferenz (Physik) – German" lang="de" hreflang="de" data-title="Interferenz (Physik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Interferents" title="Interferents – Estonian" lang="et" hreflang="et" data-title="Interferents" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%85%CE%BC%CE%B2%CE%BF%CE%BB%CE%AE_(%CF%86%CF%85%CF%83%CE%B9%CE%BA%CE%AE)" title="Συμβολή (φυσική) – Greek" lang="el" hreflang="el" data-title="Συμβολή (φυσική)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Interferencia" title="Interferencia – Spanish" lang="es" hreflang="es" data-title="Interferencia" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Interfero" title="Interfero – Esperanto" lang="eo" hreflang="eo" data-title="Interfero" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Interferentzia" title="Interferentzia – Basque" lang="eu" hreflang="eu" data-title="Interferentzia" data-language-autonym="Euskara" data-language-local-name="Basque" 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%D8%AF%D8%A7%D8%AE%D9%84_%D8%A7%D9%85%D9%88%D8%A7%D8%AC" title="تداخل امواج – Persian" lang="fa" hreflang="fa" data-title="تداخل امواج" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Interf%C3%A9rence" title="Interférence – French" lang="fr" hreflang="fr" data-title="Interférence" data-language-autonym="Français" data-language-local-name="French" 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/Trasna%C3%ADocht" title="Trasnaíocht – Irish" lang="ga" hreflang="ga" data-title="Trasnaíocht" data-language-autonym="Gaeilge" data-language-local-name="Irish" 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/Interferencia" title="Interferencia – Galician" lang="gl" hreflang="gl" data-title="Interferencia" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B0%84%EC%84%AD_(%ED%8C%8C%EB%8F%99_%EC%A0%84%ED%8C%8C)" title="간섭 (파동 전파) – Korean" lang="ko" hreflang="ko" data-title="간섭 (파동 전파)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%AC%D5%AB%D6%84%D5%B6%D5%A5%D6%80%D5%AB_%D5%AB%D5%B6%D5%BF%D5%A5%D6%80%D6%86%D5%A5%D6%80%D5%A5%D5%B6%D6%81%D5%AB%D5%A1" title="Ալիքների ինտերֆերենցիա – Armenian" lang="hy" hreflang="hy" data-title="Ալիքների ինտերֆերենցիա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A5%8D%E0%A4%AF%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A4%B0%E0%A4%A3_(%E0%A4%A4%E0%A4%B0%E0%A4%82%E0%A4%97%E0%A5%8B%E0%A4%82_%E0%A4%95%E0%A4%BE)" title="व्यतिकरण (तरंगों का) – Hindi" lang="hi" hreflang="hi" data-title="व्यतिकरण (तरंगों का)" 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/Interferencija_valova" title="Interferencija valova – Croatian" lang="hr" hreflang="hr" data-title="Interferencija valova" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Interfero" title="Interfero – Ido" lang="io" hreflang="io" data-title="Interfero" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Interferensi" title="Interferensi – Indonesian" lang="id" hreflang="id" data-title="Interferensi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Interferentia" title="Interferentia – Interlingua" lang="ia" hreflang="ia" data-title="Interferentia" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Interferenza_(fisica)" title="Interferenza (fisica) – Italian" lang="it" hreflang="it" data-title="Interferenza (fisica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%AA%D7%90%D7%91%D7%9B%D7%95%D7%AA" title="התאבכות – Hebrew" lang="he" hreflang="he" data-title="התאבכות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B5%E0%B3%8D%E0%B2%AF%E0%B2%A4%E0%B2%BF%E0%B2%95%E0%B2%B0%E0%B2%A3" title="ವ್ಯತಿಕರಣ – Kannada" lang="kn" hreflang="kn" data-title="ವ್ಯತಿಕರಣ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Ent%C3%A8ferans" title="Entèferans – Haitian Creole" lang="ht" hreflang="ht" data-title="Entèferans" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Interference" title="Interference – Latvian" lang="lv" hreflang="lv" data-title="Interference" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Interferenz_(Physik)" title="Interferenz (Physik) – Luxembourgish" lang="lb" hreflang="lb" data-title="Interferenz (Physik)" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Interferencija" title="Interferencija – Lithuanian" lang="lt" hreflang="lt" data-title="Interferencija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Interferencia" title="Interferencia – Hungarian" lang="hu" hreflang="hu" data-title="Interferencia" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%91%D1%80%D0%B0%D0%BD%D0%BE%D0%B2%D0%B0_%D0%B8%D0%BD%D1%82%D0%B5%D1%80%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D1%98%D0%B0" title="Бранова интерференција – Macedonian" lang="mk" hreflang="mk" data-title="Бранова интерференција" data-language-autonym="Македонски" data-language-local-name="Macedonian" 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%B5%E0%B5%8D%E0%B4%AF%E0%B4%A4%E0%B4%BF%E0%B4%95%E0%B4%B0%E0%B4%A3%E0%B4%82" title="വ്യതികരണം – Malayalam" lang="ml" hreflang="ml" data-title="വ്യതികരണം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Interferens" title="Interferens – Malay" lang="ms" hreflang="ms" data-title="Interferens" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D1%80%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86" title="Интерференц – Mongolian" lang="mn" hreflang="mn" data-title="Интерференц" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Interferentie_(natuurkunde)" title="Interferentie (natuurkunde) – Dutch" lang="nl" hreflang="nl" data-title="Interferentie (natuurkunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B9%B2%E6%B8%89_(%E7%89%A9%E7%90%86%E5%AD%A6)" title="干渉 (物理学) – Japanese" lang="ja" hreflang="ja" data-title="干渉 (物理学)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Interferens" title="Interferens – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Interferens" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Interferens_i_b%C3%B8lgjemekanikk" title="Interferens i bølgjemekanikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Interferens i bølgjemekanikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Tolqin_interferensiyasi" title="Tolqin interferensiyasi – Uzbek" lang="uz" hreflang="uz" data-title="Tolqin interferensiyasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%87%E0%A9%B0%E0%A8%9F%E0%A9%8D%E0%A8%B0%E0%A8%AB%E0%A9%87%E0%A8%B0%E0%A9%88%E0%A8%82%E0%A8%B8_(%E0%A8%A4%E0%A8%B0%E0%A9%B0%E0%A8%97_%E0%A8%B8%E0%A9%B0%E0%A8%9A%E0%A8%BE%E0%A8%B0)" title="ਇੰਟ੍ਰਫੇਰੈਂਸ (ਤਰੰਗ ਸੰਚਾਰ) – Punjabi" lang="pa" hreflang="pa" data-title="ਇੰਟ੍ਰਫੇਰੈਂਸ (ਤਰੰਗ ਸੰਚਾਰ)" 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/Interferencja" title="Interferencja – Polish" lang="pl" hreflang="pl" data-title="Interferencja" data-language-autonym="Polski" data-language-local-name="Polish" 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/Interfer%C3%AAncia" title="Interferência – Portuguese" lang="pt" hreflang="pt" data-title="Interferência" data-language-autonym="Português" data-language-local-name="Portuguese" 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/Interferen%C8%9B%C4%83" title="Interferență – Romanian" lang="ro" hreflang="ro" data-title="Interferență" data-language-autonym="Română" data-language-local-name="Romanian" 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%98%D0%BD%D1%82%D0%B5%D1%80%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D1%8F_%D0%B2%D0%BE%D0%BB%D0%BD" title="Интерференция волн – Russian" lang="ru" hreflang="ru" data-title="Интерференция волн" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Interference" title="Interference – Simple English" lang="en-simple" hreflang="en-simple" data-title="Interference" 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/Interferencia_(vlny)" title="Interferencia (vlny) – Slovak" lang="sk" hreflang="sk" data-title="Interferencia (vlny)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" 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/Interferenca" title="Interferenca – Slovenian" lang="sl" hreflang="sl" data-title="Interferenca" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" 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%98%D0%BD%D1%82%D0%B5%D1%80%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D1%98%D0%B0" title="Интерференција – Serbian" lang="sr" hreflang="sr" data-title="Интерференција" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Interferencija" title="Interferencija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Interferencija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Interferenssi" title="Interferenssi – Finnish" lang="fi" hreflang="fi" data-title="Interferenssi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Interferens_(v%C3%A5gr%C3%B6relse)" title="Interferens (vågrörelse) – Swedish" lang="sv" hreflang="sv" data-title="Interferens (vågrörelse)" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Interperensiya_(paglaganap_ng_alon)" title="Interperensiya (paglaganap ng alon) – Tagalog" lang="tl" hreflang="tl" data-title="Interperensiya (paglaganap ng alon)" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%85%E0%AE%B2%E0%AF%88%E0%AE%95%E0%AE%B3%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%95%E0%AF%81%E0%AE%B1%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%80%E0%AE%9F%E0%AF%81" title="அலைகளின் குறுக்கீடு – Tamil" lang="ta" hreflang="ta" data-title="அலைகளின் குறுக்கீடு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D1%80%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D1%8F_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Интерференция (физика) – Tatar" lang="tt" hreflang="tt" data-title="Интерференция (физика)" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%81%E0%B8%97%E0%B8%A3%E0%B8%81%E0%B8%AA%E0%B8%AD%E0%B8%94" title="การแทรกสอด – Thai" lang="th" hreflang="th" data-title="การแทรกสอด" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Giri%C5%9Fim" title="Girişim – Turkish" lang="tr" hreflang="tr" data-title="Girişim" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%86%D0%BD%D1%82%D0%B5%D1%80%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D1%96%D1%8F_%D1%85%D0%B2%D0%B8%D0%BB%D1%8C" title="Інтерференція хвиль – Ukrainian" lang="uk" hreflang="uk" data-title="Інтерференція хвиль" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AA%D8%AF%D8%A7%D8%AE%D9%84_(%D8%B7%D8%A8%DB%8C%D8%B9%DB%8C%D8%A7%D8%AA)" title="تداخل (طبیعیات) – Urdu" lang="ur" hreflang="ur" data-title="تداخل (طبیعیات)" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Giao_thoa" title="Giao thoa – Vietnamese" lang="vi" hreflang="vi" data-title="Giao thoa" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" 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%B9%B2%E6%B6%89%EF%BC%88%E7%89%A9%E7%90%86%E5%AD%A6%EF%BC%89" title="干涉（物理学） – Wu" lang="wuu" hreflang="wuu" data-title="干涉（物理学）" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%B9%B2%E6%B6%89" title="干涉 – Cantonese" lang="yue" hreflang="yue" data-title="干涉" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%B9%B2%E6%B6%89_(%E7%89%A9%E7%90%86%E5%AD%A6)" title="干涉 (物理学) – Chinese" lang="zh" hreflang="zh" data-title="干涉 (物理学)" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q136980#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Wave_interference" 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Phenomenon resulting from the superposition of two waves</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For interference in radio communications, see <a href="/wiki/Interference_(communication)" title="Interference (communication)">Interference (communication)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Interference pattern" redirects here. For Moiré patterns, see <a href="/wiki/Moir%C3%A9_pattern" title="Moiré pattern">Moiré pattern</a>. For the medical term, see <a href="/wiki/Interference_pattern_(electromyography)" class="mw-redirect" title="Interference pattern (electromyography)">Interference pattern (electromyography)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Technical plainlinks metadata ambox ambox-style ambox-technical" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>may be too technical for most readers to understand</b>.<span class="hide-when-compact"> Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Wave_interference&action=edit">help improve it</a> to <a href="/wiki/Wikipedia:Make_technical_articles_understandable" title="Wikipedia:Make technical articles understandable">make it understandable to non-experts</a>, without removing the technical details.