Spin (physics) - Wikipedia

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class="vector-toc-numb">1.3</span> <span>Circulation of classical fields</span> </div> </a> <ul id="toc-Circulation_of_classical_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dirac's_relativistic_electron" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dirac's_relativistic_electron"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Dirac's relativistic electron</span> </div> </a> <ul id="toc-Dirac's_relativistic_electron-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relation_to_orbital_angular_momentum" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relation_to_orbital_angular_momentum"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Relation to orbital angular momentum</span> </div> </a> <ul id="toc-Relation_to_orbital_angular_momentum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_number" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Quantum_number"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Quantum number</span> </div> </a> <button aria-controls="toc-Quantum_number-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 Quantum number subsection</span> </button> <ul id="toc-Quantum_number-sublist" class="vector-toc-list"> <li id="toc-Fermions_and_bosons" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fermions_and_bosons"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Fermions and bosons</span> </div> </a> <ul id="toc-Fermions_and_bosons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spin–statistics_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spin–statistics_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Spin–statistics theorem</span> </div> </a> <ul id="toc-Spin–statistics_theorem-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Magnetic_moments" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Magnetic_moments"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Magnetic moments</span> </div> </a> <ul id="toc-Magnetic_moments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Curie_temperature_and_loss_of_alignment" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Curie_temperature_and_loss_of_alignment"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Curie temperature and loss of alignment</span> </div> </a> <ul id="toc-Curie_temperature_and_loss_of_alignment-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Direction" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Direction"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Direction</span> </div> </a> <button aria-controls="toc-Direction-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 Direction subsection</span> </button> <ul id="toc-Direction-sublist" class="vector-toc-list"> <li id="toc-Spin_projection_quantum_number_and_multiplicity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spin_projection_quantum_number_and_multiplicity"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Spin projection quantum number and multiplicity</span> </div> </a> <ul id="toc-Spin_projection_quantum_number_and_multiplicity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vector" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vector"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Vector</span> </div> </a> <ul id="toc-Vector-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mathematical_formulation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Mathematical_formulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Mathematical formulation</span> </div> </a> <button aria-controls="toc-Mathematical_formulation-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 Mathematical formulation subsection</span> </button> <ul id="toc-Mathematical_formulation-sublist" class="vector-toc-list"> <li id="toc-Operator" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Operator"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Operator</span> </div> </a> <ul id="toc-Operator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pauli_matrices" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pauli_matrices"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Pauli matrices</span> </div> </a> <ul id="toc-Pauli_matrices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pauli_exclusion_principle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pauli_exclusion_principle"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Pauli exclusion principle</span> </div> </a> <ul id="toc-Pauli_exclusion_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rotations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rotations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4</span> <span>Rotations</span> </div> </a> <ul id="toc-Rotations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lorentz_transformations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lorentz_transformations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.5</span> <span>Lorentz transformations</span> </div> </a> <ul id="toc-Lorentz_transformations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Measurement_of_spin_along_the_x,_y,_or_z_axes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Measurement_of_spin_along_the_x,_y,_or_z_axes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.6</span> <span>Measurement of spin along the <span>x</span>, <span>y</span>, or <span>z</span> axes</span> </div> </a> <ul id="toc-Measurement_of_spin_along_the_x,_y,_or_z_axes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Measurement_of_spin_along_an_arbitrary_axis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Measurement_of_spin_along_an_arbitrary_axis"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.7</span> <span>Measurement of spin along an arbitrary axis</span> </div> </a> <ul id="toc-Measurement_of_spin_along_an_arbitrary_axis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Compatibility_of_spin_measurements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Compatibility_of_spin_measurements"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.8</span> <span>Compatibility