Bra–ket notation

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id="toc-Inner_product_and_bra–ket_identification_on_Hilbert_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inner_product_and_bra–ket_identification_on_Hilbert_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Inner product and bra–ket identification on Hilbert space</span> </div> </a> <ul id="toc-Inner_product_and_bra–ket_identification_on_Hilbert_space-sublist" class="vector-toc-list"> <li id="toc-Bras_and_kets_as_row_and_column_vectors" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Bras_and_kets_as_row_and_column_vectors"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Bras and kets as row and column vectors</span> </div> </a> <ul id="toc-Bras_and_kets_as_row_and_column_vectors-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Non-normalizable_states_and_non-Hilbert_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-normalizable_states_and_non-Hilbert_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Non-normalizable states and non-Hilbert spaces</span> </div> </a> <ul id="toc-Non-normalizable_states_and_non-Hilbert_spaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Usage_in_quantum_mechanics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Usage_in_quantum_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Usage in quantum mechanics</span> </div> </a> <button aria-controls="toc-Usage_in_quantum_mechanics-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 Usage in quantum mechanics subsection</span> </button> <ul id="toc-Usage_in_quantum_mechanics-sublist" class="vector-toc-list"> <li id="toc-Spinless_position–space_wave_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spinless_position–space_wave_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Spinless position–space wave function</span> </div> </a> <ul id="toc-Spinless_position–space_wave_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Overlap_of_states" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Overlap_of_states"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Overlap of states</span> </div> </a> <ul id="toc-Overlap_of_states-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Changing_basis_for_a_spin-1/2_particle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Changing_basis_for_a_spin-1/2_particle"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Changing basis for a spin-1/2 particle</span> </div> </a> <ul id="toc-Changing_basis_for_a_spin-1/2_particle-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Pitfalls_and_ambiguous_uses" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Pitfalls_and_ambiguous_uses"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Pitfalls and ambiguous uses</span> </div> </a> <button aria-controls="toc-Pitfalls_and_ambiguous_uses-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 Pitfalls and ambiguous uses subsection</span> </button> <ul id="toc-Pitfalls_and_ambiguous_uses-sublist" class="vector-toc-list"> <li id="toc-Separation_of_inner_product_and_vectors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Separation_of_inner_product_and_vectors"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Separation of inner product and vectors</span> </div> </a> <ul id="toc-Separation_of_inner_product_and_vectors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reuse_of_symbols" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reuse_of_symbols"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Reuse of symbols</span> </div> </a> <ul id="toc-Reuse_of_symbols-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hermitian_conjugate_of_kets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hermitian_conjugate_of_kets"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Hermitian conjugate of kets</span> </div> </a> <ul id="toc-Hermitian_conjugate_of_kets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Operations_inside_bras_and_kets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Operations_inside_bras_and_kets"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Operations inside bras and kets</span> </div> </a> <ul id="toc-Operations_inside_bras_and_kets-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Linear_operators" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Linear_operators"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Linear operators</span> </div> </a> <button aria-controls="toc-Linear_operators-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 Linear operators subsection</span> </button> <ul id="toc-Linear_operators-sublist" class="vector-toc-list"> <li id="toc-Linear_operators_acting_on_kets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linear_operators_acting_on_kets"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Linear operators acting on kets</span> </div> </a> <ul id="toc-Linear_operators_acting_on_kets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linear_operators_acting_on_bras" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linear_operators_acting_on_bras"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Linear operators acting on bras</span> </div> </a> <ul id="toc-Linear_operators_acting_on_bras-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Outer_products" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Outer_products"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Outer products</span> </div> </a> <ul id="toc-Outer_products-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hermitian_conjugate_operator" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hermitian_conjugate_operator"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Hermitian conjugate operator</span> </div> </a> <ul id="toc-Hermitian_conjugate_operator-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-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 Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Linearity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linearity"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Linearity</span> </div> </a> <ul id="toc-Linearity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Associativity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Associativity"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Associativity</span> </div> </a> <ul id="toc-Associativity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hermitian_conjugation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hermitian_conjugation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Hermitian conjugation</span> </div> </a> <ul id="toc-Hermitian_conjugation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Composite_bras_and_kets" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Composite_bras_and_kets"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Composite bras and kets</span> </div> </a> <ul id="toc-Composite_bras_and_kets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_unit_operator" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#The_unit_operator"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>The unit operator</span> </div> </a> <ul id="toc-The_unit_operator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notation_used_by_mathematicians" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notation_used_by_mathematicians"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notation used by mathematicians</span> </div> </a> <ul id="toc-Notation_used_by_mathematicians-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</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"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</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">Bra–ket notation</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 37 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-37" 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">37 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B1%D9%85%D8%B2_%D8%A8%D8%B1%D8%A7%D9%83%D9%8A%D8%AA" title="رمز براكيت – Arabic" lang="ar" hreflang="ar" data-title="رمز براكيت" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A7%8D%E0%A6%B0%E0%A6%BE-%E0%A6%95%E0%A7%87%E0%A6%9F_%E0%A6%AA%E0%A7%8D%E0%A6%B0%E0%A6%A4%E0%A6%BF%E0%A6%95" title="ব্রা-কেট প্রতিক – Bangla" lang="bn" hreflang="bn" data-title="ব্রা-কেট প্রতিক" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%91%D1%80%D0%B0_%D1%96_%D0%BA%D0%B5%D1%82" title="Бра і кет – Belarusian" lang="be" hreflang="be" data-title="Бра і кет" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Bra%E2%80%93ket_notacija" title="Bra–ket notacija – Bosnian" lang="bs" hreflang="bs" data-title="Bra–ket notacija" 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/Notaci%C3%B3_bra-ket" title="Notació bra-ket – Catalan" lang="ca" hreflang="ca" data-title="Notació bra-ket" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Diracova_notace" title="Diracova notace – Czech" lang="cs" hreflang="cs" data-title="Diracova notace" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Bra-ket-notation" title="Bra-ket-notation – Danish" lang="da" hreflang="da" data-title="Bra-ket-notation" 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/Dirac-Notation" title="Dirac-Notation – German" lang="de" hreflang="de" data-title="Dirac-Notation" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%85%CE%BC%CE%B2%CE%BF%CE%BB%CE%B9%CF%83%CE%BC%CF%8C%CF%82_%CE%9D%CF%84%CE%B9%CF%81%CE%AC%CE%BA" title="Συμβολισμός Ντιράκ – Greek" lang="el" hreflang="el" data-title="Συμβολισμός Ντιράκ" 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/Notaci%C3%B3n_bra-ket" title="Notación bra-ket – Spanish" lang="es" hreflang="es" data-title="Notación bra-ket" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bra-ket_notazio" title="Bra-ket notazio – Basque" lang="eu" hreflang="eu" data-title="Bra-ket notazio" 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/%D9%86%D8%B4%D8%A7%D9%86%E2%80%8C%DA%AF%D8%B0%D8%A7%D8%B1%DB%8C_%D8%A8%D8%B1%D8%A7-%DA%A9%D8%AA" title="نشان‌گذاری برا-کت – Persian" lang="fa" hreflang="fa" data-title="نشان‌گذاری برا-کت" 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/Notation_bra-ket" title="Notation bra-ket – French" lang="fr" hreflang="fr" data-title="Notation bra-ket" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B8%8C%EB%9D%BC-%EC%BC%93_%ED%91%9C%EA%B8%B0%EB%B2%95" title="브라-켓 표기법 – Korean" lang="ko" hreflang="ko" data-title="브라-켓 표기법" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ia badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ia.wikipedia.org/wiki/Notation_bra-ket" title="Notation bra-ket – Interlingua" lang="ia" hreflang="ia" data-title="Notation bra-ket" 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/Notazione_bra-ket" title="Notazione bra-ket – Italian" lang="it" hreflang="it" data-title="Notazione bra-ket" 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%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%93%D7%99%D7%A8%D7%90%D7%A7" title="סימון דיראק – Hebrew" lang="he" hreflang="he" data-title="סימון דיראק" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Bra-ket_notacija" title="Bra-ket notacija – Lithuanian" lang="lt" hreflang="lt" data-title="Bra-ket notacija" 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/Braket-jel%C3%B6l%C3%A9s" title="Braket-jelölés – Hungarian" lang="hu" hreflang="hu" data-title="Braket-jelölés" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AC%E0%B5%8D%E0%B4%B0%E0%B4%BE-%E0%B4%95%E0%B5%86%E0%B4%B1%E0%B5%8D%E0%B4%B1%E0%B5%8D_%E0%B4%9A%E0%B4%BF%E0%B4%B9%E0%B5%8D%E0%B4%A8%E0%B4%A8%E0%B4%99%E0%B5%8D%E0%B4%99%E0%B5%BE" title="ബ്രാ-കെറ്റ് ചിഹ്നനങ്ങൾ – Malayalam" lang="ml" hreflang="ml" data-title="ബ്രാ-കെറ്റ് ചിഹ്നനങ്ങൾ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Diracnotatie" title="Diracnotatie – Dutch" lang="nl" hreflang="nl" data-title="Diracnotatie" 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/%E3%83%96%E3%83%A9-%E3%82%B1%E3%83%83%E3%83%88%E8%A8%98%E6%B3%95" title="ブラ-ケット記法 – Japanese" lang="ja" hreflang="ja" data-title="ブラ-ケット記法" 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/Dirac-formalisme" title="Dirac-formalisme – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Dirac-formalisme" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" 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hreflang="sq" data-title="Bra-ket" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Diracov_zapis" title="Diracov zapis – Slovenian" lang="sl" hreflang="sl" data-title="Diracov zapis" 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%94%D0%B8%D1%80%D0%B0%D0%BA%D0%BE%D0%B2%D0%B0_%D0%BD%D0%BE%D1%82%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Диракова нотација – Serbian" lang="sr" hreflang="sr" data-title="Диракова нотација" 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-fi mw-list-item"><a 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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"><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"><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 class="mw-selflink selflink">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 href="/wiki/Wave_interference" title="Wave interference">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>Bra–ket notation</b>, also called <b>Dirac notation</b>, is a notation for <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> and <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operators</a> on <a href="/wiki/Complex_vector_space" class="mw-redirect" title="Complex vector space">complex vector spaces</a> together with their <a href="/wiki/Dual_space" title="Dual space">dual space</a> both in the finite-dimensional and infinite-dimensional case. It is specifically designed to ease the types of calculations that frequently come up in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>. Its use in quantum mechanics is quite widespread. </p><p>Bra–ket notation was created by <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a> in his 1939 publication <i>A New Notation for Quantum Mechanics</i>. The notation was introduced as an easier way to write quantum mechanical expressions.<sup id="cite_ref-Dirac_1-0" class="reference"><a href="#cite_note-Dirac-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The name comes from the English word "bracket". </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Quantum_mechanics">Quantum mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=1" title="Edit section: Quantum mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, bra–ket notation is used ubiquitously to denote <a href="/wiki/Quantum_state" title="Quantum state">quantum states</a>. The notation uses <a href="/wiki/Angle_bracket" class="mw-redirect" title="Angle bracket">angle brackets</a>, <span class="nounderlines" style="border: 1px solid var(--border-color-muted,#ddd); color: var(--color-base); background-color: var( --background-color-neutral-subtle, #fdfdfd); padding: 1px 1px;"><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 \langle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/465059661f2295b855415c61124fb1a960e9e888" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:0.905ex; height:2.843ex;" alt="{\displaystyle \langle }"></span></span> and <span class="nounderlines" style="border: 1px solid var(--border-color-muted,#ddd); color: var(--color-base); background-color: var( --background-color-neutral-subtle, #fdfdfd); padding: 1px 1px;"><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 \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39c8d726068b5e1a371e72e13feafb8c8cd4b0d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:0.905ex; height:2.843ex;" alt="{\displaystyle \rangle }"></span></span>, and a <a href="/wiki/Vertical_bar" title="Vertical bar">vertical bar</a> <span class="nounderlines" style="border: 1px solid var(--border-color-muted,#ddd); color: var(--color-base); background-color: var( --background-color-neutral-subtle, #fdfdfd); padding: 1px 1px;"><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 |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddcc2e0b63522a705c3c691777c9f342806b19d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:0.647ex; height:2.843ex;" alt="{\displaystyle |}"></span></span>, to construct "bras" and "kets". </p><p>A <b>ket</b> is of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |v\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |v\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a30e592038265dbec709de73cdb92e8cc55f6b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.679ex; height:2.843ex;" alt="{\displaystyle |v\rangle }"></span>. Mathematically it denotes a <a href="/wiki/Vector_space" title="Vector space">vector</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 {\boldsymbol {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b2c2d3aac4213f3996d828c6aa8f4eb464a05cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.318ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {v}}}"></span>, in an abstract (complex) <a href="/wiki/Vector_space" title="Vector space">vector space</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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, and physically it represents a state of some quantum system. </p><p>A <b>bra</b> is of the form <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 \langle f|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e99dc122035c8052d1ad9b48a7a305f8b2f3351" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.83ex; height:2.843ex;" alt="{\displaystyle \langle f|}"></span>. Mathematically it denotes a <a href="/wiki/Linear_form" title="Linear form">linear form</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 f:V\to \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:V\to \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c094ecb5e24cc406bc2a5066bc1fba7c24ca491" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.295ex; height:2.509ex;" alt="{\displaystyle f:V\to \mathbb {C} }"></span>, i.e. a <a href="/wiki/Linear_map" title="Linear map">linear