</span> <span class="date-container"><i>(<span class="date">February 2022</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure typeof="mw:File/Frame"><a href="/wiki/File:Interference_of_two_waves.svg" class="mw-file-description"><img alt="When two or more waves travel through a medium and superpose then the resultant intensity do not distributed uniformly in the space. At some places, it is maximum while at some other places it is minimum. This non uniform distribution of intensity or energy of light is known as interference." src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Interference_of_two_waves.svg/365px-Interference_of_two_waves.svg.png" decoding="async" width="365" height="122" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Interference_of_two_waves.svg/548px-Interference_of_two_waves.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Interference_of_two_waves.svg/730px-Interference_of_two_waves.svg.png 2x" data-file-width="365" data-file-height="122" /></a><figcaption>The interference of two waves. <a href="/wiki/In_phase" class="mw-redirect" title="In phase">In phase</a>: the two lower waves combine (left panel), resulting in a wave of added amplitude <b>(constructive interference)</b>. <a href="/wiki/Out_of_phase" class="mw-redirect" title="Out of phase">Out of phase</a>: (here by 180 degrees), the two lower waves combine (right panel), resulting in a wave of zero amplitude <b>(destructive interference)</b>.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Interfering_surface_waves_on_a_lake.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Interfering_surface_waves_on_a_lake.jpg/220px-Interfering_surface_waves_on_a_lake.jpg" decoding="async" width="220" height="128" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Interfering_surface_waves_on_a_lake.jpg/330px-Interfering_surface_waves_on_a_lake.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Interfering_surface_waves_on_a_lake.jpg/440px-Interfering_surface_waves_on_a_lake.jpg 2x" data-file-width="2973" data-file-height="1734" /></a><figcaption>Interfering water waves on the surface of a lake</figcaption></figure> <p>In <a href="/wiki/Physics" title="Physics">physics</a>, <b>interference</b> is a phenomenon in which two <a href="/wiki/Coherence_(physics)" title="Coherence (physics)">coherent</a> <a href="/wiki/Wave" title="Wave">waves</a> are combined by adding their intensities or displacements with due consideration for their <i><a href="/wiki/Phase_difference" class="mw-redirect" title="Phase difference">phase difference</a></i>. The resultant wave may have greater intensity (<i><b>constructive interference</b></i>) or lower amplitude (<i><b>destructive interference</b></i>) if the two waves are in phase or out of phase, respectively. Interference effects can be observed with all types of waves, for example, <a href="/wiki/Light_wave" class="mw-redirect" title="Light wave">light</a>, <a href="/wiki/Radio_wave" title="Radio wave">radio</a>, <a href="/wiki/Sound_wave" class="mw-redirect" title="Sound wave">acoustic</a>, <a href="/wiki/Surface_wave" title="Surface wave">surface water waves</a>, <a href="/wiki/Gravity_wave" title="Gravity wave">gravity waves</a>, or <a href="/wiki/Matter_wave" title="Matter wave">matter waves</a> as well as in loudspeakers as electrical waves. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Etymology">Etymology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wave_interference&action=edit&section=1" title="Edit section: Etymology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The word <i>interference</i> is derived from the <a href="/wiki/Latin" title="Latin">Latin</a> words <i>inter</i> which means "between" and <i>fere</i> which means "hit or strike", and was used in the context of wave superposition by <a href="/wiki/Thomas_Young_(scientist)" title="Thomas Young (scientist)">Thomas Young</a> in 1801.<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><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><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> <div class="mw-heading mw-heading2"><h2 id="Mechanisms">Mechanisms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wave_interference&action=edit&section=2" title="Edit section: Mechanisms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Waventerference.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Waventerference.gif/220px-Waventerference.gif" decoding="async" width="220" height="94" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Waventerference.gif/330px-Waventerference.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Waventerference.gif/440px-Waventerference.gif 2x" data-file-width="690" data-file-height="295" /></a><figcaption>Interference of right traveling (green) and left traveling (blue) waves in Two-dimensional space, resulting in final (red) wave</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Two_sources_interference.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/2/2c/Two_sources_interference.gif" decoding="async" width="200" height="200" class="mw-file-element" data-file-width="200" data-file-height="200" /></a><figcaption>Interference of waves from two point sources.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/3D_Interference_of_Laser_Light_Through_2_Pinholes_Animation.webm/220px-seek%3D3-3D_Interference_of_Laser_Light_Through_2_Pinholes_Animation.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="124" data-durationhint="72" data-mwtitle="3D_Interference_of_Laser_Light_Through_2_Pinholes_Animation.webm" data-mwprovider="wikimediacommons" resource="/wiki/File:3D_Interference_of_Laser_Light_Through_2_Pinholes_Animation.webm"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5f/3D_Interference_of_Laser_Light_Through_2_Pinholes_Animation.webm/3D_Interference_of_Laser_Light_Through_2_Pinholes_Animation.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5f/3D_Interference_of_Laser_Light_Through_2_Pinholes_Animation.webm/3D_Interference_of_Laser_Light_Through_2_Pinholes_Animation.webm.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/5/5f/3D_Interference_of_Laser_Light_Through_2_Pinholes_Animation.webm" type="video/webm; codecs="vp8"" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5f/3D_Interference_of_Laser_Light_Through_2_Pinholes_Animation.webm/3D_Interference_of_Laser_Light_Through_2_Pinholes_Animation.webm.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5f/3D_Interference_of_Laser_Light_Through_2_Pinholes_Animation.webm/3D_Interference_of_Laser_Light_Through_2_Pinholes_Animation.webm.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5f/3D_Interference_of_Laser_Light_Through_2_Pinholes_Animation.webm/3D_Interference_of_Laser_Light_Through_2_Pinholes_Animation.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5f/3D_Interference_of_Laser_Light_Through_2_Pinholes_Animation.webm/3D_Interference_of_Laser_Light_Through_2_Pinholes_Animation.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5f/3D_Interference_of_Laser_Light_Through_2_Pinholes_Animation.webm/3D_Interference_of_Laser_Light_Through_2_Pinholes_Animation.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /></video></span><figcaption>Cropped tomography scan animation of laser light interference passing through two pinholes (side edges).</figcaption></figure> <p>The <a href="/wiki/Superposition_principle#Wave_interference" title="Superposition principle">principle of superposition of waves</a> states that when two or more propagating waves of the same type are incident on the same point, the resultant <a href="/wiki/Amplitude" title="Amplitude">amplitude</a> at that point is equal to the <a href="/wiki/Euclidean_vector#Addition_and_subtraction" title="Euclidean vector">vector sum</a> of the amplitudes of the individual waves.<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> If a <a href="/wiki/Crest_(physics)" class="mw-redirect" title="Crest (physics)">crest</a> of a wave meets a crest of another wave of the same frequency at the same point, then the amplitude is the sum of the individual amplitudes—this is constructive interference. If a crest of one wave meets a trough of another wave, then the amplitude is equal to the difference in the individual amplitudes—this is known as destructive interference. In ideal mediums (water, air are almost ideal) energy is always conserved, at points of destructive interference, the wave amplitudes cancel each other out, and the energy is redistributed to other areas. For example, when two pebbles are dropped in a pond, a pattern is observable; but eventually waves continue, and only when they reach the shore is the energy absorbed away from the medium. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Interference_colours_in_soap_film_1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Interference_colours_in_soap_film_1.jpg/220px-Interference_colours_in_soap_film_1.jpg" decoding="async" width="220" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Interference_colours_in_soap_film_1.jpg/330px-Interference_colours_in_soap_film_1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/Interference_colours_in_soap_film_1.jpg/440px-Interference_colours_in_soap_film_1.jpg 2x" data-file-width="460" data-file-height="306" /></a><figcaption>Photograph of 1.5cm x 1cm region of soap film under white light. Varying film thickness and viewing geometry determine which colours undergo constructive or destructive interference. Small bubbles significantly affect surrounding film thickness.</figcaption></figure> <p>Constructive interference occurs when the <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a> difference between the waves is an <b>even multiple</b> of <span class="texhtml mvar" style="font-style:italic;">π</span> (180°), whereas destructive interference occurs when the difference is an <b>odd multiple</b> of <span class="texhtml mvar" style="font-style:italic;">π</span>. If the difference between the phases is intermediate between these two extremes, then the magnitude of the displacement of the summed waves lies between the minimum and maximum values. </p><p>Consider, for example, what happens when two identical stones are dropped into a still pool of water at different locations. Each stone generates a circular wave propagating outwards from the point where the stone was dropped. When the two waves overlap, the net displacement at a particular point is the sum of the displacements of the individual waves. At some points, these will be in phase, and will produce a maximum displacement. In other places, the waves will be in anti-phase, and there will be no net displacement at these points. Thus, parts of the surface will be stationary—these are seen in the figure above and to the right as stationary blue-green lines radiating from the centre. </p><p>Interference of light is a unique phenomenon in that we can never observe superposition of the EM field directly as we can, for example, in water. Superposition in the EM field is an assumed phenomenon and necessary to explain how two light beams pass through each other and continue on their respective paths. Prime examples of light interference are the famous <a href="/wiki/Double-slit_experiment" title="Double-slit experiment">double-slit experiment</a>, <a href="/wiki/Laser_speckle" class="mw-redirect" title="Laser speckle">laser speckle</a>, <a href="/wiki/Anti-reflective_coating" title="Anti-reflective coating">anti-reflective coatings</a> and <a href="/wiki/Interferometer" class="mw-redirect" title="Interferometer">interferometers</a>. </p><p>In addition to classical wave model for understanding optical interference, quantum <a href="/wiki/Matter_waves" class="mw-redirect" title="Matter waves">matter waves</a> also demonstrate interference. </p> <div class="mw-heading mw-heading3"><h3 id="Real-valued_wave_functions">Real-valued wave functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wave_interference&action=edit&section=3" title="Edit section: Real-valued wave functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above can be demonstrated in one dimension by deriving the formula for the sum of two