of spin measurements</span> </div> </a> <ul id="toc-Compatibility_of_spin_measurements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Higher_spins" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Higher_spins"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.9</span> <span>Higher spins</span> </div> </a> <ul id="toc-Higher_spins-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Parity" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Parity"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Parity</span> </div> </a> <ul id="toc-Parity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Measuring_spin" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Measuring_spin"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Measuring spin</span> </div> </a> <ul id="toc-Measuring_spin-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>History</span> </div> </a> <ul id="toc-History-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">12</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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">15</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">Spin (physics)</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 73 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-73" 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">73 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Spin_(fisika)" title="Spin (fisika) – Afrikaans" lang="af" hreflang="af" data-title="Spin (fisika)" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Spin" title="Spin – Alemannic" lang="gsw" hreflang="gsw" data-title="Spin" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%84%D9%81_(%D9%81%D9%8A%D8%B2%D9%8A%D8%A7%D8%A1)" title="لف (فيزياء) – Arabic" lang="ar" hreflang="ar" data-title="لف (فيزياء)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Esp%C3%ADn" title="Espín – Asturian" lang="ast" hreflang="ast" data-title="Espín" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Spin_(fizika)" title="Spin (fizika) – Azerbaijani" lang="az" hreflang="az" data-title="Spin (fizika)" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A7%8D%E0%A6%AA%E0%A6%BF%E0%A6%A8_(%E0%A6%AA%E0%A6%A6%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%A5%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8)" title="স্পিন (পদার্থবিজ্ঞান) – Bangla" lang="bn" hreflang="bn" data-title="স্পিন (পদার্থবিজ্ঞান)" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Spin" title="Spin – Minnan" lang="nan" hreflang="nan" data-title="Spin" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D0%BF%D1%96%D0%BD" title="Спін – Belarusian" lang="be" hreflang="be" data-title="Спін" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B8%D0%BD" title="Спин – Bulgarian" lang="bg" hreflang="bg" data-title="Спин" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Spin" title="Spin – Bosnian" lang="bs" hreflang="bs" data-title="Spin" 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/Esp%C3%ADn" title="Espín – Catalan" lang="ca" hreflang="ca" data-title="Espín" 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/Spin" title="Spin – Czech" lang="cs" hreflang="cs" data-title="Spin" 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/Spin_(fysik)" title="Spin (fysik) – Danish" lang="da" hreflang="da" data-title="Spin (fysik)" 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/Spin" title="Spin – German" lang="de" hreflang="de" data-title="Spin" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Spinn" title="Spinn – Estonian" lang="et" hreflang="et" data-title="Spinn" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%80%CE%B9%CE%BD" title="Σπιν – Greek" lang="el" hreflang="el" data-title="Σπιν" 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/Esp%C3%ADn" title="Espín – Spanish" lang="es" hreflang="es" data-title="Espín" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Spino_(fiziko)" title="Spino (fiziko) – Esperanto" lang="eo" hreflang="eo" data-title="Spino (fiziko)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Spin" title="Spin – Basque" lang="eu" hreflang="eu" data-title="Spin" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%B3%D9%BE%DB%8C%D9%86" title="اسپین – Persian" lang="fa" hreflang="fa" data-title="اسپین" 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/Spin" title="Spin – French" lang="fr" hreflang="fr" data-title="Spin" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Guairne" title="Guairne – Irish" lang="ga" hreflang="ga" data-title="Guairne" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Spin" title="Spin – Galician" lang="gl" hreflang="gl" data-title="Spin" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%AA%E0%AB%8D%E0%AA%B0%E0%AA%9A%E0%AA%95%E0%AB%8D%E0%AA%B0%E0%AA%A3" title="પ્રચક્રણ – Gujarati" lang="gu" hreflang="gu" data-title="પ્રચક્રણ" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8A%A4%ED%95%80_(%EB%AC%BC%EB%A6%AC%ED%95%99)" title="스핀 (물리학) – Korean" lang="ko" hreflang="ko" data-title="스핀 (물리학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8D%D5%BA%D5%AB%D5%B6" title="Սպին – Armenian" lang="hy" hreflang="hy" data-title="Սպին" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%9A%E0%A4%95%E0%A5%8D%E0%A4%B0%E0%A4%A3_(%E0%A4%AD%E0%A5%8C%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%80)" title="प्रचक्रण (भौतिकी) – Hindi" lang="hi" hreflang="hi" data-title="प्रचक्रण (भौतिकी)" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Spin" title="Spin – Croatian" lang="hr" hreflang="hr" data-title="Spin" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Vorticeso" title="Vorticeso – Ido" lang="io" hreflang="io" data-title="Vorticeso" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Spin" title="Spin – Indonesian" lang="id" hreflang="id" data-title="Spin" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spin" title="Spin – Italian" lang="it" hreflang="it" data-title="Spin" 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%A4%D7%99%D7%9F_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94)" title="ספין (פיזיקה) – Hebrew" lang="he" hreflang="he" data-title="ספין (פיזיקה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%9E%E1%83%98%E1%83%9C%E1%83%98" title="სპინი – Georgian" lang="ka" hreflang="ka" data-title="სპინი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B8%D0%BD" title="Спин – Kazakh" lang="kk" hreflang="kk" data-title="Спин" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B8%D0%BD" title="Спин – Kyrgyz" lang="ky" hreflang="ky" data-title="Спин" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Volubilitas_(quantica)" title="Volubilitas (quantica) – Latin" lang="la" hreflang="la" data-title="Volubilitas (quantica)" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Spins" title="Spins – Latvian" lang="lv" hreflang="lv" data-title="Spins" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Sukinys" title="Sukinys – Lithuanian" lang="lt" hreflang="lt" data-title="Sukinys" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Spin_(natuurkunde)" title="Spin (natuurkunde) – Limburgish" lang="li" hreflang="li" data-title="Spin (natuurkunde)" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Spin" title="Spin – Hungarian" lang="hu" hreflang="hu" data-title="Spin" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B8%D0%BD_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Спин (физика) – Macedonian" lang="mk" hreflang="mk" data-title="Спин (физика)" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Spin" title="Spin – Malay" lang="ms" hreflang="ms" data-title="Spin" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%85%E1%80%95%E1%80%84%E1%80%BA" title="စပင် – Burmese" lang="my" hreflang="my" data-title="စပင်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Spin_(kwantummechanica)" title="Spin (kwantummechanica) – Dutch" lang="nl" hreflang="nl" data-title="Spin (kwantummechanica)" 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%82%B9%E3%83%94%E3%83%B3%E8%A7%92%E9%81%8B%E5%8B%95%E9%87%8F" title="スピン角運動量 – Japanese" lang="ja" hreflang="ja" data-title="スピン角運動量" 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/Spinn" title="Spinn – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Spinn" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Spinn" title="Spinn – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Spinn" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Espin" title="Espin – Occitan" lang="oc" hreflang="oc" data-title="Espin" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A8%AA%E0%A8%BF%E0%A9%B0%E0%A8%A8_(%E0%A8%AD%E0%A9%8C%E0%A8%A4%E0%A8%BF%E0%A8%95_%E0%A8%B5%E0%A8%BF%E0%A8%97%E0%A8%BF%E0%A8%86%E0%A8%A8)" title="ਸਪਿੰਨ (ਭੌਤਿਕ ਵਿਗਿਆਨ) – Punjabi" lang="pa" hreflang="pa" data-title="ਸਪਿੰਨ (ਭੌਤਿਕ ਵਿਗਿਆਨ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Spin_(fizyka)" title="Spin (fizyka) – Polish" lang="pl" hreflang="pl" data-title="Spin (fizyka)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Spin" title="Spin – Portuguese" lang="pt" hreflang="pt" data-title="Spin" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Spin_(fizic%C4%83)" title="Spin (fizică) – Romanian" lang="ro" hreflang="ro" data-title="Spin (fizică)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B8%D0%BD" title="Спин – Russian" lang="ru" hreflang="ru" data-title="Спин" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Spini" title="Spini – Albanian" lang="sq" hreflang="sq" data-title="Spini" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Spin_(physics)" title="Spin (physics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Spin (physics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Spin_(fyzika)" title="Spin (fyzika) – Slovak" lang="sk" hreflang="sk" data-title="Spin (fyzika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Spin" title="Spin – Slovenian" lang="sl" hreflang="sl" data-title="Spin" 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%A1%D0%BF%D0%B8%D0%BD" title="Спин – Serbian" lang="sr" hreflang="sr" data-title="Спин" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Spin" title="Spin – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Spin" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Spin_(fisika)" title="Spin (fisika) – Sundanese" lang="su" hreflang="su" data-title="Spin (fisika)" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Spin" title="Spin – Finnish" lang="fi" hreflang="fi" data-title="Spin" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Spinn" title="Spinn – Swedish" lang="sv" hreflang="sv" data-title="Spinn" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AF%81%E0%AE%B4%E0%AE%B1%E0%AF%8D%E0%AE%9A%E0%AE%BF_(%E0%AE%87%E0%AE%AF%E0%AE%B1%E0%AF%8D%E0%AE%AA%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D)" title="சுழற்சி (இயற்பியல்) – Tamil" lang="ta" hreflang="ta" data-title="சுழற்சி (இயற்பியல்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Asebrin" title="Asebrin – Kabyle" lang="kab" hreflang="kab" data-title="Asebrin" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/Spin" title="Spin – Tatar" lang="tt" hreflang="tt" data-title="Spin" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AA%E0%B8%9B%E0%B8%B4%E0%B8%99_(%E0%B8%9F%E0%B8%B4%E0%B8%AA%E0%B8%B4%E0%B8%81%E0%B8%AA%E0%B9%8C)" title="สปิน (ฟิสิกส์) – Thai" lang="th" hreflang="th" data-title="สปิน (ฟิสิกส์)" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Spin_(fizik)" title="Spin (fizik) – Turkish" lang="tr" hreflang="tr" data-title="Spin (fizik)" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%BF%D1%96%D0%BD" title="Спін – Ukrainian" lang="uk" hreflang="uk" data-title="Спін" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%BA%D8%B2%D9%84_(%D8%B7%D8%A8%DB%8C%D8%B9%DB%8C%D8%A7%D8%AA)" title="غزل (طبیعیات) – Urdu" lang="ur" hreflang="ur" data-title="غزل (طبیعیات)" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Spin" title="Spin – Vietnamese" lang="vi" hreflang="vi" data-title="Spin" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%87%AA%E6%97%8B" title="自旋 – Wu" lang="wuu" hreflang="wuu" data-title="自旋" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%87%AA%E6%97%8B" title="自旋 – Cantonese" lang="yue" hreflang="yue" data-title="自旋" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%87%AA%E6%97%8B" title="自旋 – Chinese" lang="zh" hreflang="zh" data-title="自旋" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q133673#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Spin_(physics)" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Intrinsic quantum property of particles</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">This article is about the concept in quantum mechanics. For the concept in classical mechanics, see <a href="/wiki/Rotation" title="Rotation">Rotation</a>.