map</a> that maps each vector in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> to a number in the complex plane <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 \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>. Letting the linear functional <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 \langle f|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e99dc122035c8052d1ad9b48a7a305f8b2f3351" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.83ex; height:2.843ex;" alt="{\displaystyle \langle f|}"></span> act on a vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |v\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |v\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a30e592038265dbec709de73cdb92e8cc55f6b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.679ex; height:2.843ex;" alt="{\displaystyle |v\rangle }"></span> is written 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 \langle f|v\rangle \in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle f|v\rangle \in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b637dde47ba1540c82859da724d9b5f0c4629c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.381ex; height:2.843ex;" alt="{\displaystyle \langle f|v\rangle \in \mathbb {C} }"></span>. </p><p>Assume that on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> there exists an <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</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 (\cdot ,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\cdot ,\cdot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fc515c912925128800226dd0b017be508069e24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle (\cdot ,\cdot )}"></span> with <a href="/wiki/Antilinear_map" title="Antilinear map">antilinear</a> first argument, which makes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> an <a href="/wiki/Inner_product_space" title="Inner product space">inner product space</a>. Then with this <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a> each vector <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 {\boldsymbol {\phi }}\equiv |\phi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ϕ</mi> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\phi }}\equiv |\phi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/139a30dc0eab0176e1a53a36e45ef8365da822a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.69ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {\phi }}\equiv |\phi \rangle }"></span> can be identified with a corresponding linear form, by placing the vector in the anti-linear first slot of the inner product: <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 ({\boldsymbol {\phi }},\cdot )\equiv \langle \phi |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ϕ</mi> </mrow> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> <mo>≡</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\boldsymbol {\phi }},\cdot )\equiv \langle \phi |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8464d7b31db0dd84ad046caaa99a514390de065f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.18ex; height:2.843ex;" alt="{\displaystyle ({\boldsymbol {\phi }},\cdot )\equiv \langle \phi |}"></span>. The correspondence between these notations is 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 ({\boldsymbol {\phi }},{\boldsymbol {\psi }})\equiv \langle \phi |\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ϕ</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ψ</mi> </mrow> <mo stretchy="false">)</mo> <mo>≡</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\boldsymbol {\phi }},{\boldsymbol {\psi }})\equiv \langle \phi |\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07ef2ba7a77689dfc1f62e03b763810e08369760" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.713ex; height:2.843ex;" alt="{\displaystyle ({\boldsymbol {\phi }},{\boldsymbol {\psi }})\equiv \langle \phi |\psi \rangle }"></span>. The <a href="/wiki/Linear_form" title="Linear form">linear form</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 \langle \phi |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \phi |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/551efc9f54d1e2c864dcf38fb7ce534b3ed3671a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.937ex; height:2.843ex;" alt="{\displaystyle \langle \phi |}"></span> is a covector 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 |\phi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/312d43de853a9e6ca74888e63394fc8081f56a43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.937ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle }"></span>, and the set of all covectors forms a subspace of the <a href="/wiki/Dual_vector_space" class="mw-redirect" title="Dual vector space">dual vector space</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 V^{\vee }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∨</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{\vee }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c20e065987e46de2dec4d1b3c39e5fed0d6cebb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.245ex; height:2.509ex;" alt="{\displaystyle V^{\vee }}"></span>, to the initial vector space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>. The purpose of this linear form <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 \langle \phi |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \phi |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/551efc9f54d1e2c864dcf38fb7ce534b3ed3671a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.937ex; height:2.843ex;" alt="{\displaystyle \langle \phi |}"></span> can now be understood in terms of making projections onto the state <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 {\boldsymbol {\phi }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ϕ</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\phi }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76729782b5a4e4c5c3c9fd86e093db07c2685755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.302ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\phi }},}"></span> to find how linearly dependent two states are, etc. </p><p>For the vector space <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 \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"></span>, kets can be identified with column vectors, and bras with row vectors. Combinations of bras, kets, and linear operators are interpreted using <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>. If <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 \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"></span> has the standard Hermitian inner product <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 ({\boldsymbol {v}},{\boldsymbol {w}})=v^{\dagger }w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">w</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\boldsymbol {v}},{\boldsymbol {w}})=v^{\dagger }w}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6496faf43e39b568fc6457cdf5894c06732ee940" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.945ex; height:3.176ex;" alt="{\displaystyle ({\boldsymbol {v}},{\boldsymbol {w}})=v^{\dagger }w}"></span>, under this identification, the identification of kets and bras and vice versa provided by the inner product is taking the <a href="/wiki/Hermitian_conjugate" class="mw-redirect" title="Hermitian conjugate">Hermitian conjugate</a> (denoted <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 \dagger }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>†</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dagger }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fbce70d5be6fec538cd30d8bc7b7bb2d3ed2d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.032ex; height:2.676ex;" alt="{\displaystyle \dagger }"></span>). </p><p>It is common to suppress the vector or linear form from the bra–ket notation and only use a label inside the typography for the bra or ket. For example, the spin operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\sigma }}_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>σ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\sigma }}_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a4f73f53ead65d31fd72ca8d31392d317d14ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.331ex; height:2.509ex;" alt="{\displaystyle {\hat {\sigma }}_{z}}"></span> on a two-dimensional space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"></span> of <a href="/wiki/Spinor" title="Spinor">spinors</a> has eigenvalues <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 \pm {\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \pm {\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3acc5644aeef7dd2623ff7fdf8931d32651eafc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.466ex; height:3.509ex;" alt="{\textstyle \pm {\frac {1}{2}}}"></span> with eigenspinors <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 {\boldsymbol {\psi }}_{+},{\boldsymbol {\psi }}_{-}\in \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ψ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ψ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>∈</mo> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\psi }}_{+},{\boldsymbol {\psi }}_{-}\in \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3745f24f74c5c40d6be5f72f3254d9f6ee355cab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.355ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {\psi }}_{+},{\boldsymbol {\psi }}_{-}\in \Delta }"></span>. In bra–ket notation, this is typically denoted 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 {\boldsymbol {\psi }}_{+}=|+\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ψ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\psi }}_{+}=|+\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aff9f52cd35344227edc3a8a6e0f0740c1fc0393" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.731ex; height:3.009ex;" alt="{\displaystyle {\boldsymbol {\psi }}_{+}=|+\rangle }"></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 {\boldsymbol {\psi }}_{-}=|-\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ψ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\psi }}_{-}=|-\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57a048b7d2330cbfbdbc7976652e62d4ffab0af4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.731ex; height:3.009ex;" alt="{\displaystyle {\boldsymbol {\psi }}_{-}=|-\rangle }"></span>. As above, kets and bras with the same label are interpreted as kets and bras corresponding to each other using the inner product. In particular, when also identified with row and column vectors, kets and bras with the same label are identified with <a href="/wiki/Hermitian_conjugate" class="mw-redirect" title="Hermitian conjugate">Hermitian conjugate</a> column and row vectors. </p><p>Bra–ket notation was effectively established in 1939 by <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a>;<sup id="cite_ref-Dirac_1-1" class="reference"><a href="#cite_note-Dirac-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> it is thus also known as Dirac notation, despite the notation having a precursor in <a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a>'s use 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 [\phi {\mid }\psi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∣</mo> </mrow> <mi>ψ</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\phi {\mid }\psi ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/144b6bf7eb1ff9b91f70dd30486f738ee09d2570" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.839ex; height:2.843ex;" alt="{\displaystyle [\phi {\mid }\psi ]}"></span> for inner products nearly 100 years earlier.<sup id="cite_ref-Grassmann_3-0" class="reference"><a href="#cite_note-Grassmann-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><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> </p> <div class="mw-heading mw-heading2"><h2 id="Vector_spaces">Vector spaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=2" title="Edit section: Vector spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Vectors_vs_kets">Vectors vs kets</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=3" title="Edit section: Vectors vs kets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In mathematics, the term "vector" is used for an element of any vector space. In physics, however, the term "vector" tends to refer almost exclusively to quantities like <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a> or <a href="/wiki/Velocity" title="Velocity">velocity</a>, which have components that relate directly to the three dimensions of <a href="/wiki/Space" title="Space">space</a>, or relativistically, to the four of <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>. Such vectors are typically denoted with over arrows (<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 {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec3c9ce13b53e9e24c98e7cce4212627884c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}"></span>), boldface (<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 \mathbf {p} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd73e3862cb92b016721b8c492eadb4e8a577527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.009ex;" alt="{\displaystyle \mathbf {p} }"></span>) or indices (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v^{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7875457ee62423274b72c3d02311e144af146bc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.351ex; height:2.343ex;" alt="{\displaystyle v^{\mu }}"></span>). </p><p>In quantum mechanics, a quantum state is typically represented as an element of a complex <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>, for example, the infinite-dimensional vector space of all possible <a href="/wiki/Wavefunction" class="mw-redirect" title="Wavefunction">wavefunctions</a> (square integrable functions mapping each point of 3D space to a complex number) or some more abstract Hilbert space constructed more algebraically. To distinguish this type of vector from those described above, it is common and useful in physics to denote an element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> of an abstract complex vector space as a ket <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\phi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/312d43de853a9e6ca74888e63394fc8081f56a43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.937ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle }"></span>, to refer to it as a "ket" rather than as a vector, and to pronounce it "ket-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span>" or "ket-A" for <span class="texhtml"><span class="nowrap">|<i>A</i>⟩</span></span>. </p><p>Symbols, letters, numbers, or even words—whatever serves as a convenient label—can be used as the label inside a ket, with the <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 |\ \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\ \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b05d981500b4a4dd3026a9936a93d814d6eb225a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.132ex; height:2.843ex;" alt="{\displaystyle |\ \rangle }"></span> making clear that the label indicates a vector in vector space. In other words, the symbol "<span class="texhtml"><span class="nowrap">|<i>A</i>⟩</span></span>" has a recognizable mathematical meaning as to the kind of variable being represented, while just the "<span class="texhtml"><i>A</i></span>" by itself does not. For example, <span class="texhtml"><span class="nowrap">|1⟩</span> + <span class="nowrap">|2⟩</span></span> is not necessarily equal to <span class="texhtml"><span class="nowrap">|3⟩</span></span>. Nevertheless, for convenience, there is usually some logical scheme behind the labels inside kets, such as the common practice of labeling <a href="/wiki/Stationary_state" title="Stationary state">energy eigenkets</a> in quantum mechanics through a listing of their <a href="/wiki/Quantum_number" title="Quantum number">quantum numbers</a>. At its simplest, the label inside the ket is the eigenvalue of a physical operator, such 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 {\hat {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18d95a7845e4e16ffb7e18ab37a208d0ab18e0e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\hat {x}}}"></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 {\hat {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bd4c026f1b3413adc58b9b65e89e62bce92c85a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.449ex; height:2.509ex;" alt="{\displaystyle {\hat {p}}}"></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 {\hat {L}}_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {L}}_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f890c759326e0b9e75b24931dcf2a53862ab309c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.585ex; height:3.176ex;" alt="{\displaystyle {\hat {L}}_{z}}"></span>, etc. </p> <div class="mw-heading mw-heading3"><h3 id="Notation">Notation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=4" title="Edit section: Notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since kets are just vectors in a Hermitian vector space, they can be manipulated using the usual rules of linear algebra. For example: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}|A\rangle &=|B\rangle +|C\rangle \\|C\rangle &=(-1+2i)|D\rangle \\|D\rangle &=\int _{-\infty }^{\infty }e^{-x^{2}}|x\rangle \,\mathrm {d} x\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>C</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>C</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>i</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>D</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>D</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}|A\rangle &=|B\rangle +|C\rangle \\|C\rangle &=(-1+2i)|D\rangle \\|D\rangle &=\int _{-\infty }^{\infty }e^{-x^{2}}|x\rangle \,\mathrm {d} x\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/374c2d2a8af3e6b32a3e6716b1ee3acdf29d9fe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:23.45ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}|A\rangle &=|B\rangle +|C\rangle \\|C\rangle &=(-1+2i)|D\rangle \\|D\rangle &=\int _{-\infty }^{\infty }e^{-x^{2}}|x\rangle \,\mathrm {d} x\,.