waves. The equation for the amplitude of a <a href="/wiki/Sinusoid" class="mw-redirect" title="Sinusoid">sinusoidal</a> wave traveling to the right along the x-axis is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{1}(x,t)=A\cos(kx-\omega t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{1}(x,t)=A\cos(kx-\omega t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be96db09cfef0ef64ed985c3d578a5319407df02" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.076ex; height:2.843ex;" alt="{\displaystyle W_{1}(x,t)=A\cos(kx-\omega t)}"></span> where <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is the peak amplitude, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=2\pi /\lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=2\pi /\lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eecfda2366cb3a64ef6d4747e2460284155a1960" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.322ex; height:2.843ex;" alt="{\displaystyle k=2\pi /\lambda }"></span> is the <a href="/wiki/Wavenumber" title="Wavenumber">wavenumber</a> and <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 \omega =2\pi f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =2\pi f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a1bf35d395c2d52391265e4bbda0aed14f52579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.317ex; height:2.509ex;" alt="{\displaystyle \omega =2\pi f}"></span> is the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> of the wave. Suppose a second wave of the same frequency and amplitude but with a different phase is also traveling to the right <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{2}(x,t)=A\cos(kx-\omega t+\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{2}(x,t)=A\cos(kx-\omega t+\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d50d1ca5f2fb190a3801a02d17e0598552a17129" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.437ex; height:2.843ex;" alt="{\displaystyle W_{2}(x,t)=A\cos(kx-\omega t+\varphi )}"></span> where <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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> is the phase difference between the waves in <a href="/wiki/Radian" title="Radian">radians</a>. The two waves will <a href="/wiki/Superposition_principle" title="Superposition principle">superpose</a> and add: the sum of the two waves is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{1}+W_{2}=A[\cos(kx-\omega t)+\cos(kx-\omega t+\varphi )].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>A</mi> <mo stretchy="false">[</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{1}+W_{2}=A[\cos(kx-\omega t)+\cos(kx-\omega t+\varphi )].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdda8db2a7505764bd6168a596b91e4bcb8b57b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.494ex; height:2.843ex;" alt="{\displaystyle W_{1}+W_{2}=A[\cos(kx-\omega t)+\cos(kx-\omega t+\varphi )].}"></span> Using the <a href="/wiki/Trigonometric_identity" class="mw-redirect" title="Trigonometric identity">trigonometric identity</a> for the sum of two cosines: <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="{\textstyle \cos a+\cos b=2\cos \left({a-b \over 2}\right)\cos \left({a+b \over 2}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>a</mi> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <mi>b</mi> <mo>=</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>−</mo> <mi>b</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \cos a+\cos b=2\cos \left({a-b \over 2}\right)\cos \left({a+b \over 2}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e02af5c0e85ab4c2e12b959629ae8a8abf3a7b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.899ex; height:4.843ex;" alt="{\textstyle \cos a+\cos b=2\cos \left({a-b \over 2}\right)\cos \left({a+b \over 2}\right),}"></span> this can be written <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{1}+W_{2}=2A\cos \left({\varphi \over 2}\right)\cos \left(kx-\omega t+{\varphi \over 2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>φ</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>φ</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{1}+W_{2}=2A\cos \left({\varphi \over 2}\right)\cos \left(kx-\omega t+{\varphi \over 2}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c88d0b5d11ed53950262457923f7720d489a3272" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:43.754ex; height:4.843ex;" alt="{\displaystyle W_{1}+W_{2}=2A\cos \left({\varphi \over 2}\right)\cos \left(kx-\omega t+{\varphi \over 2}\right).}"></span> This represents a wave at the original frequency, traveling to the right like its components, whose amplitude is proportional to the cosine of <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 \varphi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi /2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81f148a352d693b6f5361423c3b9ff3016df5f4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.845ex; height:2.843ex;" alt="{\displaystyle \varphi /2}"></span>. </p> <ul><li><i>Constructive interference</i>: If the phase difference is an even multiple of <span class="texhtml mvar" style="font-style:italic;">π</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =\ldots ,-4\pi ,-2\pi ,0,2\pi ,4\pi ,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mo>…</mo> <mo>,</mo> <mo>−</mo> <mn>4</mn> <mi>π</mi> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo>,</mo> <mn>4</mn> <mi>π</mi> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\ldots ,-4\pi ,-2\pi ,0,2\pi ,4\pi ,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f3331966731d925b0b39aa9b376f6a04a17fc87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.412ex; height:2.676ex;" alt="{\displaystyle \varphi =\ldots ,-4\pi ,-2\pi ,0,2\pi ,4\pi ,\ldots }"></span> then <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|\cos(\varphi /2)\right|=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\cos(\varphi /2)\right|=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb64cb963690f8f546d2da4ae9454786a79d049" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.32ex; height:2.843ex;" alt="{\displaystyle \left|\cos(\varphi /2)\right|=1}"></span>, so the sum of the two waves is a wave with twice the amplitude <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{1}+W_{2}=2A\cos(kx-\omega t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{1}+W_{2}=2A\cos(kx-\omega t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed25191587b3613a062f5644cb56a1aa7b324c22" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.315ex; height:2.843ex;" alt="{\displaystyle W_{1}+W_{2}=2A\cos(kx-\omega t)}"></span></li> <li><i>Destructive interference</i>: If the phase difference is an odd multiple of <span class="texhtml mvar" style="font-style:italic;">π</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =\ldots ,-3\pi ,\,-\pi ,\,\pi ,\,3\pi ,\,5\pi ,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mo>…</mo> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mi>π</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>π</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mn>3</mn> <mi>π</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mn>5</mn> <mi>π</mi> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\ldots ,-3\pi ,\,-\pi ,\,\pi ,\,3\pi ,\,5\pi ,\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/089c897185b9232d722a554b84fc9d2846480612" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.968ex; height:2.676ex;" alt="{\displaystyle \varphi =\ldots ,-3\pi ,\,-\pi ,\,\pi ,\,3\pi ,\,5\pi ,\ldots }"></span> then <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 \cos(\varphi /2)=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\varphi /2)=0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/569e1d8cf00b3c8ccb67015aea6e9f56ab493d40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.413ex; height:2.843ex;" alt="{\displaystyle \cos(\varphi /2)=0\,}"></span>, so the sum of the two waves is zero <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{1}+W_{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{1}+W_{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bee4db9be1eb14a76d8a26eec8a8bc562fc22d7f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.597ex; height:2.509ex;" alt="{\displaystyle W_{1}+W_{2}=0}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Between_two_plane_waves">Between two plane waves</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wave_interference&action=edit&section=4" title="Edit section: Between two plane waves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Interference_of_plane_waves_3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Interference_of_plane_waves_3.svg/220px-Interference_of_plane_waves_3.svg.png" decoding="async" width="220" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Interference_of_plane_waves_3.svg/330px-Interference_of_plane_waves_3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Interference_of_plane_waves_3.svg/440px-Interference_of_plane_waves_3.svg.png 2x" data-file-width="360" data-file-height="248" /></a><figcaption>Geometrical arrangement for two plane wave interference</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Interferences_plane_waves.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Interferences_plane_waves.jpg/220px-Interferences_plane_waves.jpg" decoding="async" width="220" height="193" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Interferences_plane_waves.jpg/330px-Interferences_plane_waves.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Interferences_plane_waves.jpg/440px-Interferences_plane_waves.jpg 2x" data-file-width="684" data-file-height="600" /></a><figcaption>Interference fringes in overlapping plane waves</figcaption></figure> <p>A simple form of interference pattern is obtained if two <a href="/wiki/Plane_wave" title="Plane wave">plane waves</a> of the same frequency intersect at an angle. One wave is travelling horizontally, and the other is travelling downwards at an angle θ to the first wave. Assuming that the two waves are in phase at the point <b>B</b>, then the relative phase changes along the <i>x</i>-axis. The phase difference at the point <b>A</b> is given by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \varphi ={\frac {2\pi d}{\lambda }}={\frac {2\pi x\sin \theta }{\lambda }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>d</mi> </mrow> <mi>λ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mi>λ</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \varphi ={\frac {2\pi d}{\lambda }}={\frac {2\pi x\sin \theta }{\lambda }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9427ebee0be42c8504c832c1c2ce6317ab8b3327" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.227ex; height:5.509ex;" alt="{\displaystyle \Delta \varphi ={\frac {2\pi d}{\lambda }}={\frac {2\pi x\sin \theta }{\lambda }}.}"></span> </p><p>It can be seen that the two waves are in phase when </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x\sin \theta }{\lambda }}=0,\pm 1,\pm 2,\ldots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mi>λ</mi> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x\sin \theta }{\lambda }}=0,\pm 1,\pm 2,\ldots ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9ecf407f7d259c8167dc018c21ba88b08b5a2cb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.947ex; height:5.509ex;" alt="{\displaystyle {\frac {x\sin \theta }{\lambda }}=0,\pm 1,\pm 2,\ldots ,}"></span> </p><p>and are half a cycle out of phase when </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x\sin \theta }{\lambda }}=\pm {\frac {1}{2}},\pm {\frac {3}{2}},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mi>λ</mi> </mfrac> </mrow> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x\sin \theta }{\lambda }}=\pm {\frac {1}{2}},\pm {\frac {3}{2}},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38523a91e31be6d32103306aa594aa5948175541" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.389ex; height:5.509ex;" alt="{\displaystyle {\frac {x\sin \theta }{\lambda }}=\pm {\frac {1}{2}},\pm {\frac {3}{2}},\ldots }"></span> </p><p>Constructive interference occurs when the waves are in phase, and destructive interference when they are half a cycle out of phase. Thus, an interference fringe pattern is produced, where the separation of the maxima is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{f}={\frac {\lambda }{\sin \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{f}={\frac {\lambda }{\sin \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/781c6331ee2c45ad7ab68a4c5d1e9846f9f609af" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.613ex; height:5.509ex;" alt="{\displaystyle d_{f}={\frac {\lambda }{\sin \theta }}}"></span> </p><p>and <span class="texhtml"><i>d<sub>f</sub></i></span> is known as the fringe spacing. The fringe spacing increases with increase in <a href="/wiki/Wavelength" title="Wavelength">wavelength</a>, and with decreasing angle <span class="texhtml">θ</span>. </p><p>The fringes are observed wherever the two waves overlap and the fringe spacing is uniform throughout. </p> <div class="mw-heading mw-heading3"><h3 id="Between_two_spherical_waves">Between two spherical waves</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wave_interference&action=edit&section=5" title="Edit section: Between two spherical waves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Wavepanel.