</div> <p><b>Spin</b> is an <a href="/wiki/Intrinsic_and_extrinsic_properties" title="Intrinsic and extrinsic properties">intrinsic</a> form of <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> carried by <a href="/wiki/Elementary_particle" title="Elementary particle">elementary particles</a>, and thus by <a href="/wiki/List_of_particles#Composite_particles" title="List of particles">composite particles</a> such as <a href="/wiki/Hadron" title="Hadron">hadrons</a>, <a href="/wiki/Atomic_nucleus" title="Atomic nucleus">atomic nuclei</a>, and atoms.<sup id="cite_ref-merzbacher372_1-0" class="reference"><a href="#cite_note-merzbacher372-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-griffiths_2-0" class="reference"><a href="#cite_note-griffiths-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages: 183–184">: <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoqu00grif_190/page/n194">183</a>–184 </span></sup> Spin is quantized, and accurate models for the interaction with spin require <a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">relativistic quantum mechanics</a> or <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>. </p><p>The existence of <a href="/wiki/Electron" title="Electron">electron</a> <a href="/wiki/Spin_angular_momentum" class="mw-redirect" title="Spin angular momentum">spin angular momentum</a> is <a href="/wiki/Inferred" class="mw-redirect" title="Inferred">inferred</a> from experiments, such as the <a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach experiment</a>, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum.<sup id="cite_ref-eisberg272_3-0" class="reference"><a href="#cite_note-eisberg272-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The relativistic <a href="/wiki/Spin%E2%80%93statistics_theorem" title="Spin–statistics theorem">spin–statistics theorem</a> connects electron spin quantization to the <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a>: observations of exclusion imply half-integer spin, and observations of half-integer spin imply exclusion. </p><p>Spin is described mathematically as a vector for some particles such as photons, and as a <a href="/wiki/Spinor" title="Spinor">spinor</a> or <a href="/wiki/Bispinor" title="Bispinor">bispinor</a> for other particles such as electrons. Spinors and bispinors behave similarly to <a href="/wiki/Euclidean_vector" title="Euclidean vector">vectors</a>: they have definite magnitudes and change under rotations; however, they use an unconventional "direction". All elementary particles of a given kind have the same magnitude of spin angular momentum, though its direction may change. These are indicated by assigning the particle a <a href="/wiki/Spin_quantum_number" title="Spin quantum number">spin quantum number</a>.<sup id="cite_ref-griffiths_2-1" class="reference"><a href="#cite_note-griffiths-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages: 183–184">: <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoqu00grif_190/page/n194">183–184</a> </span></sup> </p><p>The <a href="/wiki/International_System_of_Units" title="International System of Units">SI units</a> of spin are the same as classical angular momentum (i.e., <a href="/wiki/Newton_(unit)" title="Newton (unit)">N</a>·<a href="/wiki/Metre" title="Metre">m</a>·<a href="/wiki/Second" title="Second">s</a>, <a href="/wiki/Joule" title="Joule">J</a>·s, or <a href="/wiki/Kilogram" title="Kilogram">kg</a>·m<sup>2</sup>·s<sup>−1</sup>). In quantum mechanics, angular momentum and spin angular momentum take discrete values proportional to the <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a>. In practice, spin is usually given as a <a href="/wiki/Dimensionless" class="mw-redirect" title="Dimensionless">dimensionless</a> spin quantum number by dividing the spin angular momentum by the <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a> <span class="texhtml mvar" style="font-style:italic;">ħ</span>. Often, the "spin quantum number" is simply called "spin". </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Models">Models</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=1" title="Edit section: Models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Rotating_charged_mass">Rotating charged mass</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=2" title="Edit section: Rotating charged mass"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The earliest models for electron spin imagined a rotating charged mass, but this model fails when examined in detail: the required space distribution does not match limits on the <a href="/wiki/Electron#Fundamental_properties" title="Electron">electron radius</a>: the required rotation speed exceeds the speed of light.<sup id="cite_ref-Sebens_HowSpin_4-0" class="reference"><a href="#cite_note-Sebens_HowSpin-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In the <a href="/wiki/Standard_model" class="mw-redirect" title="Standard model">Standard Model</a>, the fundamental particles are all considered "point-like": they have their effects through the field that surrounds them.