\end{aligned}}}"></span></dd></dl> <p>Note how the last line above involves infinitely many different kets, one for each real number <span class="texhtml"><i>x</i></span>. </p><p>Since the ket is an element of a vector space, a <b>bra</b> <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 \langle A|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle A|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16858cea66225d5e97d073eb640a3add588b9b80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.295ex; height:2.843ex;" alt="{\displaystyle \langle A|}"></span> is an element of its <a href="/wiki/Dual_space" title="Dual space">dual space</a>, i.e. a bra is a linear functional which is a linear map from the vector space to the complex numbers. Thus, it is useful to think of kets and bras as being elements of different vector spaces (see below however) with both being different useful concepts. </p><p>A bra <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 \langle \phi |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \phi |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/551efc9f54d1e2c864dcf38fb7ce534b3ed3671a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.937ex; height:2.843ex;" alt="{\displaystyle \langle \phi |}"></span> and a ket <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 \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"></span> (i.e. a functional and a vector), can be combined to an operator <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 \rangle \langle \phi |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle \langle \phi |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/649dc75b218cf5561251d8f9fc68192e54aae9f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.002ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle \langle \phi |}"></span> of rank one with <a href="/wiki/Outer_product" title="Outer product">outer product</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle \langle \phi |\colon |\xi \rangle \mapsto |\psi \rangle \langle \phi |\xi \rangle ~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ξ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ξ</mi> <mo fence="false" stretchy="false">⟩</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle \langle \phi |\colon |\xi \rangle \mapsto |\psi \rangle \langle \phi |\xi \rangle ~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a745c86545650f5ebd4f6024cf14e989d22a692" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.395ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle \langle \phi |\colon |\xi \rangle \mapsto |\psi \rangle \langle \phi |\xi \rangle ~.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Inner_product_and_bra–ket_identification_on_Hilbert_space"><span id="Inner_product_and_bra.E2.80.93ket_identification_on_Hilbert_space"></span>Inner product and bra–ket identification on Hilbert space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=5" title="Edit section: Inner product and bra–ket identification on Hilbert space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Main article: <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">Inner product</a></div> <p>The bra–ket notation is particularly useful in Hilbert spaces which have an inner product<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> that allows <a href="/wiki/Hermitian_conjugation" class="mw-redirect" title="Hermitian conjugation">Hermitian conjugation</a> and identifying a vector with a continuous linear functional, i.e. a ket with a bra, and vice versa (see <a href="/wiki/Riesz_representation_theorem" title="Riesz representation theorem">Riesz representation theorem</a>). The inner product on Hilbert space <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 (\ ,\ )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mtext> </mtext> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\ ,\ )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/694953cc20bfd90350b9dac4a68fda474b575399" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.005ex; height:2.843ex;" alt="{\displaystyle (\ ,\ )}"></span> (with the first argument anti linear as preferred by physicists) is fully equivalent to an (anti-linear) identification between the space of kets and that of bras in the bra ket notation: for a vector ket <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi =|\phi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =|\phi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4713674cd080efacc014754b3ba337f2b62261c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle \phi =|\phi \rangle }"></span> define a functional (i.e. bra) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\phi }=\langle \phi |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\phi }=\langle \phi |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb0a06d2fb328310b9cb3982febe57859fa87e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.387ex; height:3.009ex;" alt="{\displaystyle f_{\phi }=\langle \phi |}"></span> by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\phi ,\psi )=(|\phi \rangle ,|\psi \rangle )=:f_{\phi }(\psi )=\langle \phi |\,{\bigl (}|\psi \rangle {\bigr )}=:\langle \phi {\mid }\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>,</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> <mo>=:</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=:</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∣</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\phi ,\psi )=(|\phi \rangle ,|\psi \rangle )=:f_{\phi }(\psi )=\langle \phi |\,{\bigl (}|\psi \rangle {\bigr )}=:\langle \phi {\mid }\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/493defe5d485d41eadd72123c8a80db4d546920a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.821ex; height:3.176ex;" alt="{\displaystyle (\phi ,\psi )=(|\phi \rangle ,|\psi \rangle )=:f_{\phi }(\psi )=\langle \phi |\,{\bigl (}|\psi \rangle {\bigr )}=:\langle \phi {\mid }\psi \rangle }"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Bras_and_kets_as_row_and_column_vectors">Bras and kets as row and column vectors</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=6" title="Edit section: Bras and kets as row and column vectors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the simple case where we consider the vector space <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 \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"></span>, a ket can be identified with a <a href="/wiki/Column_vector" class="mw-redirect" title="Column vector">column vector</a>, and a bra as a <a href="/wiki/Row_vector" class="mw-redirect" title="Row vector">row vector</a>. If, moreover, we use the standard Hermitian inner product on <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 \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"></span>, the bra corresponding to a ket, in particular a bra <span class="texhtml"><span class="nowrap">⟨<i>m</i>|</span></span> and a ket <span class="texhtml"><span class="nowrap">|<i>m</i>⟩</span></span> with the same label are <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>. Moreover, conventions are set up in such a way that writing bras, kets, and linear operators next to each other simply imply <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>.<sup id="cite_ref-bra–ket_Notation_Trivializes_Matrix_Multiplication_6-0" class="reference"><a href="#cite_note-bra–ket_Notation_Trivializes_Matrix_Multiplication-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> In particular the outer product <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 \rangle \langle \phi |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle \langle \phi |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/649dc75b218cf5561251d8f9fc68192e54aae9f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.002ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle \langle \phi |}"></span> of a column and a row vector ket and bra can be identified with matrix multiplication (column vector times row vector equals matrix). </p><p>For a finite-dimensional vector space, using a fixed <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a>, the inner product can be written as a matrix multiplication of a row vector with a column vector: <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 \langle A|B\rangle \doteq A_{1}^{*}B_{1}+A_{2}^{*}B_{2}+\cdots +A_{N}^{*}B_{N}={\begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots &A_{N}^{*}\end{pmatrix}}{\begin{pmatrix}B_{1}\\B_{2}\\\vdots \\B_{N}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>≐</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle A|B\rangle \doteq A_{1}^{*}B_{1}+A_{2}^{*}B_{2}+\cdots +A_{N}^{*}B_{N}={\begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots &A_{N}^{*}\end{pmatrix}}{\begin{pmatrix}B_{1}\\B_{2}\\\vdots \\B_{N}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02d84f9f6709035c5caf4ded3ae6a926bdf83410" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:71.081ex; height:13.843ex;" alt="{\displaystyle \langle A|B\rangle \doteq A_{1}^{*}B_{1}+A_{2}^{*}B_{2}+\cdots +A_{N}^{*}B_{N}={\begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots &A_{N}^{*}\end{pmatrix}}{\begin{pmatrix}B_{1}\\B_{2}\\\vdots \\B_{N}\end{pmatrix}}}"></span> Based on this, the bras and kets can be defined as: <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 {\begin{aligned}\langle A|&\doteq {\begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots &A_{N}^{*}\end{pmatrix}}\\|B\rangle &\doteq {\begin{pmatrix}B_{1}\\B_{2}\\\vdots \\B_{N}\end{pmatrix}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>≐</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mi></mi> <mo>≐</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\langle A|&\doteq {\begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots &A_{N}^{*}\end{pmatrix}}\\|B\rangle &\doteq {\begin{pmatrix}B_{1}\\B_{2}\\\vdots \\B_{N}\end{pmatrix}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/729555b5e6a0a80149ffdffb96f404e21a59a1ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:28.446ex; height:17.176ex;" alt="{\displaystyle {\begin{aligned}\langle A|&\doteq {\begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots &A_{N}^{*}\end{pmatrix}}\\|B\rangle &\doteq {\begin{pmatrix}B_{1}\\B_{2}\\\vdots \\B_{N}\end{pmatrix}}\end{aligned}}}"></span> and then it is understood that a bra next to a ket implies matrix multiplication. </p><p>The <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> (also called <i>Hermitian conjugate</i>) of a bra is the corresponding ket and vice versa: <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 \langle A|^{\dagger }=|A\rangle ,\quad |A\rangle ^{\dagger }=\langle A|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <msup> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle A|^{\dagger }=|A\rangle ,\quad |A\rangle ^{\dagger }=\langle A|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/000fcf4e3258c32fa3699cea0265fc6c2efd7c12" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.656ex; height:3.343ex;" alt="{\displaystyle \langle A|^{\dagger }=|A\rangle ,\quad |A\rangle ^{\dagger }=\langle A|}"></span> because if one starts with the bra <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 {\begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots &A_{N}^{*}\end{pmatrix}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots &A_{N}^{*}\end{pmatrix}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9babdd2493f62a61210225cacf4226a4db5fad6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.315ex; height:3.176ex;" alt="{\displaystyle {\begin{pmatrix}A_{1}^{*}&A_{2}^{*}&\cdots &A_{N}^{*}\end{pmatrix}}\,,}"></span> then performs a <a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">complex conjugation</a>, and then a <a href="/wiki/Matrix_transpose" class="mw-redirect" title="Matrix transpose">matrix transpose</a>, one ends up with the ket <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 {\begin{pmatrix}A_{1}\\A_{2}\\\vdots \\A_{N}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}A_{1}\\A_{2}\\\vdots \\A_{N}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e51d6361c7297e90b99b070c5f3027d2aaec13e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:8.253ex; height:13.843ex;" alt="{\displaystyle {\begin{pmatrix}A_{1}\\A_{2}\\\vdots \\A_{N}\end{pmatrix}}}"></span> </p><p>Writing elements of a finite dimensional (or <a href="/wiki/Mutatis_mutandis" title="Mutatis mutandis">mutatis mutandis</a>, countably infinite) vector space as a column vector of numbers requires picking a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a>. Picking a basis is not always helpful because quantum mechanics calculations involve frequently switching between different bases (e.g. position basis, momentum basis, energy eigenbasis), and one can write something like "<span class="texhtml"><span class="nowrap">|<i>m</i>⟩</span></span>" without committing to any particular basis. In situations involving two different important basis vectors, the basis vectors can be taken in the notation explicitly and here will be referred simply as "<span class="texhtml"><span class="nowrap">|<i>−</i>⟩</span></span>" and "<span class="texhtml"><span class="nowrap">|<i>+</i>⟩</span></span>". </p> <div class="mw-heading mw-heading3"><h3 id="Non-normalizable_states_and_non-Hilbert_spaces">Non-normalizable states and non-Hilbert spaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=7" title="Edit section: Non-normalizable states and non-Hilbert spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bra–ket notation can be used even if the vector space is not a <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>. </p><p>In quantum mechanics, it is common practice to write down kets which have infinite <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a>, i.e. non-<a href="/wiki/Normalizable_wavefunction" class="mw-redirect" title="Normalizable wavefunction">normalizable wavefunctions</a>. Examples include states whose wavefunctions are <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta functions</a> or infinite <a href="/wiki/Plane_wave" title="Plane wave">plane waves</a>. These do not, technically, belong to the <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> itself. However, the definition of "Hilbert space" can be broadened to accommodate these states (see the <a href="/wiki/Gelfand%E2%80%93Naimark%E2%80%93Segal_construction" title="Gelfand–Naimark–Segal construction">Gelfand–Naimark–Segal construction</a> or <a href="/wiki/Rigged_Hilbert_space" title="Rigged Hilbert space">rigged Hilbert spaces</a>). The bra–ket notation continues to work in an analogous way in this broader context. </p><p><a href="/wiki/Banach_space" title="Banach space">Banach spaces</a> are a different generalization of Hilbert spaces. In a Banach space <span class="texhtml"><span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">B</span></span>, the vectors may be notated by kets and the continuous <a href="/wiki/Linear_functional" class="mw-redirect" title="Linear functional">linear functionals</a> by bras. Over any vector space without <a href="/wiki/Topology" title="Topology">topology</a>, we may also notate the vectors by kets and the linear functionals by bras. In these more general contexts, the bracket does not have the meaning of an inner product, because the Riesz representation theorem does not apply. </p> <div class="mw-heading mw-heading2"><h2 id="Usage_in_quantum_mechanics">Usage in quantum mechanics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=8" title="Edit section: Usage in quantum mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The mathematical structure of quantum mechanics is based in large part on <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>: </p> <ul><li><a href="/wiki/Wave_function" title="Wave function">Wave functions</a> and other quantum states can be represented as vectors in a complex Hilbert space. (The exact structure of this Hilbert space depends on the situation.) In bra–ket notation, for example, an electron might be in the "state" <span class="texhtml"><span class="nowrap">|<i>ψ</i>⟩</span></span>. (Technically, the quantum states are <i><a href="/wiki/Ray_(quantum_theory)" class="mw-redirect" title="Ray (quantum theory)">rays</a></i> of vectors in the Hilbert space, as <span class="texhtml"><i>c</i><span class="nowrap">|<i>ψ</i>⟩</span></span> corresponds to the same state for any nonzero complex number <span class="texhtml"><i>c</i></span>.)</li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Quantum superpositions</a> can be described as vector sums of the constituent states. For example, an electron in the state <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">√2</span></span>⁠</span><span class="nowrap">|1⟩</span> + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>i</i></span><span class="sr-only">/</span><span class="den">√2</span></span>⁠</span><span class="nowrap">|2⟩</span></span> is in a quantum superposition of the states <span class="texhtml"><span class="nowrap">|1⟩</span></span> and <span class="texhtml"><span class="nowrap">|2⟩</span></span>.</li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurements</a> are associated with linear operators (called <a href="/wiki/Observable" title="Observable">observables</a>) on the Hilbert space of quantum states.</li> <li>Dynamics are also described by linear operators on the Hilbert space. For example, in the <a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger picture</a>, there is a linear <a href="/wiki/Time_evolution" title="Time evolution">time evolution</a> operator <span class="texhtml"><i>U</i></span> with the property that if an electron is in state <span class="texhtml"><span class="nowrap">|<i>ψ</i>⟩</span></span> right now, at a later time it will be in the state <span class="texhtml"><i>U</i><span class="nowrap">|<i>ψ</i>⟩</span></span>, the same <span class="texhtml"><i>U</i></span> for every possible <span class="texhtml"><span class="nowrap">|<i>ψ</i>⟩</span></span>.