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/f/f1/Wavepanel.png" decoding="async" width="210" height="210" class="mw-file-element" data-file-width="210" data-file-height="210" /></a><figcaption>Optical interference between two point sources that have different wavelengths and separations of sources.</figcaption></figure> <p>A <a href="/wiki/Point_source" title="Point source">point source</a> produces a spherical wave. If the light from two point sources overlaps, the interference pattern maps out the way in which the phase difference between the two waves varies in space. This depends on the wavelength and on the separation of the point sources. The figure to the right shows interference between two spherical waves. The wavelength increases from top to bottom, and the distance between the sources increases from left to right. </p><p>When the plane of observation is far enough away, the fringe pattern will be a series of almost straight lines, since the waves will then be almost planar. </p> <div class="mw-heading mw-heading3"><h3 id="Multiple_beams">Multiple beams</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wave_interference&action=edit&section=6" title="Edit section: Multiple beams"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Interference occurs when several waves are added together provided that the phase differences between them remain constant over the observation time. </p><p>It is sometimes desirable for several waves of the same frequency and amplitude to sum to zero (that is, interfere destructively, cancel). This is the principle behind, for example, <a href="/wiki/Three-phase_electric_power" title="Three-phase electric power">3-phase power</a> and the <a href="/wiki/Diffraction_grating" title="Diffraction grating">diffraction grating</a>. In both of these cases, the result is achieved by uniform spacing of the phases. </p><p>It is easy to see that a set of waves will cancel if they have the same amplitude and their phases are spaced equally in angle. Using <a href="/wiki/Phasor" title="Phasor">phasors</a>, each wave can be represented as <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 Ae^{i\varphi _{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ae^{i\varphi _{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ac43f9e9619401fabf68575032a49353a7072c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.666ex; height:2.676ex;" alt="{\displaystyle Ae^{i\varphi _{n}}}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> waves from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26819344e55f5e671c76c07c18eb4291fcec85ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=0}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=N-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=N-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41749647b4bc797ccb5bff075fa4730e5944abc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.56ex; height:2.343ex;" alt="{\displaystyle n=N-1}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{n}-\varphi _{n-1}={\frac {2\pi }{N}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{n}-\varphi _{n-1}={\frac {2\pi }{N}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cecad95e43f4a3cc99ad326dcd6e03fe2bc52131" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.494ex; height:5.176ex;" alt="{\displaystyle \varphi _{n}-\varphi _{n-1}={\frac {2\pi }{N}}.}"></span> </p><p>To show that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{N-1}Ae^{i\varphi _{n}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>A</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{N-1}Ae^{i\varphi _{n}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a3a0579c058f1502b6192901d026576c7bfb4a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.874ex; height:7.343ex;" alt="{\displaystyle \sum _{n=0}^{N-1}Ae^{i\varphi _{n}}=0}"></span> </p><p>one merely assumes the converse, then multiplies both sides by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i{\frac {2\pi }{N}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>N</mi> </mfrac> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i{\frac {2\pi }{N}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78a53a4e178b05f699d2695f4a6686d336163b60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.798ex; height:3.509ex;" alt="{\displaystyle e^{i{\frac {2\pi }{N}}}.}"></span> </p><p>The <a href="/wiki/Fabry%E2%80%93P%C3%A9rot_interferometer" title="Fabry–Pérot interferometer">Fabry–Pérot interferometer</a> uses interference between multiple reflections. </p><p>A <a href="/wiki/Diffraction_grating" title="Diffraction grating">diffraction grating</a> can be considered to be a multiple-beam interferometer; since the peaks which it produces are generated by interference between the light transmitted by each of the elements in the grating; see <a href="/wiki/Interference_vs._diffraction" class="mw-redirect" title="Interference vs. diffraction">interference vs. diffraction</a> for further discussion. </p> <div class="mw-heading mw-heading2"><h2 id="Complex_valued_wave_functions">Complex valued wave functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wave_interference&action=edit&section=7" title="Edit section: Complex valued wave functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Mechanical_wave" title="Mechanical wave">Mechanical</a> and <a href="/wiki/Gravity_waves" class="mw-redirect" title="Gravity waves">gravity waves</a> can be directly observed: they are real-valued wave functions; <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">optical</a> and <a href="/wiki/Matter_waves" class="mw-redirect" title="Matter waves">matter waves</a> cannot be directly observed: they are complex valued <a href="/wiki/Wave_functions" class="mw-redirect" title="Wave functions">wave functions</a>. Some of the differences between real valued and complex valued wave interference include: </p> <ol type="a"> <li>The interference involves different types of mathematical functions: A classical wave is a <a href="/wiki/Real_number" title="Real number">real</a> function representing the displacement from an equilibrium position; an optical or quantum wavefunction is a <a href="/wiki/Complex_number" title="Complex number">complex</a> function. A classical wave at any point can be positive or negative; the quantum probability function is non-negative.</li> <li>Any two different real waves in the same medium interfere; complex waves must be <a href="/wiki/Coherence_(physics)" title="Coherence (physics)">coherent</a> to interfere. In practice this means the wave must come from the same source and have similar frequencies</li> <li>Real wave interference is obtained simply by adding the displacements from equilibrium (or amplitudes) of the two waves; In complex wave interference, we measure the modulus of the wavefunction squared.</li> </ol> <div class="mw-heading mw-heading3"><h3 id="Optical_wave_interference">Optical wave interference</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wave_interference&action=edit&section=8" title="Edit section: Optical wave interference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Optical_flat_interference.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Optical_flat_interference.svg/340px-Optical_flat_interference.svg.png" decoding="async" width="340" height="285" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Optical_flat_interference.svg/510px-Optical_flat_interference.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/42/Optical_flat_interference.svg/680px-Optical_flat_interference.svg.png 2x" data-file-width="1417" data-file-height="1188" /></a><figcaption>Creation of interference fringes by an <a href="/wiki/Optical_flat" title="Optical flat">optical flat</a> on a reflective surface. Light rays from a monochromatic source pass through the glass and reflect off both the bottom surface of the flat and the supporting surface. The tiny gap between the surfaces means the two reflected rays have different path lengths. In addition the ray reflected from the bottom plate undergoes a 180° phase reversal. As a result, at locations <i><b>(a)</b></i> where the path difference is an odd multiple of λ/2, the waves reinforce. At locations <i><b>(b)</b></i> where the path difference is an even multiple of λ/2 the waves cancel. Since the gap between the surfaces varies slightly in width at different points, a series of alternating bright and dark bands, <i>interference fringes</i>, are seen.</figcaption></figure> <p>Because the frequency of light waves (~10<sup>14</sup> Hz) is too high for currently available detectors to detect the variation of the electric field of the light, it is possible to observe only the <a href="/wiki/Intensity_(physics)" title="Intensity (physics)">intensity</a> of an optical interference pattern. The intensity of the light at a given point is proportional to the square of the average amplitude of the wave. This can be expressed mathematically as follows. The displacement of the two waves at a point <span class="texhtml"><b>r</b></span> is: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{1}(\mathbf {r} ,t)=A_{1}(\mathbf {r} )e^{i[\varphi _{1}(\mathbf {r} )-\omega t]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">[</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">]</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{1}(\mathbf {r} ,t)=A_{1}(\mathbf {r} )e^{i[\varphi _{1}(\mathbf {r} )-\omega t]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98aacb5c083f2a9e622b1f61dbdc7e2e5d215f6a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.891ex; height:3.343ex;" alt="{\displaystyle U_{1}(\mathbf {r} ,t)=A_{1}(\mathbf {r} )e^{i[\varphi _{1}(\mathbf {r} )-\omega t]}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{2}(\mathbf {r} ,t)=A_{2}(\mathbf {r} )e^{i[\varphi _{2}(\mathbf {r} )-\omega t]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">[</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">]</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{2}(\mathbf {r} ,t)=A_{2}(\mathbf {r} )e^{i[\varphi _{2}(\mathbf {r} )-\omega t]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/770aa815a1ce0cc0fb915490fbede1bdee62c3dc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.891ex; height:3.343ex;" alt="{\displaystyle U_{2}(\mathbf {r} ,t)=A_{2}(\mathbf {r} )e^{i[\varphi _{2}(\mathbf {r} )-\omega t]}}"></span> </p><p>where <span class="texhtml">A</span> represents the magnitude of the displacement, <span class="texhtml">φ</span> represents the phase and <span class="texhtml">ω</span> represents the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a>. </p><p>The displacement of the summed waves is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(\mathbf {r} ,t)=A_{1}(\mathbf {r} )e^{i[\varphi _{1}(\mathbf {r} )-\omega t]}+A_{2}(\mathbf {r} )e^{i[\varphi _{2}(\mathbf {r} )-\omega t]}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">[</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">]</mo> </mrow> </msup> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">[</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">]</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(\mathbf {r} ,t)=A_{1}(\mathbf {r} )e^{i[\varphi _{1}(\mathbf {r} )-\omega t]}+A_{2}(\mathbf {r} )e^{i[\varphi _{2}(\mathbf {r} )-\omega t]}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b542dc54af62dc120ead82e5091754941451a393" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.886ex; height:3.343ex;" alt="{\displaystyle U(\mathbf {r} ,t)=A_{1}(\mathbf {r} )e^{i[\varphi _{1}(\mathbf {r} )-\omega t]}+A_{2}(\mathbf {r} )e^{i[\varphi _{2}(\mathbf {r} )-\omega t]}.