<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> Any model for spin based on mass rotation would need to be consistent with that model. </p> <div class="mw-heading mw-heading3"><h3 id="Pauli's_"classically_non-describable_two-valuedness""><span id="Pauli.27s_.22classically_non-describable_two-valuedness.22"></span>Pauli's "classically non-describable two-valuedness"</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=3" title="Edit section: Pauli's "classically non-describable two-valuedness""><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Wolfgang Pauli</a>, a central figure in the history of quantum spin, initially rejected any idea that the "degree of freedom" he introduced to explain experimental observations was related to rotation. He called it "classically non-describable two-valuedness". Later, he allowed that it is related to angular momentum, but insisted on considering spin an abstract property.<sup id="cite_ref-Giulini_6-0" class="reference"><a href="#cite_note-Giulini-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> This approach allowed Pauli to develop a proof of his fundamental <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a>, a proof now called the <a href="/wiki/Spin-statistics_theorem" class="mw-redirect" title="Spin-statistics theorem">spin-statistics theorem</a>.<sup id="cite_ref-Frohlich_7-0" class="reference"><a href="#cite_note-Frohlich-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> In retrospect, this insistence and the style of his proof initiated the modern particle-physics era, where abstract quantum properties derived from symmetry properties dominate. Concrete interpretation became secondary and optional.<sup id="cite_ref-Giulini_6-1" class="reference"><a href="#cite_note-Giulini-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Circulation_of_classical_fields">Circulation of classical fields</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=4" title="Edit section: Circulation of classical fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first classical model for spin proposed a small rigid particle rotating about an axis, as ordinary use of the word may suggest. Angular momentum can be computed from a classical field as well.<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><sup id="cite_ref-PeskinSchroeder_9-0" class="reference"><a href="#cite_note-PeskinSchroeder-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 63">: 63 </span></sup> By applying <a href="/wiki/Frederik_Belinfante" title="Frederik Belinfante">Frederik Belinfante</a>'s approach to calculating the angular momentum of a field, Hans C. Ohanian showed that "spin is essentially a wave property ... generated by a circulating flow of charge in the wave field of the electron".<sup id="cite_ref-ohanian_10-0" class="reference"><a href="#cite_note-ohanian-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> This same concept of spin can be applied to gravity waves in water: "spin is generated by subwavelength circular motion of water particles".<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>Unlike classical wavefield circulation, which allows continuous values of angular momentum, quantum wavefields allow only discrete values.<sup id="cite_ref-ohanian_10-1" class="reference"><a href="#cite_note-ohanian-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Consequently, energy transfer to or from spin states always occurs in fixed quantum steps. Only a few steps are allowed: for many qualitative purposes, the complexity of the spin quantum wavefields can be ignored and the system properties can be discussed in terms of "integer" or "half-integer" spin models as discussed in <a href="#Quantum_numbers">quantum numbers</a> below. </p> <div class="mw-heading mw-heading3"><h3 id="Dirac's_relativistic_electron"><span id="Dirac.27s_relativistic_electron"></span>Dirac's relativistic electron</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=5" title="Edit section: Dirac's relativistic electron"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Quantitative calculations of spin properties for electrons requires the Dirac <a href="/wiki/Dirac_equation" title="Dirac equation">relativistic wave equation</a>.<sup id="cite_ref-Frohlich_7-1" class="reference"><a href="#cite_note-Frohlich-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Relation_to_orbital_angular_momentum">Relation to orbital angular momentum</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=6" title="Edit section: Relation to orbital angular momentum"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As the name suggests, spin was originally conceived as the rotation of a particle around some axis. Historically <a href="/wiki/Angular_momentum_operator#Orbital_angular_momentum" title="Angular momentum operator">orbital angular momentum</a> related to particle orbits.<sup id="cite_ref-Whittaker_1989_p.87_12-0" class="reference"><a href="#cite_note-Whittaker_1989_p.87-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 131">: 131 </span></sup> While the names based on mechanical models have survived, the physical explanation has not. <a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">Quantization</a> fundamentally alters the character of both spin and orbital angular momentum. </p><p>Since elementary particles are point-like, self-rotation is not well-defined for them. However, spin implies that the phase of the particle depends on the angle 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 e^{iS\theta }\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>S</mi> <mi>θ</mi> </mrow> </msup> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{iS\theta }\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a05cb230c347426a76962ce3e06dee8192f0be6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.942ex; height:3.009ex;" alt="{\displaystyle e^{iS\theta }\ ,}"></span> for rotation of angle <span class="texhtml mvar" style="font-style:italic;">θ</span> around the axis parallel to the spin <span class="texhtml mvar" style="font-style:italic;">S</span>. This is equivalent to the quantum-mechanical interpretation of <a href="/wiki/Momentum" title="Momentum">momentum</a> as phase dependence in the position, and of <a href="/wiki/Angular_momentum_operator#Orbital_angular_momentum" title="Angular momentum operator">orbital angular momentum</a> as phase dependence in the angular position. </p><p>For fermions, the picture is less clear: From the <a href="/wiki/Ehrenfest_theorem" title="Ehrenfest theorem">Ehrenfest theorem</a>, the <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> is equal to the derivative of the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> to its <a href="/wiki/Conjugate_momentum" class="mw-redirect" title="Conjugate momentum">conjugate momentum</a>, which is the total <a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">angular momentum operator</a> <span class="nowrap"><span class="texhtml"><b>J</b> = <b>L</b> + <b>S</b></span> .