</li> <li><a href="/wiki/Normalizable_wave_function" class="mw-redirect" title="Normalizable wave function">Wave function normalization</a> is scaling a wave function so that its <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a> is 1.</li></ul> <p>Since virtually every calculation in quantum mechanics involves vectors and linear operators, it can involve, and often does involve, bra–ket notation. A few examples follow: </p> <div class="mw-heading mw-heading3"><h3 id="Spinless_position–space_wave_function"><span id="Spinless_position.E2.80.93space_wave_function"></span>Spinless position–space wave function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=9" title="Edit section: Spinless position–space wave function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:463px;max-width:463px"><div class="trow"><div class="tsingle" style="width:227px;max-width:227px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Discrete_complex_vector_components.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Discrete_complex_vector_components.svg/225px-Discrete_complex_vector_components.svg.png" decoding="async" width="225" height="216" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Discrete_complex_vector_components.svg/338px-Discrete_complex_vector_components.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Discrete_complex_vector_components.svg/450px-Discrete_complex_vector_components.svg.png 2x" data-file-width="210" data-file-height="202" /></a></span></div><div class="thumbcaption">Discrete components <span class="texhtml"><i>A</i><sub><i>k</i></sub></span> of a complex vector <span class="texhtml"><span class="nowrap">|<i>A</i>⟩</span> = Σ<sub><i>k</i></sub> <i>A</i><sub><i>k</i></sub> <span class="nowrap">|<i>e<sub>k</sub></i>⟩</span></span>, which belongs to a <i>countably infinite</i>-dimensional Hilbert space; there are countably infinitely many <span class="texhtml"><i>k</i></span> values and basis vectors <span class="texhtml"><span class="nowrap">|<i>e<sub>k</sub></i>⟩</span></span>.</div></div><div class="tsingle" style="width:232px;max-width:232px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Continuous_complex_vector_components.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Continuous_complex_vector_components.svg/230px-Continuous_complex_vector_components.svg.png" decoding="async" width="230" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Continuous_complex_vector_components.svg/345px-Continuous_complex_vector_components.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Continuous_complex_vector_components.svg/460px-Continuous_complex_vector_components.svg.png 2x" data-file-width="214" data-file-height="200" /></a></span></div><div class="thumbcaption">Continuous components <span class="texhtml"><i>ψ</i>(<i>x</i>)</span> of a complex vector <span class="texhtml"><span class="nowrap">|<i>ψ</i>⟩</span> = ∫ d<i>x</i> <i>ψ</i>(<i>x</i>)<span class="nowrap">|<i>x</i>⟩</span></span>, which belongs to an <i>uncountably infinite</i>-dimensional Hilbert space; there are infinitely many <span class="texhtml"><i>x</i></span> values and basis vectors <span class="texhtml"><span class="nowrap">|<i>x</i>⟩</span></span>.</div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Components of complex vectors plotted against index number; discrete <span class="texhtml"><i>k</i></span> and continuous <span class="texhtml"><i>x</i></span>. Two particular components out of infinitely many are highlighted.</div></div></div></div> <p>The Hilbert space of a <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a>-0 point particle is spanned by a "position <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a>" <span class="texhtml">{ <span class="nowrap">|<b>r</b>⟩</span> }</span>, where the label <span class="texhtml"><b>r</b></span> extends over the set of all points in <a href="/wiki/Position_space" class="mw-redirect" title="Position space">position space</a>. This label is the eigenvalue of the position operator acting on such a basis state, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {r} }}|\mathbf {r} \rangle =\mathbf {r} |\mathbf {r} \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {r} }}|\mathbf {r} \rangle =\mathbf {r} |\mathbf {r} \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc58225cb7e18eb82edeb57732c38649b890fe28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.67ex; height:2.843ex;" alt="{\displaystyle {\hat {\mathbf {r} }}|\mathbf {r} \rangle =\mathbf {r} |\mathbf {r} \rangle }"></span>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2024)">citation needed</span></a></i>]</sup> Since there are an <a href="/wiki/Uncountably_infinite" class="mw-redirect" title="Uncountably infinite">uncountably infinite</a> number of vector components in the basis, this is an uncountably infinite-dimensional Hilbert space.<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> The dimensions of the Hilbert space (usually infinite) and position space (usually 1, 2 or 3) are not to be conflated. </p><p>Starting from any ket <span class="texhtml"><span class="nowrap">|Ψ⟩</span></span> in this Hilbert space, one may <i>define</i> a complex scalar function of <span class="texhtml"><b>r</b></span>, known as a <a href="/wiki/Wavefunction" class="mw-redirect" title="Wavefunction">wavefunction</a>,<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="Wave functions, by definition, belong to a separable (i.e. countable) Hilbert space (April 2024)">clarification needed</span></a></i>]</sup> <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 \Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |\Psi \rangle \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |\Psi \rangle \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5083954f9aea058e7277ddde8e33eeb74b31bd31" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.428ex; height:3.843ex;" alt="{\displaystyle \Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |\Psi \rangle \,.}"></span> </p><p>On the left-hand side, <span class="texhtml">Ψ(<b>r</b>)</span> is a function mapping any point in space to a complex number; on the right-hand side, <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 \left|\Psi \right\rangle =\int d^{3}\mathbf {r} \,\Psi (\mathbf {r} )\left|\mathbf {r} \right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mi mathvariant="normal">Ψ</mi> <mo>⟩</mo> </mrow> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mspace width="thinmathspace" /> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|\Psi \right\rangle =\int d^{3}\mathbf {r} \,\Psi (\mathbf {r} )\left|\mathbf {r} \right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30c06981517835f65cbe97b4cad2ba27a088292" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.561ex; height:5.676ex;" alt="{\displaystyle \left|\Psi \right\rangle =\int d^{3}\mathbf {r} \,\Psi (\mathbf {r} )\left|\mathbf {r} \right\rangle }"></span> is a ket consisting of a superposition of kets with relative coefficients specified by that function. </p><p>It is then customary to define linear operators acting on wavefunctions in terms of linear operators acting on kets, by <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 {\hat {A}}(\mathbf {r} )~\Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |{\hat {A}}|\Psi \rangle \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</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> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {A}}(\mathbf {r} )~\Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |{\hat {A}}|\Psi \rangle \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee44915430cd969e92588ef7582130c565e33a83" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.119ex; height:3.843ex;" alt="{\displaystyle {\hat {A}}(\mathbf {r} )~\Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |{\hat {A}}|\Psi \rangle \,.}"></span> </p><p>For instance, the <a href="/wiki/Momentum" title="Momentum">momentum</a> operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {p} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {p} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bda303998274a5d448b3493c8a62b31d7ff7143" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.485ex; height:2.676ex;" alt="{\displaystyle {\hat {\mathbf {p} }}}"></span> has the following coordinate representation, <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 {\hat {\mathbf {p} }}(\mathbf {r} )~\Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |{\hat {\mathbf {p} }}|\Psi \rangle =-i\hbar \nabla \Psi (\mathbf {r} )\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <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> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mi mathvariant="normal">∇</mi> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {p} }}(\mathbf {r} )~\Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |{\hat {\mathbf {p} }}|\Psi \rangle =-i\hbar \nabla \Psi (\mathbf {r} )\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d37cd44be674d9a5701043e73f133551f10cbbb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.209ex; height:3.843ex;" alt="{\displaystyle {\hat {\mathbf {p} }}(\mathbf {r} )~\Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |{\hat {\mathbf {p} }}|\Psi \rangle =-i\hbar \nabla \Psi (\mathbf {r} )\,.}"></span> </p><p>One occasionally even encounters an expression such 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 \nabla |\Psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <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 \nabla |\Psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8782adf1c0317eb51f6b09997c404696ac9a60d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.296ex; height:2.843ex;" alt="{\displaystyle \nabla |\Psi \rangle }"></span>, though this is something of an <a href="/wiki/Abuse_of_notation" title="Abuse of notation">abuse of notation</a>. The differential operator must be understood to be an abstract operator, acting on kets, that has the effect of differentiating wavefunctions once the expression is projected onto the position basis, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \langle \mathbf {r} |\Psi \rangle \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \langle \mathbf {r} |\Psi \rangle \,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/360ed2bcdad0a486c71f6e3c7dd969996f680c95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.336ex; height:2.843ex;" alt="{\displaystyle \nabla \langle \mathbf {r} |\Psi \rangle \,,}"></span> even though, in the momentum basis, this operator amounts to a mere multiplication operator (by <span class="texhtml"><i>iħ</i><b>p</b></span>). That is, to say, <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 \langle \mathbf {r} |{\hat {\mathbf {p} }}=-i\hbar \nabla \langle \mathbf {r} |~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mi mathvariant="normal">∇</mi> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \mathbf {r} |{\hat {\mathbf {p} }}=-i\hbar \nabla \langle \mathbf {r} |~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1d29d0ec1eafa970d91b3fc67bdd373c47ae7c1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.971ex; height:2.843ex;" alt="{\displaystyle \langle \mathbf {r} |{\hat {\mathbf {p} }}=-i\hbar \nabla \langle \mathbf {r} |~,}"></span> or <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 {\hat {\mathbf {p} }}=\int d^{3}\mathbf {r} ~|\mathbf {r} \rangle (-i\hbar \nabla )\langle \mathbf {r} |~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {p} }}=\int d^{3}\mathbf {r} ~|\mathbf {r} \rangle (-i\hbar \nabla )\langle \mathbf {r} |~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/819abf9668a8d09f0b88b09af68a864704fad209" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.316ex; height:5.676ex;" alt="{\displaystyle {\hat {\mathbf {p} }}=\int d^{3}\mathbf {r} ~|\mathbf {r} \rangle (-i\hbar \nabla )\langle \mathbf {r} |~.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Overlap_of_states">Overlap of states</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=10" title="Edit section: Overlap of states"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In quantum mechanics the expression <span class="texhtml"><span class="nowrap">⟨<i>φ</i>|<i>ψ</i>⟩</span></span> is typically interpreted as the <a href="/wiki/Probability_amplitude" title="Probability amplitude">probability amplitude</a> for the state <span class="texhtml"><i>ψ</i></span> to <a href="/wiki/Wavefunction_collapse" class="mw-redirect" title="Wavefunction collapse">collapse</a> into the state <span class="texhtml"><i>φ</i></span>. Mathematically, this means the coefficient for the projection of <span class="texhtml"><i>ψ</i></span> onto <span class="texhtml"><i>φ</i></span>. It is also described as the projection of state <span class="texhtml"><i>ψ</i></span> onto state <span class="texhtml"><i>φ</i></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Changing_basis_for_a_spin-1/2_particle"><span id="Changing_basis_for_a_spin-1.2F2_particle"></span>Changing basis for a spin-1/2 particle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=11" title="Edit section: Changing basis for a spin-1/2 particle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A stationary <a href="/wiki/Spin-1/2" title="Spin-1/2">spin-<style data-mw-deduplicate="TemplateStyles:r1154941027">'"`UNIQ--templatestyles-000000C4-QINU`"'</style><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span></a> particle has a two-dimensional Hilbert space. One <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> 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 |{\uparrow }_{z}\rangle \,,\;|{\downarrow }_{z}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↑</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\uparrow }_{z}\rangle \,,\;|{\downarrow }_{z}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b132e14d26d7a9e9411faadb6593500bfd0a83d2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.497ex; height:2.843ex;" alt="{\displaystyle |{\uparrow }_{z}\rangle \,,\;|{\downarrow }_{z}\rangle }"></span> where <span class="texhtml"><span class="nowrap">|↑<sub><i>z</i></sub>⟩</span></span> is the state with a definite value of the <a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">spin operator <span class="texhtml"><i>S<sub>z</sub></i></span></a> equal to +<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span> and <span class="texhtml"><span class="nowrap">|↓<sub><i>z</i></sub>⟩</span></span> is the state with a definite value of the <a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">spin operator <span class="texhtml"><i>S<sub>z</sub></i></span></a> equal to −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>. </p><p>Since these are a basis, <i>any</i> quantum state of the particle can be expressed as a <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> (i.e., <a href="/wiki/Quantum_superposition" title="Quantum superposition">quantum superposition</a>) of these two states: <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 |\psi \rangle =a_{\psi }|{\uparrow }_{z}\rangle +b_{\psi }|{\downarrow }_{z}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↑</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle =a_{\psi }|{\uparrow }_{z}\rangle +b_{\psi }|{\downarrow }_{z}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1bb75e6b7c2cfa89450923588e3db54d0b3bbe3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.267ex; height:3.009ex;" alt="{\displaystyle |\psi \rangle =a_{\psi }|{\uparrow }_{z}\rangle +b_{\psi }|{\downarrow }_{z}\rangle }"></span> where <span class="texhtml"><i>a<sub>ψ</sub></i></span> and <span class="texhtml"><i>b<sub>ψ</sub></i></span> are complex numbers. </p><p>A <i>different</i> basis for the same Hilbert space 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 |{\uparrow }_{x}\rangle \,,\;|{\downarrow }_{x}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↑</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\uparrow }_{x}\rangle \,,\;|{\downarrow }_{x}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ffb1687127391d7d6aa1613850783ed4ab6b0b3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.839ex; height:2.843ex;" alt="{\displaystyle |{\uparrow }_{x}\rangle \,,\;|{\downarrow }_{x}\rangle }"></span> defined in terms of <span class="texhtml"><i>S<sub>x</sub></i></span> rather than <span class="texhtml"><i>S<sub>z</sub></i></span>. </p><p>Again, <i>any</i> state of the particle can be expressed as a linear combination of these two: <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 |\psi \rangle =c_{\psi }|{\uparrow }_{x}\rangle +d_{\psi }|{\downarrow }_{x}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↑</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">↓</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle =c_{\psi }|{\uparrow }_{x}\rangle +d_{\psi }|{\downarrow }_{x}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b52c5fb67a87f569e6b920f91f76f4046a0e57e1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.597ex; height:3.009ex;" alt="{\displaystyle |\psi \rangle =c_{\psi }|{\uparrow }_{x}\rangle +d_{\psi }|{\downarrow }_{x}\rangle }"></span> </p><p>In vector form, you might write <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 |\psi \rangle \doteq {\begin{pmatrix}a_{\psi }\\b_{\psi }\end{pmatrix}}\quad {\text{or}}\quad |\psi \rangle \doteq {\begin{pmatrix}c_{\psi }\\d_{\psi }\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>≐</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>or</mtext> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>≐</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle \doteq {\begin{pmatrix}a_{\psi }\\b_{\psi }\end{pmatrix}}\quad {\text{or}}\quad |\psi \rangle \doteq {\begin{pmatrix}c_{\psi }\\d_{\psi }\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/860c4ab91191f92a745eaaa58d03ece6cd4912e5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.434ex; height:6.509ex;" alt="{\displaystyle |\psi \rangle \doteq {\begin{pmatrix}a_{\psi }\\b_{\psi }\end{pmatrix}}\quad {\text{or}}\quad |\psi \rangle \doteq {\begin{pmatrix}c_{\psi }\\d_{\psi }\end{pmatrix}}}"></span> depending on which basis you are using. In other words, the "coordinates" of a vector depend on the basis used. </p><p>There is a mathematical relationship between <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_{\psi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{\psi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f2fb80b10b5a3afdcc677b6c94ca3d256da4378" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.532ex; height:2.343ex;" alt="{\displaystyle a_{\psi }}"></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 b_{\psi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{\psi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6446fc300f037b679ac554d52ddaa687085e221" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.3ex; height:2.843ex;" alt="{\displaystyle b_{\psi }}"></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 c_{\psi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{\psi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6438fd5d86fd72a57413a2d49bc81df46b7a0740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.309ex; height:2.343ex;" alt="{\displaystyle c_{\psi }}"></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 d_{\psi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{\psi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d7ca083d4b2969c740df02e42f56ef804bd5a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.511ex; height:2.843ex;" alt="{\displaystyle d_{\psi }}"></span>; see <a href="/wiki/Change_of_basis" title="Change of basis">change of basis</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Pitfalls_and_ambiguous_uses">Pitfalls and ambiguous uses</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=12" title="Edit section: Pitfalls and ambiguous uses"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are some conventions and uses of notation that may be confusing or ambiguous for the non-initiated or early student. </p> <div class="mw-heading mw-heading3"><h3 id="Separation_of_inner_product_and_vectors">Separation of inner product and vectors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=13" title="Edit section: Separation of inner product and vectors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A cause for confusion is that the notation does not separate the inner-product operation from the notation for a (bra) vector. If a (dual space) bra-vector is constructed as a linear combination of other bra-vectors (for instance when expressing it in some basis) the notation creates some ambiguity and hides mathematical details. We can compare bra–ket notation to using bold for vectors, such 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 {\boldsymbol {\psi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ψ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\psi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7482208ebcc9b05c4c3bfec994cf25901c1f6c8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.762ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\psi }}}"></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 (\cdot ,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\cdot ,\cdot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fc515c912925128800226dd0b017be508069e24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle (\cdot ,\cdot )}"></span> for the inner product. Consider the following dual space bra-vector in the basis <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{|e_{n}\rangle \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{|e_{n}\rangle \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/912d988c05c2e62e2c25b8571d5e37d879b6e554" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.178ex; height:2.843ex;" alt="{\displaystyle \{|e_{n}\rangle \}}"></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 \langle \psi |=\sum _{n}\langle e_{n}|\psi _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi |=\sum _{n}\langle e_{n}|\psi _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dded403548e8f37d8acded72dac61dcc47a67789" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.103ex; height:5.509ex;" alt="{\displaystyle \langle \psi |=\sum _{n}\langle e_{n}|\psi _{n}}"></span> </p><p>It has to be determined by convention if the complex numbers <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 _{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\psi _{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e889c99c882e72863f4c9608efc4518f448fd514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.057ex; height:2.843ex;" alt="{\displaystyle \{\psi _{n}\}}"></span> are inside or outside of the inner product, and each convention gives different results. </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 \langle \psi |\equiv ({\boldsymbol {\psi }},\cdot )=\sum _{n}({\boldsymbol {e}}_{n},\cdot )\,\psi _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ψ</mi> </mrow> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi |\equiv ({\boldsymbol {\psi }},\cdot )=\sum _{n}({\boldsymbol {e}}_{n},\cdot )\,\psi _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3424f2a34330a3b0969241da7e034cd42c674e34" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.983ex; height:5.509ex;" alt="{\displaystyle \langle \psi |\equiv ({\boldsymbol {\psi }},\cdot )=\sum _{n}({\boldsymbol {e}}_{n},\cdot )\,\psi _{n}}"></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 \langle \psi |\equiv ({\boldsymbol {\psi }},\cdot )=\sum _{n}({\boldsymbol {e}}_{n}\psi _{n},\cdot )=\sum _{n}({\boldsymbol {e}}_{n},\cdot )\,\psi _{n}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ψ</mi> </mrow> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi |\equiv ({\boldsymbol {\psi }},\cdot )=\sum _{n}({\boldsymbol {e}}_{n}\psi _{n},\cdot )=\sum _{n}({\boldsymbol {e}}_{n},\cdot )\,\psi _{n}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0462808704239d95b2528af0cfb3b06880fd51a7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.165ex; height:5.509ex;" alt="{\displaystyle \langle \psi |\equiv ({\boldsymbol {\psi }},\cdot )=\sum _{n}({\boldsymbol {e}}_{n}\psi _{n},\cdot )=\sum _{n}({\boldsymbol {e}}_{n},\cdot )\,\psi _{n}^{*}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Reuse_of_symbols">Reuse of symbols</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=14" title="Edit section: Reuse of symbols"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is common to use the same symbol for <i>labels</i> and <i>constants</i>. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\alpha }}|\alpha \rangle =\alpha |\alpha \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\alpha }}|\alpha \rangle =\alpha |\alpha \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f81561e8cde519acc3235e5a05c47c8fcf7c5ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.152ex; height:2.843ex;" alt="{\displaystyle {\hat {\alpha }}|\alpha \rangle =\alpha |\alpha \rangle }"></span>, where the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> is used simultaneously as the <i>name of the operator</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/682d943d1947245b587f282aba6c88f0870fb302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:2.176ex;" alt="{\displaystyle {\hat {\alpha }}}"></span>, its <i>eigenvector</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\alpha \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f42032e642ee1c9d27adb318d34c7cc85f7a95d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.039ex; height:2.843ex;" alt="{\displaystyle |\alpha \rangle }"></span> and the associated <i>eigenvalue</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>. Sometimes the <i>hat</i> is also dropped for operators, and one can see notation such 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 A|a\rangle =a|a\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A|a\rangle =a|a\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6315dff46ddc2777664075fe063caffe49a1083b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.634ex; height:2.843ex;" alt="{\displaystyle A|a\rangle =a|a\rangle }"></span>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Hermitian_conjugate_of_kets">Hermitian conjugate of kets</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=15" title="Edit section: Hermitian conjugate of kets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is common to see the usage <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 \rangle ^{\dagger }=\langle \psi |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msup> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle ^{\dagger }=\langle \psi |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9a6f15eb28ff3b506d3483acafed4ccf7793e06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.19ex; height:3.176ex;" alt="{\displaystyle |\psi \rangle ^{\dagger }=\langle \psi |}"></span>, where the dagger (<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 \dagger }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>†</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dagger }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fbce70d5be6fec538cd30d8bc7b7bb2d3ed2d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.032ex; height:2.676ex;" alt="{\displaystyle \dagger }"></span>) corresponds to the Hermitian conjugate. This is however not correct in a technical sense, since the ket, <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 \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"></span>, represents a <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vector</a> in a complex Hilbert-space <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 {\mathcal {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}"></span>, and the bra, <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 \langle \psi |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d787823b57ed5783859bc23b2428db128fb2306e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle \langle \psi |}"></span>, is a <a href="/wiki/Linear_functional" class="mw-redirect" title="Linear functional">linear functional</a> on vectors in <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 {\mathcal {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}"></span>. In other words, <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 \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"></span> is just a vector, while <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 \langle \psi |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d787823b57ed5783859bc23b2428db128fb2306e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle \langle \psi |}"></span> is the combination of a vector and an inner product. </p> <div class="mw-heading mw-heading3"><h3 id="Operations_inside_bras_and_kets">Operations inside bras and kets</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=16" title="Edit section: Operations inside bras and kets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This is done for a fast notation of scaling vectors. For instance, if the vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\alpha \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f42032e642ee1c9d27adb318d34c7cc85f7a95d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.039ex; height:2.843ex;" alt="{\displaystyle |\alpha \rangle }"></span> is scaled 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 1/{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a0bbdb60fcb73ac67d9970a5eb0808b87fd37d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:3.176ex;" alt="{\displaystyle 1/{\sqrt {2}}}"></span>, it may be denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha /{\sqrt {2}}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\alpha /{\sqrt {2}}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2668a35424afbc0d99d56d8a361ab2c64123281a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.3ex; height:3.176ex;" alt="{\displaystyle |\alpha /{\sqrt {2}}\rangle }"></span>. This can be ambiguous since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> is simply a label for a state, and not a mathematical object on which operations can be performed. This usage is more common when denoting vectors as tensor products, where part of the labels are moved <b>outside</b> the designed slot, e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha \rangle =|\alpha /{\sqrt {2}}\rangle _{1}\otimes |\alpha /{\sqrt {2}}\rangle _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\alpha \rangle =|\alpha /{\sqrt {2}}\rangle _{1}\otimes |\alpha /{\sqrt {2}}\rangle _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/996adbde3959d6e63897e7a68233b554f130a4e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.686ex; height:3.176ex;" alt="{\displaystyle |\alpha \rangle =|\alpha /{\sqrt {2}}\rangle _{1}\otimes |\alpha /{\sqrt {2}}\rangle _{2}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Linear_operators">Linear operators</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=17" title="Edit section: Linear operators"><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/Linear_operator" class="mw-redirect" title="Linear operator">Linear operator</a></div> <div class="mw-heading mw-heading3"><h3 id="Linear_operators_acting_on_kets">Linear operators acting on kets</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=18" title="Edit section: Linear operators acting on kets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A linear operator is a map that inputs a ket and outputs a ket. (In order to be called "linear", it is required to have <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">certain properties</a>.) In other words, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f595a6c73d1183d6a1b2ac21fe47ac28c1483821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.843ex;" alt="{\displaystyle {\hat {A}}}"></span> is a linear operator 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 \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"></span> is a ket-vector, 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 {\hat {A}}|\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {A}}|\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb12cc68ebe19c31fbf93c23e5c04e385a5a18a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.84ex; height:3.343ex;" alt="{\displaystyle {\hat {A}}|\psi \rangle }"></span> is another ket-vector. </p><p>In an <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>-dimensional Hilbert space, we can impose a basis on the space and represent <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 \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"></span> in terms of its coordinates as a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\times 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>×</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\times 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a5fc06dc20829afcff1139b5ea311efae47e4a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.066ex; height:2.176ex;" alt="{\displaystyle N\times 1}"></span> <a href="/wiki/Column_vector" class="mw-redirect" title="Column vector">column vector</a>. Using the same basis 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 {\hat {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f595a6c73d1183d6a1b2ac21fe47ac28c1483821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.843ex;" alt="{\displaystyle {\hat {A}}}"></span>, it is represented by an <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\times N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>×</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\times N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99a86c5231bb3cbb863d9d428ebe9ac8db8d4ffb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.968ex; height:2.176ex;" alt="{\displaystyle N\times N}"></span> complex matrix. The ket-vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {A}}|\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {A}}|\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb12cc68ebe19c31fbf93c23e5c04e385a5a18a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.84ex; height:3.343ex;" alt="{\displaystyle {\hat {A}}|\psi \rangle }"></span> can now be computed by matrix multiplication. </p><p>Linear operators are ubiquitous in the theory of quantum mechanics. For example, observable physical quantities are represented by <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">self-adjoint operators</a>, such as <a href="/wiki/Energy" title="Energy">energy</a> or <a href="/wiki/Momentum" title="Momentum">momentum</a>, whereas transformative processes are represented by <a href="/wiki/Unitary_operator" title="Unitary operator">unitary</a> linear operators such as rotation or the progression of time. </p> <div class="mw-heading mw-heading3"><h3 id="Linear_operators_acting_on_bras">Linear operators acting on bras</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=19" title="Edit section: Linear operators acting on bras"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Operators can also be viewed as acting on bras <i>from the right hand side</i>. Specifically, if <span class="texhtml"><i><b>A</b></i></span> is a linear operator and <span class="texhtml"><span class="nowrap">⟨<i>φ</i>|</span></span> is a bra, then <span class="texhtml"><span class="nowrap">⟨<i>φ</i>|</span><i><b>A</b></i></span> is another bra defined by the rule <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 {\bigl (}\langle \phi |{\boldsymbol {A}}{\bigr )}|\psi \rangle =\langle \phi |{\bigl (}{\boldsymbol {A}}|\psi \rangle {\bigr )}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}\langle \phi |{\boldsymbol {A}}{\bigr )}|\psi \rangle =\langle \phi |{\bigl (}{\boldsymbol {A}}|\psi \rangle {\bigr )}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbd152cec54e5f3763ea064f62a61b30632292fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.434ex; height:3.176ex;" alt="{\displaystyle {\bigl (}\langle \phi |{\boldsymbol {A}}{\bigr )}|\psi \rangle =\langle \phi |{\bigl (}{\boldsymbol {A}}|\psi \rangle {\bigr )}\,,}"></span> (in other words, a <a href="/wiki/Function_composition" title="Function composition">function composition</a>). This expression is commonly written as (cf. <a href="/wiki/Energy_inner_product" class="mw-redirect" title="Energy inner product">energy inner product</a>) <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 \langle \phi |{\boldsymbol {A}}|\psi \rangle \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \phi |{\boldsymbol {A}}|\psi \rangle \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45f155af2ed3d88a5f3ff22d7de82dc87384991c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.055ex; height:2.843ex;" alt="{\displaystyle \langle \phi |{\boldsymbol {A}}|\psi \rangle \,.