}"></span> </p><p>The intensity of the light at <span class="texhtml"><b>r</b></span> is given by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(\mathbf {r} )=\int U(\mathbf {r} ,t)U^{*}(\mathbf {r} ,t)\,dt\propto A_{1}^{2}(\mathbf {r} )+A_{2}^{2}(\mathbf {r} )+2A_{1}(\mathbf {r} )A_{2}(\mathbf {r} )\cos[\varphi _{1}(\mathbf {r} )-\varphi _{2}(\mathbf {r} )].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>∝</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(\mathbf {r} )=\int U(\mathbf {r} ,t)U^{*}(\mathbf {r} ,t)\,dt\propto A_{1}^{2}(\mathbf {r} )+A_{2}^{2}(\mathbf {r} )+2A_{1}(\mathbf {r} )A_{2}(\mathbf {r} )\cos[\varphi _{1}(\mathbf {r} )-\varphi _{2}(\mathbf {r} )].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7d47a3560c51fa16a533d69ee2523a20b9a159c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:78.48ex; height:5.676ex;" alt="{\displaystyle I(\mathbf {r} )=\int U(\mathbf {r} ,t)U^{*}(\mathbf {r} ,t)\,dt\propto A_{1}^{2}(\mathbf {r} )+A_{2}^{2}(\mathbf {r} )+2A_{1}(\mathbf {r} )A_{2}(\mathbf {r} )\cos[\varphi _{1}(\mathbf {r} )-\varphi _{2}(\mathbf {r} )].}"></span> </p><p>This can be expressed in terms of the intensities of the individual waves as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(\mathbf {r} )=I_{1}(\mathbf {r} )+I_{2}(\mathbf {r} )+2{\sqrt {I_{1}(\mathbf {r} )I_{2}(\mathbf {r} )}}\cos[\varphi _{1}(\mathbf {r} )-\varphi _{2}(\mathbf {r} )].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </msqrt> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(\mathbf {r} )=I_{1}(\mathbf {r} )+I_{2}(\mathbf {r} )+2{\sqrt {I_{1}(\mathbf {r} )I_{2}(\mathbf {r} )}}\cos[\varphi _{1}(\mathbf {r} )-\varphi _{2}(\mathbf {r} )].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f41265d6a2a8696fb6e681c5c64043ad1cfd644" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:55.554ex; height:4.843ex;" alt="{\displaystyle I(\mathbf {r} )=I_{1}(\mathbf {r} )+I_{2}(\mathbf {r} )+2{\sqrt {I_{1}(\mathbf {r} )I_{2}(\mathbf {r} )}}\cos[\varphi _{1}(\mathbf {r} )-\varphi _{2}(\mathbf {r} )].}"></span> </p><p>Thus, the interference pattern maps out the difference in phase between the two waves, with maxima occurring when the phase difference is a multiple of 2<span class="texhtml mvar" style="font-style:italic;">π</span>. If the two beams are of equal intensity, the maxima are four times as bright as the individual beams, and the minima have zero intensity. </p><p>Classically the two waves must have the same <a href="/wiki/Polarization_(waves)" title="Polarization (waves)">polarization</a> to give rise to interference fringes since it is not possible for waves of different polarizations to cancel one another out or add together. Instead, when waves of different polarization are added together, they give rise to a wave of a different <a href="/wiki/Polarization_(waves)#Polarization_state" title="Polarization (waves)">polarization state</a>. </p><p>Quantum mechanically the theories of Paul Dirac and Richard Feynman offer a more modern approach. Dirac showed that every quanta or photon of light acts on its own which he famously stated as "every photon interferes with itself". Richard Feynman showed that by evaluating a path integral where all possible paths are considered, that a number of higher probability paths will emerge. In thin films for example, film thickness which is not a multiple of light wavelength will not allow the quanta to traverse, only reflection is possible. </p> <div class="mw-heading mw-heading4"><h4 id="Light_source_requirements">Light source requirements</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wave_interference&action=edit&section=9" title="Edit section: Light source requirements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The discussion above assumes that the waves which interfere with one another are monochromatic, i.e. have a single frequency—this requires that they are infinite in time. This is not, however, either practical or necessary. Two identical waves of finite duration whose frequency is fixed over that period will give rise to an interference pattern while they overlap. Two identical waves which consist of a narrow spectrum of frequency waves of finite duration (but shorter than their coherence time), will give a series of fringe patterns of slightly differing spacings, and provided the spread of spacings is significantly less than the average fringe spacing, a fringe pattern will again be observed during the time when the two waves overlap. </p><p>Conventional light sources emit waves of differing frequencies and at different times from different points in the source. If the light is split into two waves and then re-combined, each individual light wave may generate an interference pattern with its other half, but the individual fringe patterns generated will have different phases and spacings, and normally no overall fringe pattern will be observable. However, single-element light sources, such as <a href="/wiki/Sodium-vapor_lamp" title="Sodium-vapor lamp">sodium-</a> or <a href="/wiki/Mercury-vapor_lamp" title="Mercury-vapor lamp">mercury-vapor lamps</a> have emission lines with quite narrow frequency spectra. When these are spatially and colour filtered, and then split into two waves, they can be superimposed to generate interference fringes.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> All interferometry prior to the invention of the laser was done using such sources and had a wide range of successful applications. </p><p>A <a href="/wiki/Laser_beam" class="mw-redirect" title="Laser beam">laser beam</a> generally approximates much more closely to a monochromatic source, and thus it is much more straightforward to generate interference fringes using a laser. The ease with which interference fringes can be observed with a laser beam can sometimes cause problems in that stray reflections may give spurious interference fringes which can result in errors. </p><p>Normally, a single laser beam is used in interferometry, though interference has been observed using two independent lasers whose frequencies were sufficiently matched to satisfy the phase requirements.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> This has also been observed for widefield interference between two incoherent laser sources.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>It is also possible to observe interference fringes using white light. A white light fringe pattern can be considered to be made up of a 'spectrum' of fringe patterns each of slightly different spacing. If all the fringe patterns are in phase in the centre, then the fringes will increase in size as the wavelength decreases and the summed intensity will show three to four fringes of varying colour. Young describes this very elegantly in his discussion of two slit interference. Since white light fringes are obtained only when the two waves have travelled equal distances from the light source, they can be very useful in interferometry, as they allow the zero path difference fringe to be identified.<sup id="cite_ref-Born_and_Wolf_8-0" class="reference"><a href="#cite_note-Born_and_Wolf-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Optical_arrangements">Optical arrangements</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wave_interference&action=edit&section=10" title="Edit section: Optical arrangements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To generate interference fringes, light from the source has to be divided into two waves which then have to be re-combined. Traditionally, interferometers have been classified as either amplitude-division or wavefront-division systems. </p><p>In an amplitude-division system, a <a href="/wiki/Beam_splitter" title="Beam splitter">beam splitter</a> is used to divide the light into two beams travelling in different directions, which are then superimposed to produce the interference pattern. The <a href="/wiki/Michelson_interferometer" title="Michelson interferometer">Michelson interferometer</a> and the <a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a> are examples of amplitude-division systems. </p><p>In wavefront-division systems, the wave is divided in space—examples are <a href="/wiki/Young%27s_double_slit_experiment" class="mw-redirect" title="Young's double slit experiment">Young's double slit interferometer</a> and <a href="/wiki/Lloyd%27s_mirror" title="Lloyd's mirror">Lloyd's mirror</a>. </p><p>Interference can also be seen in everyday phenomena such as <a href="/wiki/Iridescence" title="Iridescence">iridescence</a> and <a href="/wiki/Structural_coloration" title="Structural coloration">structural coloration</a>. For example, the colours seen in a soap bubble arise from interference of light reflecting off the front and back surfaces of the thin soap film. Depending on the thickness of the film, different colours interfere constructively and destructively. </p> <ul class="gallery mw-gallery-packed"> <li class="gallerycaption">Iridiscence caused by thin-film interference</li> <li class="gallerybox" style="width: 162px"> <div class="thumb" style="width: 160px;"><span typeof="mw:File"><a href="/wiki/File:Samsung_Galaxy_A50_back_2.jpg" class="mw-file-description" title="Smartphone with iridescent back panel"><img alt="Smartphone with iridescent back panel" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Samsung_Galaxy_A50_back_2.jpg/240px-Samsung_Galaxy_A50_back_2.jpg" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Samsung_Galaxy_A50_back_2.jpg/360px-Samsung_Galaxy_A50_back_2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Samsung_Galaxy_A50_back_2.jpg/480px-Samsung_Galaxy_A50_back_2.jpg 2x" data-file-width="2048" data-file-height="1536" /></a></span></div> <div class="gallerytext">Smartphone with iridescent back panel</div> </li> <li class="gallerybox" style="width: 136.66666666667px"> <div class="thumb" style="width: 134.66666666667px;"><span typeof="mw:File"><a href="/wiki/File:Dieselrainbow.jpg" class="mw-file-description" title="An oil spill"><img alt="An oil spill" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Dieselrainbow.jpg/202px-Dieselrainbow.jpg" decoding="async" width="135" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Dieselrainbow.jpg/303px-Dieselrainbow.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Dieselrainbow.jpg/403px-Dieselrainbow.jpg 2x" data-file-width="976" data-file-height="871" /></a></span></div> <div class="gallerytext">An oil spill</div> </li> <li class="gallerybox" style="width: 162px"> <div class="thumb" style="width: 160px;"><span typeof="mw:File"><a href="/wiki/File:Soap_bubble_sky.jpg" class="mw-file-description" title="White light interference in a soap bubble."><img alt="White light interference in a soap bubble." src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Soap_bubble_sky.jpg/240px-Soap_bubble_sky.jpg" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Soap_bubble_sky.jpg/360px-Soap_bubble_sky.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/Soap_bubble_sky.jpg/480px-Soap_bubble_sky.jpg 2x" data-file-width="1024" data-file-height="768" /></a></span></div> <div class="gallerytext">White light interference in a soap bubble.