</span> Therefore, if the Hamiltonian <span class="texhtml mvar" style="font-style:italic;">H</span> has any dependence on the spin <span class="texhtml mvar" style="font-style:italic;">S</span>, then <span class="texhtml"> <span style="font-size:120%"><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"> ∂ <i>H</i> </span><span class="sr-only">/</span><span class="den"> ∂ <i>S</i> </span></span>⁠</span> </span> </span> must be non-zero; consequently, for <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a>, the existence of spin in the Hamiltonian will produce an actual angular velocity, and hence an actual physical rotation – that is, a change in the phase-angle, <span class="texhtml mvar" style="font-style:italic;">θ</span>, over time. However, whether this holds true for free electron is ambiguous, since for an electron, <span class="texhtml mvar" style="font-style:italic;">| S |</span>² is a constant <span class="nowrap"> <span class="texhtml"><span style="font-size:85%;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 2 </span></span>⁠</span> </span><a href="/wiki/Planck_constant" title="Planck constant">ℏ</a></span> ,</span> and one might decide that since it cannot change, no <a href="/wiki/Partial_derivative" title="Partial derivative">partial</a> (<span class="texhtml">∂</span>) can exist. Therefore it is a matter of interpretation whether the Hamiltonian must include such a term, and whether this aspect of <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> extends into <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> (any particle's intrinsic spin angular momentum, <span class="texhtml mvar" style="font-style:italic;"><b>S</b></span>, is a <a href="/wiki/Quantum_number" title="Quantum number">quantum number</a> arising from a "<a href="/wiki/Spinor" title="Spinor">spinor</a>" in the mathematical solution to the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a>, rather than being a more nearly physical quantity, like <a href="/wiki/Orbital_angular_momentum" class="mw-redirect" title="Orbital angular momentum">orbital angular momentum</a> <span class="texhtml mvar" style="font-style:italic;"><b>L</b></span>). Nevertheless, spin appears in the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a>, and thus the relativistic Hamiltonian of the electron, treated as a <a href="/wiki/Dirac_field" class="mw-redirect" title="Dirac field">Dirac field</a>, can be interpreted as including a dependence in the spin <span class="texhtml mvar" style="font-style:italic;">S</span>.<sup id="cite_ref-PeskinSchroeder_9-1" class="reference"><a href="#cite_note-PeskinSchroeder-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Quantum_number">Quantum number</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=7" title="Edit section: Quantum number"><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/Spin_quantum_number" title="Spin quantum number">Spin quantum number</a></div> <p>Spin obeys the mathematical laws of <a href="/wiki/Angular_momentum_quantization" class="mw-redirect" title="Angular momentum quantization">angular momentum quantization</a>. The specific properties of spin angular momenta include: </p> <ul><li>Spin quantum numbers may take either <a href="/wiki/Half-integer" title="Half-integer">half-integer</a> or integer values.</li> <li>Although the direction of its spin can be changed, the magnitude of the spin of an elementary particle cannot be changed.</li> <li>The spin of a charged particle is associated with a <a href="/wiki/Magnetic_dipole_moment" class="mw-redirect" title="Magnetic dipole moment">magnetic dipole moment</a> with a <a href="/wiki/G-factor_(physics)" title="G-factor (physics)"><span class="texhtml mvar" style="font-style:italic;">g</span>-factor</a> that differs from 1. (In the classical context, this would imply the <a href="/wiki/Gyromagnetic_ratio#Gyromagnetic_ratio_for_a_classical_rotating_body" title="Gyromagnetic ratio">internal charge and mass distributions differing</a> for a rotating object.<sup id="cite_ref-Sebens_HowSpin_4-1" class="reference"><a href="#cite_note-Sebens_HowSpin-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>)</li></ul> <p>The conventional definition of the <b>spin quantum number</b> is <span class="texhtml"><i>s</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>n</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>, where <span class="texhtml mvar" style="font-style:italic;">n</span> can be any <a href="/wiki/Non-negative_integer" class="mw-redirect" title="Non-negative integer">non-negative integer</a>. Hence the allowed values of <span class="texhtml mvar" style="font-style:italic;">s</span> are 0, <a href="/wiki/Spin-1/2" title="Spin-1/2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></a>, 1, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, 2, etc. The value of <span class="texhtml mvar" style="font-style:italic;">s</span> for an <a href="/wiki/Elementary_particle" title="Elementary particle">elementary particle</a> depends only on the type of particle and cannot be altered in any known way (in contrast to the <i>spin direction</i> described below). The spin angular momentum <span class="texhtml mvar" style="font-style:italic;">S</span> of any physical system is <a href="/wiki/Angular_momentum_quantization" class="mw-redirect" title="Angular momentum quantization">quantized</a>. The allowed values of <span class="texhtml mvar" style="font-style:italic;">S</span> are <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 S=\hbar \,{\sqrt {s(s+1)}}={\frac {h}{2\pi }}\,{\sqrt {{\frac {n}{2}}{\frac {(n+2)}{2}}}}={\frac {h}{4\pi }}\,{\sqrt {n(n+2)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\hbar \,{\sqrt {s(s+1)}}={\frac {h}{2\pi }}\,{\sqrt {{\frac {n}{2}}{\frac {(n+2)}{2}}}}={\frac {h}{4\pi }}\,{\sqrt {n(n+2)}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c85d7682e45045d420df3a4c6d4b7105501d151c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:54.41ex; height:7.676ex;" alt="{\displaystyle S=\hbar \,{\sqrt {s(s+1)}}={\frac {h}{2\pi }}\,{\sqrt {{\frac {n}{2}}{\frac {(n+2)}{2}}}}={\frac {h}{4\pi }}\,{\sqrt {n(n+2)}},}"></span> where <span class="texhtml mvar" style="font-style:italic;">h</span> is the <a href="/wiki/Planck_constant" title="Planck constant">Planck constant</a>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \hbar ={\frac {h}{2\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \hbar ={\frac {h}{2\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/788c1a8a0024c8e4f4e46bb932e1dabb5c1ae78d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.005ex; height:3.676ex;" alt="{\textstyle \hbar ={\frac {h}{2\pi }}}"></span> is the reduced Planck constant. In contrast, <a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">orbital angular momentum</a> can only take on integer values of <span class="texhtml mvar" style="font-style:italic;">s</span>; i.e., even-numbered values of <span class="texhtml mvar" style="font-style:italic;">n</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Fermions_and_bosons">Fermions and bosons</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=8" title="Edit section: Fermions and bosons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Those particles with half-integer spins, such as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, are known as <a href="/wiki/Fermion" title="Fermion">fermions</a>, while those particles with integer spins, such as 0, 1, 2, are known as <a href="/wiki/Bosons" class="mw-redirect" title="Bosons">bosons</a>. The two families of particles obey different rules and <i>broadly</i> have different roles in the world around us. A key distinction between the two families is that fermions obey the <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a>: that is, there cannot be two identical fermions simultaneously having the same quantum numbers (meaning, roughly, having the same position, velocity and spin direction). Fermions obey the rules of <a href="/wiki/Fermi%E2%80%93Dirac_statistics" title="Fermi–Dirac statistics">Fermi–Dirac statistics</a>. In contrast, bosons obey the rules of <a href="/wiki/Bose%E2%80%93Einstein_statistics" title="Bose–Einstein statistics">Bose–Einstein statistics</a> and have no such restriction, so they may "bunch together" in identical states. Also, composite particles can have spins different from their component particles. For example, a <a href="/wiki/Helium-4" title="Helium-4">helium-4</a> atom in the ground state has spin 0 and behaves like a boson, even though the <a href="/wiki/Quarks" class="mw-redirect" title="Quarks">quarks</a> and electrons which make it up are all fermions. </p><p>This has some profound consequences: </p> <ul><li><a href="/wiki/Quarks" class="mw-redirect" title="Quarks">Quarks</a> and <a href="/wiki/Leptons" class="mw-redirect" title="Leptons">leptons</a> (including <a href="/wiki/Electrons" class="mw-redirect" title="Electrons">electrons</a> and <a href="/wiki/Neutrinos" class="mw-redirect" title="Neutrinos">neutrinos</a>), which make up what is classically known as <a href="/wiki/Matter" title="Matter">matter</a>, are all fermions with <a href="/wiki/Spin-1/2" title="Spin-1/2">spin <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></a>. The common idea that "matter takes up space" actually comes from the <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a> acting on these particles to prevent the fermions from being in the same quantum state. Further compaction would require electrons to occupy the same energy states, and therefore a kind of <a href="/wiki/Pressure" title="Pressure">pressure</a> (sometimes known as <a href="/wiki/Degenerate_matter" title="Degenerate matter">degeneracy pressure of electrons</a>) acts to resist the fermions being overly close.<div class="paragraphbreak" style="margin-top:0.5em"></div> Elementary fermions with other spins (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, etc.) are not known to exist.</li> <li>Elementary particles which are thought of as <a href="/wiki/Force_carrier" title="Force carrier">carrying forces</a> are all bosons with spin 1. They include the <a href="/wiki/Photon" title="Photon">photon</a>, which carries the <a href="/wiki/Electromagnetic_force" class="mw-redirect" title="Electromagnetic force">electromagnetic force</a>, the <a href="/wiki/Gluon" title="Gluon">gluon</a> (<a href="/wiki/Strong_force" class="mw-redirect" title="Strong force">strong force</a>), and the <a href="/wiki/W_and_Z_bosons" title="W and Z bosons">W and Z bosons</a> (<a href="/wiki/Weak_force" class="mw-redirect" title="Weak force">weak force</a>). The ability of bosons to occupy the same quantum state is used in the <a href="/wiki/Laser" title="Laser">laser</a>, which aligns many photons having the same quantum number (the same direction and frequency), <a href="/wiki/Superfluid" class="mw-redirect" title="Superfluid">superfluid</a> <a href="/wiki/Liquid_helium" title="Liquid helium">liquid helium</a> resulting from helium-4 atoms being bosons, and <a href="/wiki/Superconductivity" title="Superconductivity">superconductivity</a>, where <a href="/wiki/Cooper_pair" title="Cooper pair">pairs of electrons</a> (which individually are fermions) act as single composite bosons.