}"></span> </p><p>In an <span class="texhtml"><i>N</i></span>-dimensional Hilbert space, <span class="texhtml"><span class="nowrap">⟨<i>φ</i>|</span></span> can be written as a <span class="texhtml">1 × <i>N</i></span> <a href="/wiki/Row_vector" class="mw-redirect" title="Row vector">row vector</a>, and <span class="texhtml"><i><b>A</b></i></span> (as in the previous section) is an <span class="texhtml"><i>N</i> × <i>N</i></span> matrix. Then the bra <span class="texhtml"><span class="nowrap">⟨<i>φ</i>|</span><i><b>A</b></i></span> can be computed by normal matrix multiplication. </p><p>If the same state vector appears on both bra and ket side, <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 \langle \psi |{\boldsymbol {A}}|\psi \rangle \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi |{\boldsymbol {A}}|\psi \rangle \,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a230d2ed59ef2b1c74481053dbc923ece0f221d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.183ex; height:2.843ex;" alt="{\displaystyle \langle \psi |{\boldsymbol {A}}|\psi \rangle \,,}"></span> then this expression gives the <a href="/wiki/Expectation_value_(quantum_mechanics)" title="Expectation value (quantum mechanics)">expectation value</a>, or mean or average value, of the observable represented by operator <span class="texhtml"><i><b>A</b></i></span> for the physical system in the state <span class="texhtml"><span class="nowrap">|<i>ψ</i>⟩</span></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Outer_products">Outer products</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=20" title="Edit section: Outer products"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A convenient way to define linear operators on a Hilbert space <span class="texhtml"><span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">H</span></span> is given by the <a href="/wiki/Outer_product" title="Outer product">outer product</a>: if <span class="texhtml"><span class="nowrap">⟨<i>ϕ</i>|</span></span> is a bra and <span class="texhtml"><span class="nowrap">|<i>ψ</i>⟩</span></span> is a ket, the outer product <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 |\phi \rangle \,\langle \psi |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle \,\langle \psi |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95b74792aef11aa55eed386a84774253a0f727dd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.389ex; height:2.843ex;" alt="{\displaystyle |\phi \rangle \,\langle \psi |}"></span> denotes the <a href="/wiki/Finite-rank_operator" title="Finite-rank operator">rank-one operator</a> with the rule <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 {\bigl (}|\phi \rangle \langle \psi |{\bigr )}(x)=\langle \psi |x\rangle |\phi \rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}|\phi \rangle \langle \psi |{\bigr )}(x)=\langle \psi |x\rangle |\phi \rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cebecf934fc1c210641f7c18d349510f566525ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.252ex; height:3.176ex;" alt="{\displaystyle {\bigl (}|\phi \rangle \langle \psi |{\bigr )}(x)=\langle \psi |x\rangle |\phi \rangle .}"></span> </p><p>For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication: <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 |\phi \rangle \,\langle \psi |\doteq {\begin{pmatrix}\phi _{1}\\\phi _{2}\\\vdots \\\phi _{N}\end{pmatrix}}{\begin{pmatrix}\psi _{1}^{*}&\psi _{2}^{*}&\cdots &\psi _{N}^{*}\end{pmatrix}}={\begin{pmatrix}\phi _{1}\psi _{1}^{*}&\phi _{1}\psi _{2}^{*}&\cdots &\phi _{1}\psi _{N}^{*}\\\phi _{2}\psi _{1}^{*}&\phi _{2}\psi _{2}^{*}&\cdots &\phi _{2}\psi _{N}^{*}\\\vdots &\vdots &\ddots &\vdots \\\phi _{N}\psi _{1}^{*}&\phi _{N}\psi _{2}^{*}&\cdots &\phi _{N}\psi _{N}^{*}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≐</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> <mtd> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> <mtd> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> <mtd> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle \,\langle \psi |\doteq {\begin{pmatrix}\phi _{1}\\\phi _{2}\\\vdots \\\phi _{N}\end{pmatrix}}{\begin{pmatrix}\psi _{1}^{*}&\psi _{2}^{*}&\cdots &\psi _{N}^{*}\end{pmatrix}}={\begin{pmatrix}\phi _{1}\psi _{1}^{*}&\phi _{1}\psi _{2}^{*}&\cdots &\phi _{1}\psi _{N}^{*}\\\phi _{2}\psi _{1}^{*}&\phi _{2}\psi _{2}^{*}&\cdots &\phi _{2}\psi _{N}^{*}\\\vdots &\vdots &\ddots &\vdots \\\phi _{N}\psi _{1}^{*}&\phi _{N}\psi _{2}^{*}&\cdots &\phi _{N}\psi _{N}^{*}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13e29b13218d7dccc8c9cfc8b811ff8c4682356d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:73.407ex; height:15.176ex;" alt="{\displaystyle |\phi \rangle \,\langle \psi |\doteq {\begin{pmatrix}\phi _{1}\\\phi _{2}\\\vdots \\\phi _{N}\end{pmatrix}}{\begin{pmatrix}\psi _{1}^{*}&\psi _{2}^{*}&\cdots &\psi _{N}^{*}\end{pmatrix}}={\begin{pmatrix}\phi _{1}\psi _{1}^{*}&\phi _{1}\psi _{2}^{*}&\cdots &\phi _{1}\psi _{N}^{*}\\\phi _{2}\psi _{1}^{*}&\phi _{2}\psi _{2}^{*}&\cdots &\phi _{2}\psi _{N}^{*}\\\vdots &\vdots &\ddots &\vdots \\\phi _{N}\psi _{1}^{*}&\phi _{N}\psi _{2}^{*}&\cdots &\phi _{N}\psi _{N}^{*}\end{pmatrix}}}"></span> The outer product is an <span class="texhtml"><i>N</i> × <i>N</i></span> matrix, as expected for a linear operator. </p><p>One of the uses of the outer product is to construct <a href="/wiki/Projection_operator" class="mw-redirect" title="Projection operator">projection operators</a>. Given a ket <span class="texhtml"><span class="nowrap">|<i>ψ</i>⟩</span></span> of norm 1, the orthogonal projection onto the <a href="/wiki/Linear_subspace" title="Linear subspace">subspace</a> spanned by <span class="texhtml"><span class="nowrap">|<i>ψ</i>⟩</span></span> 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 |\psi \rangle \,\langle \psi |\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle \,\langle \psi |\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01f4a571eea93d3cc8cbb56a48d48830985c61b5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.55ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle \,\langle \psi |\,.}"></span> This is an <a href="/wiki/Idempotent" class="mw-redirect" title="Idempotent">idempotent</a> in the algebra of observables that acts on the Hilbert space. </p> <div class="mw-heading mw-heading3"><h3 id="Hermitian_conjugate_operator">Hermitian conjugate operator</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=21" title="Edit section: Hermitian conjugate operator"><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/Hermitian_conjugate" class="mw-redirect" title="Hermitian conjugate">Hermitian conjugate</a></div> <p>Just as kets and bras can be transformed into each other (making <span class="texhtml"><span class="nowrap">|<i>ψ</i>⟩</span></span> into <span class="texhtml"><span class="nowrap">⟨<i>ψ</i>|</span></span>), the element from the dual space corresponding to <span class="texhtml"><i>A</i><span class="nowrap">|<i>ψ</i>⟩</span></span> is <span class="texhtml"><span class="nowrap">⟨<i>ψ</i>|</span><i>A</i><sup>†</sup></span>, where <span class="texhtml"><i>A</i><sup>†</sup></span> denotes the Hermitian conjugate (or adjoint) of the operator <span class="texhtml"><i>A</i></span>. In other words, <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 |\phi \rangle =A|\psi \rangle \quad {\text{if and only if}}\quad \langle \phi |=\langle \psi |A^{\dagger }\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>if and only if</mtext> </mrow> <mspace width="1em" /> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\phi \rangle =A|\psi \rangle \quad {\text{if and only if}}\quad \langle \phi |=\langle \psi |A^{\dagger }\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/822a04676c6d84ad6398da3e328589a0d9016a6b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.864ex; height:3.176ex;" alt="{\displaystyle |\phi \rangle =A|\psi \rangle \quad {\text{if and only if}}\quad \langle \phi |=\langle \psi |A^{\dagger }\,.}"></span> </p><p>If <span class="texhtml"><i>A</i></span> is expressed as an <span class="texhtml"><i>N</i> × <i>N</i></span> matrix, then <span class="texhtml"><i>A</i><sup>†</sup></span> is its conjugate transpose. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=22" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bra–ket notation was designed to facilitate the formal manipulation of linear-algebraic expressions. Some of the properties that allow this manipulation are listed herein. In what follows, <span class="texhtml"><i>c</i><sub>1</sub></span> and <span class="texhtml"><i>c</i><sub>2</sub></span> denote arbitrary <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, <span class="texhtml"><i>c</i>*</span> denotes the <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> of <span class="texhtml"><i>c</i></span>, <span class="texhtml"><i>A</i></span> and <span class="texhtml"><i>B</i></span> denote arbitrary linear operators, and these properties are to hold for any choice of bras and kets. </p> <div class="mw-heading mw-heading3"><h3 id="Linearity">Linearity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=23" title="Edit section: Linearity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Since bras are linear functionals, <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 \langle \phi |{\bigl (}c_{1}|\psi _{1}\rangle +c_{2}|\psi _{2}\rangle {\bigr )}=c_{1}\langle \phi |\psi _{1}\rangle +c_{2}\langle \phi |\psi _{2}\rangle \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \phi |{\bigl (}c_{1}|\psi _{1}\rangle +c_{2}|\psi _{2}\rangle {\bigr )}=c_{1}\langle \phi |\psi _{1}\rangle +c_{2}\langle \phi |\psi _{2}\rangle \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34fa5b322e779b0dd9f2fabfbea07bfdadc6b927" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.18ex; height:3.176ex;" alt="{\displaystyle \langle \phi |{\bigl (}c_{1}|\psi _{1}\rangle +c_{2}|\psi _{2}\rangle {\bigr )}=c_{1}\langle \phi |\psi _{1}\rangle +c_{2}\langle \phi |\psi _{2}\rangle \,.}"></span></li> <li>By the definition of addition and scalar multiplication of linear functionals in the dual space,<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> <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 {\bigl (}c_{1}\langle \phi _{1}|+c_{2}\langle \phi _{2}|{\bigr )}|\psi \rangle =c_{1}\langle \phi _{1}|\psi \rangle +c_{2}\langle \phi _{2}|\psi \rangle \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}c_{1}\langle \phi _{1}|+c_{2}\langle \phi _{2}|{\bigr )}|\psi \rangle =c_{1}\langle \phi _{1}|\psi \rangle +c_{2}\langle \phi _{2}|\psi \rangle \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d313ff5b460c66854cd2a40c1bd41f8f8a11b38" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.052ex; height:3.176ex;" alt="{\displaystyle {\bigl (}c_{1}\langle \phi _{1}|+c_{2}\langle \phi _{2}|{\bigr )}|\psi \rangle =c_{1}\langle \phi _{1}|\psi \rangle +c_{2}\langle \phi _{2}|\psi \rangle \,.}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Associativity">Associativity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=24" title="Edit section: Associativity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given any expression involving complex numbers, bras, kets, inner products, outer products, and/or linear operators (but not addition), written in bra–ket notation, the parenthetical groupings do not matter (i.e., the <a href="/wiki/Associative_property" title="Associative property">associative property</a> holds). For example: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\langle \psi |{\bigl (}A|\phi \rangle {\bigr )}={\bigl (}\langle \psi |A{\bigr )}|\phi \rangle \,&{\stackrel {\text{def}}{=}}\,\langle \psi |A|\phi \rangle \\{\bigl (}A|\psi \rangle {\bigr )}\langle \phi |=A{\bigl (}|\psi \rangle \langle \phi |{\bigr )}\,&{\stackrel {\text{def}}{=}}\,A|\psi \rangle \langle \phi |\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\langle \psi |{\bigl (}A|\phi \rangle {\bigr )}={\bigl (}\langle \psi |A{\bigr )}|\phi \rangle \,&{\stackrel {\text{def}}{=}}\,\langle \psi |A|\phi \rangle \\{\bigl (}A|\psi \rangle {\bigr )}\langle \phi |=A{\bigl (}|\psi \rangle \langle \phi |{\bigr )}\,&{\stackrel {\text{def}}{=}}\,A|\psi \rangle \langle \phi |\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70a7cba5bab9b7ece342d1a9be531e90375d9925" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:34.265ex; height:8.176ex;" alt="{\displaystyle {\begin{aligned}\langle \psi |{\bigl (}A|\phi \rangle {\bigr )}={\bigl (}\langle \psi |A{\bigr )}|\phi \rangle \,&{\stackrel {\text{def}}{=}}\,\langle \psi |A|\phi \rangle \\{\bigl (}A|\psi \rangle {\bigr )}\langle \phi |=A{\bigl (}|\psi \rangle \langle \phi |{\bigr )}\,&{\stackrel {\text{def}}{=}}\,A|\psi \rangle \langle \phi |\end{aligned}}}"></span></dd></dl> <p>and so forth. The expressions on the right (with no parentheses whatsoever) are allowed to be written unambiguously <i>because</i> of the equalities on the left. Note that the associative property does <i>not</i> hold for expressions that include nonlinear operators, such as the <a href="/wiki/Antilinear" class="mw-redirect" title="Antilinear">antilinear</a> <a href="/wiki/T-symmetry" title="T-symmetry">time reversal operator</a> in physics. </p> <div class="mw-heading mw-heading3"><h3 id="Hermitian_conjugation">Hermitian conjugation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=25" title="Edit section: Hermitian conjugation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bra–ket notation makes it particularly easy to compute the Hermitian conjugate (also called <i>dagger</i>, and denoted <span class="texhtml">†</span>) of expressions. The formal rules are: </p> <ul><li>The Hermitian conjugate of a bra is the corresponding ket, and vice versa.</li> <li>The Hermitian conjugate of a complex number is its complex conjugate.</li> <li>The Hermitian conjugate of the Hermitian conjugate of anything (linear operators, bras, kets, numbers) is itself—i.e., <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 \left(x^{\dagger }\right)^{\dagger }=x\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(x^{\dagger }\right)^{\dagger }=x\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4db0a7eb51dd26c18c258fb243847434f2820f2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.846ex; height:3.843ex;" alt="{\displaystyle \left(x^{\dagger }\right)^{\dagger }=x\,.}"></span></li> <li>Given any combination of complex numbers, bras, kets, inner products, outer products, and/or linear operators, written in bra–ket notation, its Hermitian conjugate can be computed by reversing the order of the components, and taking the Hermitian conjugate of each.</li></ul> <p>These rules are sufficient to formally write the Hermitian conjugate of any such expression; some examples are as follows: </p> <ul><li>Kets: <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 {\bigl (}c_{1}|\psi _{1}\rangle +c_{2}|\psi _{2}\rangle {\bigr )}^{\dagger }=c_{1}^{*}\langle \psi _{1}|+c_{2}^{*}\langle \psi _{2}|\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>=</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}c_{1}|\psi _{1}\rangle +c_{2}|\psi _{2}\rangle {\bigr )}^{\dagger }=c_{1}^{*}\langle \psi _{1}|+c_{2}^{*}\langle \psi _{2}|\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3d98a0ebf29b86f115b67cc1d0c69afcbc8e046" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.625ex; height:3.843ex;" alt="{\displaystyle {\bigl (}c_{1}|\psi _{1}\rangle +c_{2}|\psi _{2}\rangle {\bigr )}^{\dagger }=c_{1}^{*}\langle \psi _{1}|+c_{2}^{*}\langle \psi _{2}|\,.}"></span></li> <li>Inner products: <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 \langle \phi |\psi \rangle ^{*}=\langle \psi |\phi \rangle \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msup> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \phi |\psi \rangle ^{*}=\langle \psi |\phi \rangle \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f158793f9d1b526f14c1886b7a74232bb103df3a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.896ex; height:2.843ex;" alt="{\displaystyle \langle \phi |\psi \rangle ^{*}=\langle \psi |\phi \rangle \,.