</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Quantum_interference">Quantum interference</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wave_interference&action=edit&section=11" title="Edit section: Quantum interference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit experiment</a> and <a href="/wiki/Matter_wave" title="Matter wave">Matter wave</a></div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist nowraplinks" style="width:19.0em;"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1799e4a910c7d26396922a20ef5ceec25ca1871c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.882ex; height:5.509ex;" alt="{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }"></span><div class="sidebar-caption" style="font-size:90%;padding-top:0.4em;font-style:italic;"><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a></div></td></tr><tr><td class="sidebar-above hlist nowrap" style="display:block;margin-bottom:0.4em;"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a></li></ul></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Background</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a class="mw-selflink selflink">Interference</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)">Complementarity</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_number" title="Quantum number">Quantum number</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">State</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li></ul></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Experiments</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell's inequality</a></li> <li><a href="/wiki/CHSH_inequality" title="CHSH inequality">CHSH inequality</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Leggett_inequality" title="Leggett inequality">Leggett inequality</a></li> <li><a href="/wiki/Leggett%E2%80%93Garg_inequality" title="Leggett–Garg inequality">Leggett–Garg inequality</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li></ul> </div> <ul><li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a> <ul><li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice</a></li></ul></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed-choice</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Formulations</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Overview</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase-space</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Sum-over-histories (path integral)</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective-collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Advanced topics</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Yakir_Aharonov" title="Yakir Aharonov">Aharonov</a></li> <li><a href="/wiki/John_Stewart_Bell" title="John Stewart Bell">Bell</a></li> <li><a href="/wiki/Hans_Bethe" title="Hans Bethe">Bethe</a></li> <li><a href="/wiki/Patrick_Blackett" title="Patrick Blackett">Blackett</a></li> <li><a href="/wiki/Felix_Bloch" title="Felix Bloch">Bloch</a></li> <li><a href="/wiki/David_Bohm" title="David Bohm">Bohm</a></li> <li><a href="/wiki/Niels_Bohr" title="Niels Bohr">Bohr</a></li> <li><a href="/wiki/Max_Born" title="Max Born">Born</a></li> <li><a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Bose</a></li> <li><a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">de Broglie</a></li> <li><a href="/wiki/Arthur_Compton" title="Arthur Compton">Compton</a></li> <li><a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a></li> <li><a href="/wiki/Clinton_Davisson" title="Clinton Davisson">Davisson</a></li> <li><a href="/wiki/Peter_Debye" title="Peter Debye">Debye</a></li> <li><a href="/wiki/Paul_Ehrenfest" title="Paul Ehrenfest">Ehrenfest</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Hugh_Everett_III" title="Hugh Everett III">Everett</a></li> <li><a href="/wiki/Vladimir_Fock" title="Vladimir Fock">Fock</a></li> <li><a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a></li> <li><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman</a></li> <li><a href="/wiki/Roy_J._Glauber" title="Roy J. Glauber">Glauber</a></li> <li><a href="/wiki/Martin_Gutzwiller" title="Martin Gutzwiller">Gutzwiller</a></li> <li><a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Jordan</a></li> <li><a href="/wiki/Hans_Kramers" title="Hans Kramers">Kramers</a></li> <li><a href="/wiki/Willis_Lamb" title="Willis Lamb">Lamb</a></li> <li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li> <li><a href="/wiki/Max_von_Laue" title="Max von Laue">Laue</a></li> <li><a href="/wiki/Henry_Moseley" title="Henry Moseley">Moseley</a></li> <li><a href="/wiki/Robert_Andrews_Millikan" title="Robert Andrews Millikan">Millikan</a></li> <li><a href="/wiki/Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Onnes</a></li> <li><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/Isidor_Isaac_Rabi" title="Isidor Isaac Rabi">Rabi</a></li> <li><a href="/wiki/C._V._Raman" title="C. V. Raman">Raman</a></li> <li><a href="/wiki/Johannes_Rydberg" title="Johannes Rydberg">Rydberg</a></li> <li><a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödinger</a></li> <li><a href="/wiki/Michelle_Simmons" title="Michelle Simmons">Simmons</a></li> <li><a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Sommerfeld</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl</a></li> <li><a href="/wiki/Wilhelm_Wien" title="Wilhelm Wien">Wien</a></li> <li><a href="/wiki/Eugene_Wigner" title="Eugene Wigner">Wigner</a></li> <li><a href="/wiki/Pieter_Zeeman" title="Pieter Zeeman">Zeeman</a></li> <li><a href="/wiki/Anton_Zeilinger" title="Anton Zeilinger">Zeilinger</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar" style="border-top:1px solid #aaa;padding-top:0.1em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_mechanics" title="Template:Quantum mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics" title="Template talk:Quantum mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics" title="Special:EditPage/Template:Quantum mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Quantum interference</b> – the observed <a href="/wiki/Matter_wave" title="Matter wave"> wave-behavior of matter</a><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> – resembles <a class="mw-selflink-fragment" href="#Optical_interference">optical interference</a>. Let <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,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f78f255e641e6cd2cd4bcb63183af3a47d84808e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.821ex; height:2.843ex;" alt="{\displaystyle \Psi (x,t)}"></span> be a <a href="/wiki/Wave_function" title="Wave function">wavefunction</a> solution of the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> for a quantum mechanical object. Then the <a href="/wiki/Probability_amplitude" title="Probability amplitude">probability</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 P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89833156eff2c51bfb8750db3306a0544ce34e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.884ex; height:2.843ex;" alt="{\displaystyle P(x)}"></span> of observing the object at position <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> 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 P(x)=|\Psi (x,t)|^{2}=\Psi ^{*}(x,t)\Psi (x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</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> <mo>=</mo> <msup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=|\Psi (x,t)|^{2}=\Psi ^{*}(x,t)\Psi (x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66c9b04f667e119388887f2f134cbfa5a2468f06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.946ex; height:3.343ex;" alt="{\displaystyle P(x)=|\Psi (x,t)|^{2}=\Psi ^{*}(x,t)\Psi (x,t)}"></span> where * indicates <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugation</a>. Quantum interference concerns the issue of this probability when the wavefunction is expressed as a sum or <a href="/wiki/Quantum_superposition" title="Quantum superposition">linear superposition</a> of two terms <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,t)=\Psi _{A}(x,t)+\Psi _{B}(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (x,t)=\Psi _{A}(x,t)+\Psi _{B}(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6ed299f6a8db6991e06abfde261c42ede2365fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.345ex; height:2.843ex;" alt="{\displaystyle \Psi (x,t)=\Psi _{A}(x,t)+\Psi _{B}(x,t)}"></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=|\Psi (x,t)|^{2}=|\Psi _{A}(x,t)|^{2}+|\Psi _{B}(x,t)|^{2}+(\Psi _{A}^{*}(x,t)\Psi _{B}(x,t)+\Psi _{A}(x,t)\Psi _{B}^{*}(x,t))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</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> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</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> <mo>+</mo> <mo stretchy="false">(</mo> <msubsup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msubsup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=|\Psi (x,t)|^{2}=|\Psi _{A}(x,t)|^{2}+|\Psi _{B}(x,t)|^{2}+(\Psi _{A}^{*}(x,t)\Psi _{B}(x,t)+\Psi _{A}(x,t)\Psi _{B}^{*}(x,t))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee706e4c276ac3263a1fa53fc3eb81d1f2dab20" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:85.034ex; height:3.509ex;" alt="{\displaystyle P(x)=|\Psi (x,t)|^{2}=|\Psi _{A}(x,t)|^{2}+|\Psi _{B}(x,t)|^{2}+(\Psi _{A}^{*}(x,t)\Psi _{B}(x,t)+\Psi _{A}(x,t)\Psi _{B}^{*}(x,t))}"></span> </p><p>Usually, <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 _{A}(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{A}(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afbf95c5a27d9dc2601a3f211137e9c2cf6b0c79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.285ex; height:2.843ex;" alt="{\displaystyle \Psi _{A}(x,t)}"></span> and <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 _{B}(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{B}(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25f35fa93977e783e5a20810276725056ff9c4f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.3ex; height:2.843ex;" alt="{\displaystyle \Psi _{B}(x,t)}"></span> correspond to distinct situations A and B. When this is the case, the equation <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,t)=\Psi _{A}(x,t)+\Psi _{B}(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (x,t)=\Psi _{A}(x,t)+\Psi _{B}(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6ed299f6a8db6991e06abfde261c42ede2365fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.345ex; height:2.843ex;" alt="{\displaystyle \Psi (x,t)=\Psi _{A}(x,t)+\Psi _{B}(x,t)}"></span> indicates that the object can be in situation A or situation B. The above equation can then be interpreted as: The probability of finding the object at <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> is the probability of finding the object at <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> when it is in situation A plus the probability of finding the object at <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> when it is in situation B plus an extra term. This extra term, which is called the <i>quantum interference term</i>, 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 \Psi _{A}^{*}(x,t)\Psi _{B}(x,t)+\Psi _{A}(x,t)\Psi _{B}^{*}(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msubsup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{A}^{*}(x,t)\Psi _{B}(x,t)+\Psi _{A}(x,t)\Psi _{B}^{*}(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/483eafcd38f5a10cfb014e6796d47ce82307bc43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.012ex; height:3.009ex;" alt="{\displaystyle \Psi _{A}^{*}(x,t)\Psi _{B}(x,t)+\Psi _{A}(x,t)\Psi _{B}^{*}(x,t)}"></span> in the above equation. As in the <a class="mw-selflink-fragment" href="#Mechanisms">classical wave case</a> above, the quantum interference term can add (constructive interference) or subtract (destructive interference) from <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 _{A}(x,t)|^{2}+|\Psi _{B}(x,t)|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</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> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</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 |\Psi _{A}(x,t)|^{2}+|\Psi _{B}(x,t)|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546c48512e2f39361ec8dabc50daef5db409e1c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.122ex; height:3.343ex;" alt="{\displaystyle |\Psi _{A}(x,t)|^{2}+|\Psi _{B}(x,t)|^{2}}"></span> in the above equation depending on whether the quantum interference term is positive or negative. If this term is absent for all <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>, then there is no quantum mechanical interference associated with situations A and B. </p><p>The best known example of quantum interference is the <a href="/wiki/Double-slit_experiment" title="Double-slit experiment">double-slit experiment</a>. In this experiment, <a href="/wiki/Matter_wave" title="Matter wave">matter waves</a> from electrons, atoms or molecules approach a barrier with two slits in it. One slit becomes <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 _{A}(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{A}(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afbf95c5a27d9dc2601a3f211137e9c2cf6b0c79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.285ex; height:2.843ex;" alt="{\displaystyle \Psi _{A}(x,t)}"></span> and the other becomes <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 _{B}(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{B}(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25f35fa93977e783e5a20810276725056ff9c4f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.3ex; height:2.843ex;" alt="{\displaystyle \Psi _{B}(x,t)}"></span>. The interference pattern occurs on the far side, observed by detectors suitable to the particles originating the <a href="/wiki/Matter_wave" title="Matter wave">matter wave</a>.<sup id="cite_ref-Bach_Pope_Liou_Batelaan_2013_p=033018_10-0" class="reference"><a href="#cite_note-Bach_Pope_Liou_Batelaan_2013_p=033018-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The pattern matches the optical double slit pattern. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wave_interference&action=edit&section=12" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Beat">Beat</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wave_interference&action=edit&section=13" title="Edit section: Beat"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Beat_(acoustics)" title="Beat (acoustics)">Beat (acoustics)</a></div> <p>In <a href="/wiki/Acoustics" title="Acoustics">acoustics</a>, a <b>beat</b> is an <a href="/wiki/Interference_(wave_propagation)" class="mw-redirect" title="Interference (wave propagation)">interference</a> pattern between two <a href="/wiki/Sound" title="Sound">sounds</a> of slightly different <a href="/wiki/Frequency" title="Frequency">frequencies</a>, <i>perceived</i> as a periodic variation in <a href="/wiki/Amplitude_(music)" class="mw-redirect" title="Amplitude (music)">volume</a> whose rate is the <a href="/wiki/Difference_(mathematics)" class="mw-redirect" title="Difference (mathematics)">difference</a> of the two frequencies. </p><p>With <a href="/wiki/Musical_tuning" title="Musical tuning">tuning</a> instruments that can produce sustained tones, beats can be readily recognized. Tuning two tones to a <a href="/wiki/Unison" title="Unison">unison</a> will present a peculiar effect: when the two tones are close in pitch but not identical, the difference in frequency generates the beating. The volume varies like in a <a href="/wiki/Tremolo" title="Tremolo">tremolo</a> as the sounds alternately interfere constructively and destructively. As the two tones gradually approach unison, the beating slows down and may become so slow as to be imperceptible. As the two tones get further apart, their beat frequency starts to approach the range of human pitch perception,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> the beating starts to sound like a note, and a <a href="/wiki/Combination_tone" title="Combination tone">combination tone</a> is produced. This combination tone can also be referred to as a <a href="/wiki/Missing_fundamental" title="Missing fundamental">missing fundamental</a>, as the beat frequency of any two tones is equivalent to the frequency of their implied fundamental frequency. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Interferometry" title="Interferometry">Interferometry</a></div> <div class="mw-heading mw-heading3"><h3 id="Optical_interferometry">Optical interferometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wave_interference&action=edit&section=14" title="Edit section: Optical interferometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Optical_interferometry" class="mw-redirect" title="Optical interferometry">Optical interferometry</a></div> <p>Interferometry has played an important role in the advancement of physics, and also has a wide range of applications in physical and engineering measurement. </p><p><a href="/wiki/Thomas_Young_(scientist)" title="Thomas Young (scientist)">Thomas Young</a>'s double slit interferometer in 1803 demonstrated interference fringes when two small holes were illuminated by light from another small hole which was illuminated by sunlight. Young was able to estimate the wavelength of different colours in the spectrum from the spacing of the fringes. The experiment played a major role in the general acceptance of the wave theory of light.<sup id="cite_ref-Born_and_Wolf_8-1" class="reference"><a href="#cite_note-Born_and_Wolf-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> In quantum mechanics, this experiment is considered to demonstrate the inseparability of the wave and particle natures of light and other quantum particles (<a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">wave–particle duality</a>). <a href="/wiki/Richard_Feynman" title="Richard Feynman">Richard Feynman</a> was fond of saying that all of quantum mechanics can be gleaned from carefully thinking through the implications of this single experiment.<sup id="cite_ref-Greene_1999_12-0" class="reference"><a href="#cite_note-Greene_1999-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>The results of the <a href="/wiki/Michelson%E2%80%93Morley_experiment" title="Michelson–Morley experiment">Michelson–Morley experiment</a> are generally considered to be the first strong evidence against the theory of a <a href="/wiki/Luminiferous_aether" title="Luminiferous aether">luminiferous aether</a> and in favor of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>. </p><p>Interferometry has been used in defining and calibrating <a href="/wiki/Length_measurement" title="Length measurement">length standards</a>. When the metre was defined as the distance between two marks on a platinum-iridium bar, <a href="/wiki/Albert_Abraham_Michelson" class="mw-redirect" title="Albert Abraham Michelson">Michelson</a> and Benoît used interferometry to measure the wavelength of the red <a href="/wiki/Cadmium" title="Cadmium">cadmium</a> line in the new standard, and also showed that it could be used as a length standard. Sixty years later, in 1960, the metre in the new <a href="/wiki/SI" class="mw-redirect" title="SI">SI</a> system was defined to be equal to 1,650,763.73 wavelengths of the orange-red emission line in the electromagnetic spectrum of the krypton-86 atom in a vacuum. This definition was replaced in 1983 by defining the metre as the distance travelled by light in vacuum during a specific time interval. Interferometry is still fundamental in establishing the <a href="/wiki/Calibration" title="Calibration">calibration</a> chain in length measurement. </p><p>Interferometry is used in the calibration of <a href="/wiki/Gauge_block" title="Gauge block">slip gauges</a> (called gauge blocks in the US) and in <a href="/wiki/Coordinate-measuring_machine" title="Coordinate-measuring machine">coordinate-measuring machines</a>. It is also used in the testing of optical components.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Radio_interferometry">Radio interferometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wave_interference&action=edit&section=15" title="Edit section: Radio interferometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Astronomical_interferometer" title="Astronomical interferometer">Astronomical interferometer</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:USA.NM.VeryLargeArray.02.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/USA.NM.VeryLargeArray.02.jpg/220px-USA.NM.VeryLargeArray.02.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/USA.NM.VeryLargeArray.02.jpg/330px-USA.NM.VeryLargeArray.02.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/USA.NM.VeryLargeArray.02.jpg/440px-USA.NM.VeryLargeArray.02.jpg 2x" data-file-width="640" data-file-height="480" /></a><figcaption>The <a href="/wiki/Very_Large_Array" title="Very Large Array">Very Large Array</a>, an <a href="/wiki/Interferometer" class="mw-redirect" title="Interferometer">interferometric array</a> formed from many smaller <a href="/wiki/Telescope" title="Telescope">telescopes</a>, like many larger <a href="/wiki/Radio_telescope" title="Radio telescope">radio telescopes</a>.</figcaption></figure> <p>In 1946, a technique called <a href="/wiki/Astronomical_interferometer" title="Astronomical interferometer">astronomical interferometry</a> was developed. Astronomical radio interferometers usually consist either of arrays of parabolic dishes or two-dimensional arrays of omni-directional antennas. All of the telescopes in the array are widely separated and are usually connected together using <a href="/wiki/Coaxial_cable" title="Coaxial cable">coaxial cable</a>, <a href="/wiki/Waveguide" title="Waveguide">waveguide</a>, <a href="/wiki/Optical_fiber" title="Optical fiber">optical fiber</a>, or other type of <a href="/wiki/Transmission_line" title="Transmission line">transmission line</a>. Interferometry increases the total signal collected, but its primary purpose is to vastly increase the resolution through a process called <a href="/wiki/Aperture_synthesis" title="Aperture synthesis">Aperture synthesis</a>. This technique works by superposing (interfering) the signal waves from the different telescopes on the principle that waves that coincide with the same phase will add to each other while two waves that have opposite phases will cancel each other out. This creates a combined telescope that is equivalent in resolution (though not in sensitivity) to a single antenna whose diameter is equal to the spacing of the antennas farthest apart in the array. </p> <div class="mw-heading mw-heading3"><h3 id="Acoustic_interferometry">Acoustic interferometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wave_interference&action=edit&section=16" title="Edit section: Acoustic interferometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <a href="/wiki/Acoustic_interferometer" title="Acoustic interferometer">acoustic interferometer</a> is an instrument for measuring the physical characteristics of sound waves in a <a href="/wiki/Gas" title="Gas">gas</a> or liquid, such <a href="/wiki/Velocity" title="Velocity">velocity</a>, wavelength, <a href="/wiki/Absorption_(acoustics)" title="Absorption (acoustics)">absorption</a>, or <a href="/wiki/Electrical_impedance" title="Electrical impedance">impedance</a>. A vibrating <a href="/wiki/Crystal" title="Crystal">crystal</a> creates ultrasonic waves that are radiated into the medium. The waves strike a reflector placed parallel to the crystal, reflected back to the source and measured. </p><p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wave_interference&action=edit&section=17" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 14em;"> <ul><li><a href="/wiki/Active_noise_control" title="Active noise control">Active noise control</a></li> <li><a href="/wiki/Beat_(acoustics)" title="Beat (acoustics)">Beat (acoustics)</a></li> <li><a href="/wiki/Coherence_(physics)" title="Coherence (physics)">Coherence (physics)</a></li> <li><a href="/wiki/Diffraction" title="Diffraction">Diffraction</a></li> <li><a href="/wiki/Haidinger_fringes" class="mw-redirect" title="Haidinger fringes">Haidinger fringes</a></li> <li><a href="/wiki/Interference_lithography" title="Interference lithography">Interference lithography</a></li> <li><a href="/wiki/Interference_visibility" class="mw-redirect" title="Interference visibility">Interference visibility</a></li> <li><a href="/wiki/Interferometer" class="mw-redirect" title="Interferometer">Interferometer</a></li> <li><a href="/wiki/Lloyd%27s_Mirror" class="mw-redirect" title="Lloyd's Mirror">Lloyd's Mirror</a></li> <li><a href="/wiki/Moir%C3%A9_pattern" title="Moiré pattern">Moiré pattern</a></li> <li><a href="/wiki/Multipath_interference" class="mw-redirect" title="Multipath interference">Multipath interference</a></li> <li><a href="/wiki/Newton%27s_rings" title="Newton's rings">Newton's rings</a></li> <li><a href="/wiki/Optical_path_length" title="Optical path length">Optical path length</a></li> <li><a href="/wiki/Thin-film_interference" title="Thin-film interference">Thin-film interference</a></li> <li><a href="/wiki/Rayleigh_roughness_criterion" class="mw-redirect" title="Rayleigh roughness criterion">Rayleigh roughness criterion</a></li> <li><a href="/wiki/Upfade" class="mw-redirect" title="Upfade">Upfade</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wave_interference&action=edit&section=18" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFOn_the_mechanism_of_the_eye_/_by_Thomas_Young.YoungUniversity_College1801" class="citation book cs1">On the mechanism of the eye / by Thomas Young.; Young, Thomas; University College, London Library Services (1801). <a rel="nofollow" class="external text" href="http://archive.org/details/b21641778_0"><i>Young, Thomas, 1773-1829</i></a>. University College London (UCL) UCL Library Services. London : printed by W. Bulmer and Co., Cleveland Row, St. James's.