}"></span> Note that <span class="texhtml"><span class="nowrap">⟨<i>φ</i>|<i>ψ</i>⟩</span></span> is a scalar, so the Hermitian conjugate is just the complex conjugate, i.e., <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 {\bigl (}\langle \phi |\psi \rangle {\bigr )}^{\dagger }=\langle \phi |\psi \rangle ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msup> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}\langle \phi |\psi \rangle {\bigr )}^{\dagger }=\langle \phi |\psi \rangle ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1832b655aabc9305ef22b36f47ce126c6b6e1649" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.954ex; height:3.843ex;" alt="{\displaystyle {\bigl (}\langle \phi |\psi \rangle {\bigr )}^{\dagger }=\langle \phi |\psi \rangle ^{*}}"></span></li> <li>Matrix elements: <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 {\begin{aligned}\langle \phi |A|\psi \rangle ^{\dagger }&=\left\langle \psi \left|A^{\dagger }\right|\phi \right\rangle \\\left\langle \phi \left|A^{\dagger }B^{\dagger }\right|\psi \right\rangle ^{\dagger }&=\langle \psi |BA|\phi \rangle \,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msup> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>⟨</mo> <mrow> <mi>ψ</mi> <mrow> <mo>|</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>|</mo> </mrow> <mi>ϕ</mi> </mrow> <mo>⟩</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mrow> <mo>⟨</mo> <mrow> <mi>ϕ</mi> <mrow> <mo>|</mo> <mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mrow> <mo>|</mo> </mrow> <mi>ψ</mi> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\langle \phi |A|\psi \rangle ^{\dagger }&=\left\langle \psi \left|A^{\dagger }\right|\phi \right\rangle \\\left\langle \phi \left|A^{\dagger }B^{\dagger }\right|\psi \right\rangle ^{\dagger }&=\langle \psi |BA|\phi \rangle \,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5614085861ac4a66f5c80f15cb3a9d6dd04099a0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:27.948ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}\langle \phi |A|\psi \rangle ^{\dagger }&=\left\langle \psi \left|A^{\dagger }\right|\phi \right\rangle \\\left\langle \phi \left|A^{\dagger }B^{\dagger }\right|\psi \right\rangle ^{\dagger }&=\langle \psi |BA|\phi \rangle \,.\end{aligned}}}"></span></li> <li>Outer products: <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 {\Big (}{\bigl (}c_{1}|\phi _{1}\rangle \langle \psi _{1}|{\bigr )}+{\bigl (}c_{2}|\phi _{2}\rangle \langle \psi _{2}|{\bigr )}{\Big )}^{\dagger }={\bigl (}c_{1}^{*}|\psi _{1}\rangle \langle \phi _{1}|{\bigr )}+{\bigl (}c_{2}^{*}|\psi _{2}\rangle \langle \phi _{2}|{\bigr )}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Big (}{\bigl (}c_{1}|\phi _{1}\rangle \langle \psi _{1}|{\bigr )}+{\bigl (}c_{2}|\phi _{2}\rangle \langle \psi _{2}|{\bigr )}{\Big )}^{\dagger }={\bigl (}c_{1}^{*}|\psi _{1}\rangle \langle \phi _{1}|{\bigr )}+{\bigl (}c_{2}^{*}|\psi _{2}\rangle \langle \phi _{2}|{\bigr )}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33f1af662ef009ad5e9a619e864fb1ade4d284fc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:62.755ex; height:5.343ex;" alt="{\displaystyle {\Big (}{\bigl (}c_{1}|\phi _{1}\rangle \langle \psi _{1}|{\bigr )}+{\bigl (}c_{2}|\phi _{2}\rangle \langle \psi _{2}|{\bigr )}{\Big )}^{\dagger }={\bigl (}c_{1}^{*}|\psi _{1}\rangle \langle \phi _{1}|{\bigr )}+{\bigl (}c_{2}^{*}|\psi _{2}\rangle \langle \phi _{2}|{\bigr )}\,.}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Composite_bras_and_kets">Composite bras and kets</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=26" title="Edit section: Composite bras and kets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Two Hilbert spaces <span class="texhtml"><i>V</i></span> and <span class="texhtml"><i>W</i></span> may form a third space <span class="texhtml"><i>V</i> ⊗ <i>W</i></span> by a <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a>. In quantum mechanics, this is used for describing composite systems. If a system is composed of two subsystems described in <span class="texhtml"><i>V</i></span> and <span class="texhtml"><i>W</i></span> respectively, then the Hilbert space of the entire system is the tensor product of the two spaces. (The exception to this is if the subsystems are actually <a href="/wiki/Identical_particles" class="mw-redirect" title="Identical particles">identical particles</a>. In that case, the situation is a little more complicated.) </p><p>If <span class="texhtml"><span class="nowrap">|<i>ψ</i>⟩</span></span> is a ket in <span class="texhtml"><i>V</i></span> and <span class="texhtml"><span class="nowrap">|<i>φ</i>⟩</span></span> is a ket in <span class="texhtml"><i>W</i></span>, the tensor product of the two kets is a ket in <span class="texhtml"><i>V</i> ⊗ <i>W</i></span>. This is written in various notations: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle |\phi \rangle \,,\quad |\psi \rangle \otimes |\phi \rangle \,,\quad |\psi \phi \rangle \,,\quad |\psi ,\phi \rangle \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo>,</mo> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle |\phi \rangle \,,\quad |\psi \rangle \otimes |\phi \rangle \,,\quad |\psi \phi \rangle \,,\quad |\psi ,\phi \rangle \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b0724e44de71be5456cd51a6f1086176c0eed3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.042ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle |\phi \rangle \,,\quad |\psi \rangle \otimes |\phi \rangle \,,\quad |\psi \phi \rangle \,,\quad |\psi ,\phi \rangle \,.}"></span></dd></dl> <p>See <a href="/wiki/Quantum_entanglement#Quantum_mechanical_framework" title="Quantum entanglement">quantum entanglement</a> and the <a href="/wiki/EPR_paradox#Mathematical_formulation" class="mw-redirect" title="EPR paradox">EPR paradox</a> for applications of this product. </p> <div class="mw-heading mw-heading2"><h2 id="The_unit_operator">The unit operator</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=27" title="Edit section: The unit operator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider a complete <a href="/wiki/Orthonormal" class="mw-redirect" title="Orthonormal">orthonormal</a> system (<i>basis</i>), <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 \{e_{i}\ |\ i\in \mathbb {N} \}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo fence="false" stretchy="false">}</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{e_{i}\ |\ i\in \mathbb {N} \}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b920d5ad8a81fa28fcc5fd6263ec07456b5c943a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.371ex; height:2.843ex;" alt="{\displaystyle \{e_{i}\ |\ i\in \mathbb {N} \}\,,}"></span> for a Hilbert space <span class="texhtml"><i>H</i></span>, with respect to the norm from an inner product <span class="texhtml"><span class="nowrap">⟨·,·⟩</span></span>. </p><p>From basic <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>, it is known that any ket <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 \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"></span> can also be written as <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 |\psi \rangle =\sum _{i\in \mathbb {N} }\langle e_{i}|\psi \rangle |e_{i}\rangle ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </munder> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle =\sum _{i\in \mathbb {N} }\langle e_{i}|\psi \rangle |e_{i}\rangle ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52d39619e5e29712be9eb66a44d58be3761c369d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.452ex; height:5.676ex;" alt="{\displaystyle |\psi \rangle =\sum _{i\in \mathbb {N} }\langle e_{i}|\psi \rangle |e_{i}\rangle ,}"></span> with <span class="texhtml"><span class="nowrap">⟨·|·⟩</span></span> the inner product on the Hilbert space. </p><p>From the commutativity of kets with (complex) scalars, it follows 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 _{i\in \mathbb {N} }|e_{i}\rangle \langle e_{i}|=\mathbb {I} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i\in \mathbb {N} }|e_{i}\rangle \langle e_{i}|=\mathbb {I} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e86676f7992a0aeb92dd8e179c2c189100f88559" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.615ex; height:5.676ex;" alt="{\displaystyle \sum _{i\in \mathbb {N} }|e_{i}\rangle \langle e_{i}|=\mathbb {I} }"></span> must be the <i>identity operator</i>, which sends each vector to itself. </p><p>This, then, can be inserted in any expression without affecting its value; for example <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 {\begin{aligned}\langle v|w\rangle &=\langle v|\left(\sum _{i\in \mathbb {N} }|e_{i}\rangle \langle e_{i}|\right)|w\rangle \\&=\langle v|\left(\sum _{i\in \mathbb {N} }|e_{i}\rangle \langle e_{i}|\right)\left(\sum _{j\in \mathbb {N} }|e_{j}\rangle \langle e_{j}|\right)|w\rangle \\&=\langle v|e_{i}\rangle \langle e_{i}|e_{j}\rangle \langle e_{j}|w\rangle \,,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>w</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\langle v|w\rangle &=\langle v|\left(\sum _{i\in \mathbb {N} }|e_{i}\rangle \langle e_{i}|\right)|w\rangle \\&=\langle v|\left(\sum _{i\in \mathbb {N} }|e_{i}\rangle \langle e_{i}|\right)\left(\sum _{j\in \mathbb {N} }|e_{j}\rangle \langle e_{j}|\right)|w\rangle \\&=\langle v|e_{i}\rangle \langle e_{i}|e_{j}\rangle \langle e_{j}|w\rangle \,,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe4c785ef6b2cedbbcc5422962f9048db53ddbc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:44.959ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}\langle v|w\rangle &=\langle v|\left(\sum _{i\in \mathbb {N} }|e_{i}\rangle \langle e_{i}|\right)|w\rangle \\&=\langle v|\left(\sum _{i\in \mathbb {N} }|e_{i}\rangle \langle e_{i}|\right)\left(\sum _{j\in \mathbb {N} }|e_{j}\rangle \langle e_{j}|\right)|w\rangle \\&=\langle v|e_{i}\rangle \langle e_{i}|e_{j}\rangle \langle e_{j}|w\rangle \,,\end{aligned}}}"></span> where, in the last line, the <a href="/wiki/Einstein_summation_convention" class="mw-redirect" title="Einstein summation convention">Einstein summation convention</a> has been used to avoid clutter. </p><p>In quantum mechanics, it often occurs that little or no information about the inner product <span class="texhtml"><span class="nowrap">⟨<i>ψ</i>|<i>φ</i>⟩</span></span> of two arbitrary (state) kets is present, while it is still possible to say something about the expansion coefficients <span class="texhtml"><span class="nowrap">⟨<i>ψ</i>|<i>e<sub>i</sub></i>⟩</span> = <span class="nowrap">⟨<i>e<sub>i</sub></i>|<i>ψ</i>⟩</span>*</span> and <span class="texhtml"><span class="nowrap">⟨<i>e<sub>i</sub></i>|<i>φ</i>⟩</span></span> of those vectors with respect to a specific (orthonormalized) basis. In this case, it is particularly useful to insert the unit operator into the bracket one time or more. </p><p>For more information, see <a href="/wiki/Resolution_of_the_identity" class="mw-redirect" title="Resolution of the identity">Resolution of the identity</a>,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> <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 {\mathbb {I} }=\int \!dx~|x\rangle \langle x|=\int \!dp~|p\rangle \langle p|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">I</mi> </mrow> </mrow> <mo>=</mo> <mo>∫</mo> <mspace width="negativethinmathspace" /> <mi>d</mi> <mi>x</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo>∫</mo> <mspace width="negativethinmathspace" /> <mi>d</mi> <mi>p</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {I} }=\int \!dx~|x\rangle \langle x|=\int \!dp~|p\rangle \langle p|,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ef5e42f1bedaba8fae7862afeb932b8e4cd5c85" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.432ex; height:5.676ex;" alt="{\displaystyle {\mathbb {I} }=\int \!dx~|x\rangle \langle x|=\int \!dp~|p\rangle \langle p|,}"></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 |p\rangle =\int dx{\frac {e^{ixp/\hbar }|x\rangle }{\sqrt {2\pi \hbar }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>∫</mo> <mi>d</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <msqrt> <mn>2</mn> <mi>π</mi> <mi class="MJX-variant">ℏ</mi> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |p\rangle =\int dx{\frac {e^{ixp/\hbar }|x\rangle }{\sqrt {2\pi \hbar }}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/452664bf6a393ee6c94ff6750d0ce9cd6098da0d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.706ex; height:7.009ex;" alt="{\displaystyle |p\rangle =\int dx{\frac {e^{ixp/\hbar }|x\rangle }{\sqrt {2\pi \hbar }}}.}"></span> </p><p>Since <span class="texhtml"><span class="nowrap">⟨<i>x</i><span class="nowrap" style="padding-left:0.15em;">′</span>|<i>x</i>⟩</span> = <i>δ</i>(<i>x</i> − <i>x</i><span class="nowrap" style="padding-left:0.15em;">′</span>)</span>, plane waves follow, <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 \langle x|p\rangle ={\frac {e^{ixp/\hbar }}{\sqrt {2\pi \hbar }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>x</mi> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> </mrow> </msup> <msqrt> <mn>2</mn> <mi>π</mi> <mi class="MJX-variant">ℏ</mi> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x|p\rangle ={\frac {e^{ixp/\hbar }}{\sqrt {2\pi \hbar }}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/669214e68214a51461d88c6399aa0e10caa9f07e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.273ex; height:6.676ex;" alt="{\displaystyle \langle x|p\rangle ={\frac {e^{ixp/\hbar }}{\sqrt {2\pi \hbar }}}.}"></span> </p><p>In his book (1958), Ch. III.20, Dirac defines the <i>standard ket</i> which, up to a normalization, is the translationally invariant momentum eigenstate <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 |\varpi \rangle =\lim _{p\to 0}|p\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϖ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle |\varpi \rangle =\lim _{p\to 0}|p\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef4c36822ded733d03857bd159cfd51cb65c69c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.436ex; height:3.009ex;" alt="{\textstyle |\varpi \rangle =\lim _{p\to 0}|p\rangle }"></span> in the momentum representation, i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {p}}|\varpi \rangle =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϖ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {p}}|\varpi \rangle =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0137b3457b0140f0a484078b9cb268b6ec0ba7e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.186ex; height:2.843ex;" alt="{\displaystyle {\hat {p}}|\varpi \rangle =0}"></span>. Consequently, the corresponding wavefunction is a constant, <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 \langle x|\varpi \rangle {\sqrt {2\pi \hbar }}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϖ</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> <mi class="MJX-variant">ℏ</mi> </msqrt> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x|\varpi \rangle {\sqrt {2\pi \hbar }}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e43ed5b97b681bcf4764564c4c6a052d2528b9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.708ex; height:3.176ex;" alt="{\displaystyle \langle x|\varpi \rangle {\sqrt {2\pi \hbar }}=1}"></span>, and <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 |x\rangle =\delta ({\hat {x}}-x)|\varpi \rangle {\sqrt {2\pi \hbar }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϖ</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> <mi class="MJX-variant">ℏ</mi> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x\rangle =\delta ({\hat {x}}-x)|\varpi \rangle {\sqrt {2\pi \hbar }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/503d61aa2dfe9448ee11ae2258a66edd870ae8a1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.197ex; height:3.176ex;" alt="{\displaystyle |x\rangle =\delta ({\hat {x}}-x)|\varpi \rangle {\sqrt {2\pi \hbar }},}"></span> as well as <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\rangle =\exp(ip{\hat {x}}/\hbar )|\varpi \rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi class="MJX-variant">ℏ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϖ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |p\rangle =\exp(ip{\hat {x}}/\hbar )|\varpi \rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fab09ee2de86d34c9910d941d44a081fad47058" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.074ex; height:2.843ex;" alt="{\displaystyle |p\rangle =\exp(ip{\hat {x}}/\hbar )|\varpi \rangle .}"></span> </p><p>Typically, when all matrix elements of an operator such as <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 \langle x|A|y\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x|A|y\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da3e7779ca9d3a4f64eb0edc6067710560138a5d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.331ex; height:2.843ex;" alt="{\displaystyle \langle x|A|y\rangle }"></span> are available, this resolution serves to reconstitute the full operator, <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 \int dx\,dy\,|x\rangle \langle x|A|y\rangle \langle y|=A\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>A</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int dx\,dy\,|x\rangle \langle x|A|y\rangle \langle y|=A\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a34182f6b8650083f6cecb734eeb3400226bd490" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.067ex; height:5.676ex;" alt="{\displaystyle \int dx\,dy\,|x\rangle \langle x|A|y\rangle \langle y|=A\,.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Notation_used_by_mathematicians">Notation used by mathematicians</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=28" title="Edit section: Notation used by mathematicians"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The object physicists are considering when using bra–ket notation is a Hilbert space (a <a href="/wiki/Complete_metric_space" title="Complete metric space">complete</a> inner product space). </p><p>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 ({\mathcal {H}},\langle \cdot ,\cdot \rangle )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mo>,</mo> <mo fence="false" stretchy="false">⟨</mo> <mo>⋅</mo> <mo>,</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathcal {H}},\langle \cdot ,\cdot \rangle )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3965bf28dc09229ad57a101951cfe7f1ead0494d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.944ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {H}},\langle \cdot ,\cdot \rangle )}"></span> be a Hilbert space and <span class="texhtml"><i>h</i> ∈ <span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">H</span></span> a vector in <span class="texhtml"><span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">H</span></span>. What physicists would denote by <span class="texhtml"><span class="nowrap">|<i>h</i>⟩</span></span> is the vector itself. That 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 |h\rangle \in {\mathcal {H}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>h</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |h\rangle \in {\mathcal {H}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed430fdf5a987c47f27b258123fc83ba769fd902" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.342ex; height:2.843ex;" alt="{\displaystyle |h\rangle \in {\mathcal {H}}.