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Young%2C+Thomas%2C+1773-1829.&rft.pub=London+%3A+printed+by+W.+Bulmer+and+Co.%2C+Cleveland+Row%2C+St.+James%27s&rft.date=1801&rft.au=On+the+mechanism+of+the+eye+%2F+by+Thomas+Young.&rft.au=Young%2C+Thomas&rft.au=University+College%2C+London+Library+Services&rft_id=http%3A%2F%2Farchive.org%2Fdetails%2Fb21641778_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWave+interference" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJones2001" class="citation book cs1">Jones, Peter Ward (2001). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1093/gmo/9781561592630.article.20622"><i>Oxford University Press</i></a>. Oxford Music Online. Oxford University Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fgmo%2F9781561592630.article.20622">10.1093/gmo/9781561592630.article.20622</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Oxford+University+Press&rft.series=Oxford+Music+Online&rft.pub=Oxford+University+Press&rft.date=2001&rft_id=info%3Adoi%2F10.1093%2Fgmo%2F9781561592630.article.20622&rft.aulast=Jones&rft.aufirst=Peter+Ward&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1093%2Fgmo%2F9781561592630.article.20622&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWave+interference" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKipnis1991" class="citation book cs1">Kipnis, Nahum (1991). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-3-0348-8652-9"><i>History of the Principle of Interference of Light</i></a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-0348-8652-9">10.1007/978-3-0348-8652-9</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-0348-9717-4" title="Special:BookSources/978-3-0348-9717-4"><bdi>978-3-0348-9717-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+the+Principle+of+Interference+of+Light&rft.date=1991&rft_id=info%3Adoi%2F10.1007%2F978-3-0348-8652-9&rft.isbn=978-3-0348-9717-4&rft.aulast=Kipnis&rft.aufirst=Nahum&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-3-0348-8652-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWave+interference" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Ockenga, Wymke. <a rel="nofollow" class="external text" href="http://www.leica-microsystems.com/science-lab/phase-contrast/">Phase contrast</a>. Leika Science Lab, 09 June 2011. "If two waves interfere, the amplitude of the resulting light wave will be equal to the vector sum of the amplitudes of the two interfering waves."</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteel1986" class="citation book cs1">Steel, W. H. (1986). <i>Interferometry</i>. 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(2014). <a rel="nofollow" class="external text" href="https://www.osapublishing.org/oe/abstract.cfm?uri=oe-22-22-27094">"Widefield two laser interferometry"</a>. <i>Optics Express</i>. <b>22</b> (22): 27094–27101. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014OExpr..2227094P">2014OExpr..2227094P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1364%2FOE.22.027094">10.1364/OE.22.027094</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/25401860">25401860</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Optics+Express&rft.atitle=Widefield+two+laser+interferometry&rft.volume=22&rft.issue=22&rft.pages=27094-27101&rft.date=2014&rft_id=info%3Apmid%2F25401860&rft_id=info%3Adoi%2F10.1364%2FOE.22.027094&rft_id=info%3Abibcode%2F2014OExpr..2227094P&rft.aulast=Patel&rft.aufirst=R.&rft.au=Achamfuo-Yeboah%2C+S.&rft.au=Light+R.&rft.au=Clark+M.&rft_id=https%3A%2F%2Fwww.osapublishing.org%2Foe%2Fabstract.cfm%3Furi%3Doe-22-22-27094&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWave+interference" class="Z3988"></span></span> </li> <li id="cite_note-Born_and_Wolf-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Born_and_Wolf_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Born_and_Wolf_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBornWolf1999" class="citation book cs1"><a href="/wiki/Max_Born" title="Max Born">Born, Max</a>; Wolf, Emil (1999). <i><a href="/wiki/Principles_of_Optics" title="Principles of Optics">Principles of Optics</a></i>. 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Dutton. p. 22. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0525949695" title="Special:BookSources/978-0525949695"><bdi>978-0525949695</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=This+is+Your+Brain+on+Music%3A+The+Science+of+a+Human+Obsession&rft.pages=22&rft.pub=Dutton&rft.date=2006&rft.isbn=978-0525949695&rft.aulast=Levitin&rft.aufirst=Daniel+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWave+interference" class="Z3988"></span></span> </li> <li id="cite_note-Greene_1999-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Greene_1999_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreene1999" class="citation book cs1"><a href="/wiki/Brian_Greene" title="Brian Greene">Greene, Brian</a> (1999). <a href="/wiki/The_Elegant_Universe:_Superstrings,_Hidden_Dimensions,_and_the_Quest_for_the_Ultimate_Theory" class="mw-redirect" title="The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory"><i>The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory</i></a>. New York: W.W. Norton. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/elegantuniverses0000gree/page/97">97–109</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-393-04688-5" title="Special:BookSources/978-0-393-04688-5"><bdi>978-0-393-04688-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Elegant+Universe%3A+Superstrings%2C+Hidden+Dimensions%2C+and+the+Quest+for+the+Ultimate+Theory&rft.place=New+York&rft.pages=97-109&rft.pub=W.W.+Norton&rft.date=1999&rft.isbn=978-0-393-04688-5&rft.aulast=Greene&rft.aufirst=Brian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWave+interference" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">RS Longhurst, <i>Geometrical and Physical Optics</i>, 1968, Longmans, London.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wave_interference&action=edit&section=19" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output 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href="http://iwant2study.org/lookangejss/04waves_11superposition/ejss_model_wave1d01/wave1d01_Simulation.xhtml">Easy JavaScript Simulation Model of One Dimensional Wave Interference</a></li> <li><a rel="nofollow" class="external text" href="http://www.citycollegiate.com/interference1.htm">Expressions of position and fringe spacing</a></li> <li><a rel="nofollow" class="external text" href="http://www.phy.hk/wiki/englishhtm/Interference.htm">Java simulation of interference of water waves 1</a></li> <li><a rel="nofollow" class="external text" href="http://www.phy.hk/wiki/englishhtm/Interference2.htm">Java simulation of interference of water waves 2</a></li> <li><a rel="nofollow" class="external text" href="http://www.acoustics.salford.ac.uk/feschools/waves/super2.htm">Flash animations demonstrating interference</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090624132120/http://www.acoustics.salford.ac.uk/feschools/waves/super2.htm">Archived</a> 2009-06-24 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output 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td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Quantum_mechanics" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_mechanics_topics" title="Template:Quantum mechanics topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics_topics" title="Template talk:Quantum mechanics topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics_topics" title="Special:EditPage/Template:Quantum mechanics topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_mechanics" style="font-size:114%;margin:0 4em"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a> <ul><li><a href="/wiki/Timeline_of_quantum_mechanics" title="Timeline of quantum mechanics">Timeline</a></li></ul></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamentals</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Born_rule" title="Born rule">Born rule</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li> <li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)"> Complementarity</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a> <ul><li><a href="/wiki/Ground_state" title="Ground state">Ground state</a></li> <li><a href="/wiki/Excited_state" title="Excited state">Excited state</a></li> <li><a href="/wiki/Degenerate_energy_levels" title="Degenerate energy levels">Degenerate levels</a></li> <li><a href="/wiki/Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li></ul></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a class="mw-selflink selflink">Interference</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Formulations</a></li> <li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Equations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Weyl_equation" title="Weyl equation">Weyl</a></li> <li><a href="/wiki/Majorana_equation" title="Majorana equation">Majorana</a></li> <li><a href="/wiki/Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell test</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li> <li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed choice</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_nanoscience" class="mw-redirect" title="Quantum nanoscience">Science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_biology" title="Quantum biology">Quantum biology</a></li> <li><a href="/wiki/Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li> <li><a href="/wiki/Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li> <li><a href="/wiki/Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li> <li><a href="/wiki/Quantum_mind" title="Quantum mind">Quantum mind</a></li> <li><a href="/wiki/Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li> <li><a href="/wiki/Quantum_spacetime" title="Quantum spacetime">Quantum spacetime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li> <li><a href="/wiki/Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">Quantum bus</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a> <ul><li><a href="/wiki/Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li> <li><a href="/wiki/Quantum_computing" title="Quantum computing">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li> <li><a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">Quantum electronics</a></li> <li><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li> <li><a href="/wiki/Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li> <li><a href="/wiki/Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li> <li><a href="/wiki/Quantum_machine" title="Quantum machine">Quantum machine</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li> <li><a href="/wiki/Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li> <li><a href="/wiki/Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">Quantum network</a></li> <li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li> <li><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Extensions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li> <li><a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a> <ul><li><a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></li></ul></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat_in_popular_culture" title="Schrödinger's cat in popular culture">in popular culture</a></li></ul></li> <li><a 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