}"></span> </p><p>Let <span class="texhtml"><span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">H</span>*</span> be the dual space of <span class="texhtml"><span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">H</span></span>. This is the space of linear functionals on <span class="texhtml"><span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">H</span></span>. The <a href="/wiki/Embedding" title="Embedding">embedding</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 \Phi :{\mathcal {H}}\hookrightarrow {\mathcal {H}}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mo stretchy="false">↪</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi :{\mathcal {H}}\hookrightarrow {\mathcal {H}}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/491a9049cda35c719900d193cd9d21074cf7e9fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.504ex; height:2.343ex;" alt="{\displaystyle \Phi :{\mathcal {H}}\hookrightarrow {\mathcal {H}}^{*}}"></span> is defined 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 \Phi (h)=\varphi _{h}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (h)=\varphi _{h}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fe592acea5a211128575132ab57ac030cf0ae6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.624ex; height:2.843ex;" alt="{\displaystyle \Phi (h)=\varphi _{h}}"></span>, where for every <span class="texhtml"><i>h</i> ∈ <span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">H</span></span> the linear functional <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 _{h}:{\mathcal {H}}\to \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{h}:{\mathcal {H}}\to \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db6a7b7c636bcf2f4621c8c848c4bef6f8bbc851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.892ex; height:2.676ex;" alt="{\displaystyle \varphi _{h}:{\mathcal {H}}\to \mathbb {C} }"></span> satisfies for every <span class="texhtml"><i>g</i> ∈ <span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">H</span></span> the functional 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 \varphi _{h}(g)=\langle h,g\rangle =\langle h\mid g\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>h</mi> <mo>,</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>h</mi> <mo>∣</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{h}(g)=\langle h,g\rangle =\langle h\mid g\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be2d609647227bb81bc36b99736abb1ef2ee9c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.321ex; height:2.843ex;" alt="{\displaystyle \varphi _{h}(g)=\langle h,g\rangle =\langle h\mid g\rangle }"></span>. Notational confusion arises when identifying <span class="texhtml"><i>φ<sub>h</sub></i></span> and <span class="texhtml"><i>g</i></span> with <span class="texhtml"><span class="nowrap">⟨<i>h</i>|</span></span> and <span class="texhtml"><span class="nowrap">|<i>g</i>⟩</span></span> respectively. This is because of literal symbolic substitutions. 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 \varphi _{h}=H=\langle h\mid }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>=</mo> <mi>H</mi> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>h</mi> <mo stretchy="false">∣</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{h}=H=\langle h\mid }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76f43d8f30a636a4aadeec9b49e95da34b34bd15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.85ex; height:2.843ex;" alt="{\displaystyle \varphi _{h}=H=\langle h\mid }"></span> and let <span class="texhtml"><i>g</i> = G = <span class="nowrap">|<i>g</i>⟩</span></span>. This gives <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 _{h}(g)=H(g)=H(G)=\langle h|(G)=\langle h|{\bigl (}|g\rangle {\bigr )}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{h}(g)=H(g)=H(G)=\langle h|(G)=\langle h|{\bigl (}|g\rangle {\bigr )}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f0ca2cc6275aae53ddec334a807f22f87b75cc1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.955ex; height:3.176ex;" alt="{\displaystyle \varphi _{h}(g)=H(g)=H(G)=\langle h|(G)=\langle h|{\bigl (}|g\rangle {\bigr )}\,.}"></span> </p><p>One ignores the parentheses and removes the double bars. </p><p>Moreover, mathematicians usually write the dual entity not at the first place, as the physicists do, but at the second one, and they usually use not an <a href="/wiki/Asterisk" title="Asterisk">asterisk</a> but an overline (which the physicists reserve for averages and the <a href="/wiki/Dirac_equation#Conservation_of_probability_current" title="Dirac equation">Dirac spinor adjoint</a>) to denote <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> numbers; i.e., for scalar products mathematicians usually write <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 \langle \phi ,\psi \rangle =\int \phi (x)\cdot {\overline {\psi (x)}}\,\mathrm {d} x\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ϕ</mi> <mo>,</mo> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>∫</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \phi ,\psi \rangle =\int \phi (x)\cdot {\overline {\psi (x)}}\,\mathrm {d} x\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5eb1d9db3820f05c8702e7b5814a65e498b915e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.435ex; height:5.676ex;" alt="{\displaystyle \langle \phi ,\psi \rangle =\int \phi (x)\cdot {\overline {\psi (x)}}\,\mathrm {d} x\,,}"></span> whereas physicists would write for the same quantity <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 \langle \psi |\phi \rangle =\int dx\,\psi ^{*}(x)\phi (x)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ϕ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>∫</mo> <mi>d</mi> <mi>x</mi> <mspace width="thinmathspace" /> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi |\phi \rangle =\int dx\,\psi ^{*}(x)\phi (x)~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8d57e3bbe42ac32c9074664f3255cbd0865ccda" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.425ex; height:5.676ex;" alt="{\displaystyle \langle \psi |\phi \rangle =\int dx\,\psi ^{*}(x)\phi (x)~.}"></span> </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=Bra%E2%80%93ket_notation&action=edit&section=29" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output 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diagrams (quantum mechanics)">Angular momentum diagrams (quantum mechanics)</a></li> <li><a href="/wiki/N-slit_interferometric_equation" title="N-slit interferometric equation"><span class="texhtml mvar" style="font-style:italic;">n</span>-slit interferometric equation</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bra%E2%80%93ket_notation&action=edit&section=30" title="Edit section: Notes"><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"><ol class="references"> <li id="cite_note-Dirac-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Dirac_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Dirac_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFDirac1939">Dirac 1939</a></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"><a href="#CITEREFShankar1994">Shankar 1994</a>, Chapter 1</span> </li> <li id="cite_note-Grassmann-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Grassmann_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGrassmann1862">Grassmann 1862</a></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"><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=VtBRKw1Ab7E&t=2561">Lecture 2 | Quantum Entanglements, Part 1 (Stanford)</a>, Leonard Susskind on complex numbers, complex conjugate, bra, ket. 2006-10-02.</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"><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=VtBRKw1Ab7E&t=2834">Lecture 2 | Quantum Entanglements, Part 1 (Stanford)</a>, Leonard Susskind on inner product, 2006-10-02.</span> </li> <li id="cite_note-bra–ket_Notation_Trivializes_Matrix_Multiplication-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-bra–ket_Notation_Trivializes_Matrix_Multiplication_6-0">^</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 class="citation web cs1"><a rel="nofollow" class="external text" href="http://algassert.com/post/1629">"Gidney, Craig (2017). Bra–Ket Notation Trivializes Matrix Multiplication"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Gidney%2C+Craig+%282017%29.+Bra%E2%80%93Ket+Notation+Trivializes+Matrix+Multiplication&rft_id=http%3A%2F%2Falgassert.com%2Fpost%2F1629&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABra%E2%80%93ket+notation" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFSakuraiNapolitano2021">Sakurai & Napolitano 2021</a> Sec 1.2</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFSakuraiNapolitano2021">Sakurai & Napolitano 2021</a> Sec 1.2, 1.3</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://bohr.physics.berkeley.edu/classes/221/1112/notes/hilbert.pdf">Lecture notes by Robert Littlejohn</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120617144946/http://bohr.physics.berkeley.edu/classes/221/1112/notes/hilbert.pdf">Archived</a> 2012-06-17 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, eqns 12 and 13</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFSakuraiNapolitano2021">Sakurai & Napolitano 2021</a> Sec 1.2, 1.3</span> </li> </ol></div></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=Bra%E2%80%93ket_notation&action=edit&section=31" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDirac1939" class="citation journal cs1">Dirac, P. A. M. (1939). "A new notation for quantum mechanics". <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>. <b>35</b> (3): 416–418. <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/1939PCPS...35..416D">1939PCPS...35..416D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0305004100021162">10.1017/S0305004100021162</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121466183">121466183</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Proceedings+of+the+Cambridge+Philosophical+Society&rft.atitle=A+new+notation+for+quantum+mechanics&rft.volume=35&rft.issue=3&rft.pages=416-418&rft.date=1939&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121466183%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0305004100021162&rft_id=info%3Abibcode%2F1939PCPS...35..416D&rft.aulast=Dirac&rft.aufirst=P.+A.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABra%E2%80%93ket+notation" class="Z3988"></span>. Also see his standard text, <i>The Principles of Quantum Mechanics</i>, IV edition, Clarendon Press (1958), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0198520115" title="Special:BookSources/978-0198520115">978-0198520115</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrassmann1862" class="citation book cs1">Grassmann, H. (1862). <i>Extension Theory</i>. History of Mathematics Sources. 2000 translation by Lloyd C. Kannenberg. American Mathematical Society, London Mathematical Society.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Extension+Theory&rft.series=History+of+Mathematics+Sources&rft.pub=American+Mathematical+Society%2C+London+Mathematical+Society&rft.date=1862&rft.aulast=Grassmann&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABra%E2%80%93ket+notation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCajori1929" class="citation book cs1"><a href="/wiki/Florian_Cajori" title="Florian Cajori">Cajori, Florian</a> (1929). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00cajo_0/page/134"><i>A History Of Mathematical Notations Volume II</i></a>. <a href="/wiki/Open_Court_Publishing_Company" title="Open Court Publishing Company">Open Court Publishing</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00cajo_0/page/134">134</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-67766-8" title="Special:BookSources/978-0-486-67766-8"><bdi>978-0-486-67766-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+Of+Mathematical+Notations+Volume+II&rft.pages=134&rft.pub=Open+Court+Publishing&rft.date=1929&rft.isbn=978-0-486-67766-8&rft.aulast=Cajori&rft.aufirst=Florian&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema00cajo_0%2Fpage%2F134&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABra%E2%80%93ket+notation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShankar1994" class="citation book cs1">Shankar, R. (1994). <i>Principles of Quantum Mechanics</i> (2nd ed.). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-306-44790-8" title="Special:BookSources/0-306-44790-8"><bdi>0-306-44790-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Quantum+Mechanics&rft.edition=2nd&rft.date=1994&rft.isbn=0-306-44790-8&rft.aulast=Shankar&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABra%E2%80%93ket+notation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynmanLeightonSands1965" class="citation book cs1">Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1965). <a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/III_toc.html"><i>The Feynman Lectures on Physics</i></a>. Vol. III. Reading, MA: Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-02118-8" title="Special:BookSources/0-201-02118-8"><bdi>0-201-02118-8</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+Feynman+Lectures+on+Physics&rft.place=Reading%2C+MA&rft.pub=Addison-Wesley&rft.date=1965&rft.isbn=0-201-02118-8&rft.aulast=Feynman&rft.aufirst=Richard+P.&rft.au=Leighton%2C+Robert+B.&rft.au=Sands%2C+Matthew&rft_id=https%3A%2F%2Ffeynmanlectures.caltech.edu%2FIII_toc.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABra%E2%80%93ket+notation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSakuraiNapolitano2021" class="citation book cs1">Sakurai, J J; Napolitano, J (2021). <i><a href="/wiki/Modern_Quantum_Mechanics" title="Modern Quantum Mechanics">Modern Quantum Mechanics</a></i> (3rd ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-108-42241-3" title="Special:BookSources/978-1-108-42241-3"><bdi>978-1-108-42241-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Quantum+Mechanics&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=2021&rft.isbn=978-1-108-42241-3&rft.aulast=Sakurai&rft.aufirst=J+J&rft.au=Napolitano%2C+J&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABra%E2%80%93ket+notation" class="Z3988"></span></li></ul> <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=Bra%E2%80%93ket_notation&action=edit&section=32" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Richard Fitzpatrick, <a rel="nofollow" class="external text" href="https://farside.ph.utexas.edu/teaching/qm/lectures/">"Quantum Mechanics: A graduate level course"</a>, The University of Texas at Austin. Includes: <ul><li>1. <a rel="nofollow" class="external text" href="http://farside.ph.utexas.edu/teaching/qm/lectures/node7.html">Ket space</a></li> <li>2. <a rel="nofollow" class="external text" href="http://farside.ph.utexas.edu/teaching/qm/lectures/node8.html">Bra space</a></li> <li>3. <a rel="nofollow" class="external text" href="http://farside.ph.utexas.edu/teaching/qm/lectures/node9.html">Operators</a></li> <li>4. <a rel="nofollow" class="external text" href="http://farside.ph.utexas.edu/teaching/qm/lectures/node10.html">The outer product</a></li> <li>5. <a rel="nofollow" class="external text" href="http://farside.ph.utexas.edu/teaching/qm/lectures/node11.html">Eigenvalues and eigenvectors</a></li></ul></li> <li>Robert Littlejohn, <a rel="nofollow" class="external text" href="http://bohr.physics.berkeley.edu/classes/221/0708/notes/hilbert.pdf">Lecture notes on "The Mathematical Formalism of Quantum mechanics", including bra–ket notation.</a> University of California, Berkeley.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGieres2000" class="citation journal cs1">Gieres, F. (2000). "Mathematical surprises and Dirac's formalism in quantum mechanics". <i>Rep. Prog. Phys</i>. <b>63</b> (12): 1893–1931. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/9907069">quant-ph/9907069</a></span>. <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/2000RPPh...63.1893G">2000RPPh...63.1893G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0034-4885%2F63%2F12%2F201">10.1088/0034-4885/63/12/201</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:10854218">10854218</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Rep.+Prog.+Phys.&rft.atitle=Mathematical+surprises+and+Dirac%27s+formalism+in+quantum+mechanics&rft.volume=63&rft.issue=12&rft.pages=1893-1931&rft.date=2000&rft_id=info%3Aarxiv%2Fquant-ph%2F9907069&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A10854218%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0034-4885%2F63%2F12%2F201&rft_id=info%3Abibcode%2F2000RPPh...63.1893G&rft.aulast=Gieres&rft.aufirst=F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABra%E2%80%93ket+notation" class="Z3988"></span></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 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class="nowraplinks hlist mw-collapsible autocollapse 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 class="mw-selflink selflink">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 href="/wiki/Wave_interference" title="Wave interference">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 href="/wiki/Wigner%27s_friend" title="Wigner's friend">Wigner's friend</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/Quantum_mysticism" title="Quantum mysticism">Quantum mysticism</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" 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Bra–ket notation - Wikipedia