<div class="paragraphbreak" style="margin-top:0.5em"></div> Elementary bosons with other spins (0, 2, 3, etc.) were not historically known to exist, although they have received considerable theoretical treatment and are well established within their respective mainstream theories. In particular, theoreticians have proposed the <a href="/wiki/Graviton" title="Graviton">graviton</a> (predicted to exist by some <a href="/wiki/Quantum_gravity" title="Quantum gravity">quantum gravity</a> theories) with spin 2, and the <a href="/wiki/Higgs_boson" title="Higgs boson">Higgs boson</a> (explaining <a href="/wiki/Electroweak_symmetry_breaking" class="mw-redirect" title="Electroweak symmetry breaking">electroweak symmetry breaking</a>) with spin 0. Since 2013, the Higgs boson with spin 0 has been considered proven to exist.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> It is the first <a href="/wiki/Scalar_boson" title="Scalar boson">scalar elementary particle</a> (spin 0) known to exist in nature.</li> <li>Atomic nuclei have <a href="/wiki/Spin_quantum_number#Nuclear_spin" title="Spin quantum number">nuclear spin</a> which may be either half-integer or integer, so that the nuclei may be either fermions or bosons.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Spin–statistics_theorem"><span id="Spin.E2.80.93statistics_theorem"></span>Spin–statistics theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=9" title="Edit section: Spin–statistics theorem"><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/Spin%E2%80%93statistics_theorem" title="Spin–statistics theorem">spin–statistics theorem</a></div> <p>The <a href="/wiki/Spin%E2%80%93statistics_theorem" title="Spin–statistics theorem">spin–statistics theorem</a> splits particles into two groups: <a href="/wiki/Bosons" class="mw-redirect" title="Bosons">bosons</a> and <a href="/wiki/Fermions" class="mw-redirect" title="Fermions">fermions</a>, where bosons obey <a href="/wiki/Bose%E2%80%93Einstein_statistics" title="Bose–Einstein statistics">Bose–Einstein statistics</a>, and fermions obey <a href="/wiki/Fermi%E2%80%93Dirac_statistics" title="Fermi–Dirac statistics">Fermi–Dirac statistics</a> (and therefore the <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a>). Specifically, the theorem requires that particles with half-integer spins obey the <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a> while particles with integer spin do not. As an example, <a href="/wiki/Electron" title="Electron">electrons</a> have half-integer spin and are fermions that obey the Pauli exclusion principle, while photons have integer spin and do not. The theorem was derived by <a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Wolfgang Pauli</a> in 1940; it relies on both quantum mechanics and the theory of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>. Pauli described this connection between spin and statistics as "one of the most important applications of the special relativity theory".<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Magnetic_moments">Magnetic moments</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=10" title="Edit section: Magnetic moments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" id="Spin_magnetic_moment"></span> </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Neutron_spin_dipole_field.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Neutron_spin_dipole_field.jpg/220px-Neutron_spin_dipole_field.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Neutron_spin_dipole_field.jpg/330px-Neutron_spin_dipole_field.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/15/Neutron_spin_dipole_field.jpg/440px-Neutron_spin_dipole_field.jpg 2x" data-file-width="450" data-file-height="450" /></a><figcaption>Schematic diagram depicting the spin of the neutron as the black arrow and magnetic field lines associated with the <a href="/wiki/Neutron_magnetic_moment" class="mw-redirect" title="Neutron magnetic moment">neutron magnetic moment</a>. The neutron has a negative magnetic moment. While the spin of the neutron is upward in this diagram, the magnetic field lines at the center of the dipole are downward.</figcaption></figure> <p>Particles with spin can possess a <a href="/wiki/Magnetic_dipole_moment" class="mw-redirect" title="Magnetic dipole moment">magnetic dipole moment</a>, just like a rotating <a href="/wiki/Electric_charge" title="Electric charge">electrically charged</a> body in <a href="/wiki/Classical_electrodynamics" class="mw-redirect" title="Classical electrodynamics">classical electrodynamics</a>. These magnetic moments can be experimentally observed in several ways, e.g. by the deflection of particles by inhomogeneous <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic fields</a> in a <a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach experiment</a>, or by measuring the magnetic fields generated by the particles themselves. </p><p>The intrinsic magnetic moment <span class="texhtml"><i><b>μ</b></i></span> of a <a href="/wiki/Spin-1/2" title="Spin-1/2">spin-<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></a> particle with charge <span class="texhtml mvar" style="font-style:italic;">q</span>, mass <span class="texhtml mvar" style="font-style:italic;">m</span>, and spin angular momentum <span class="texhtml"><b>S</b></span> is<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\mu }}={\frac {g_{\text{s}}q}{2m}}\mathbf {S} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">μ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mi>q</mi> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\mu }}={\frac {g_{\text{s}}q}{2m}}\mathbf {S} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/328316c6a8d0b3e6f0962d6e462a37c584c87130" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.915ex; height:4.843ex;" alt="{\displaystyle {\boldsymbol {\mu }}={\frac {g_{\text{s}}q}{2m}}\mathbf {S} ,}"></span></dd></dl> <p>where the <a href="/wiki/Dimensionless_quantity" title="Dimensionless quantity">dimensionless quantity</a> <span class="texhtml"><i>g</i><sub>s</sub></span> is called the spin <a href="/wiki/G-factor_(physics)#Electron_spin_g-factor" title="G-factor (physics)"><span class="texhtml mvar" style="font-style:italic;">g</span>-factor</a>. For exclusively orbital rotations, it would be 1 (assuming that the mass and the charge occupy spheres of equal radius). </p><p>The electron, being a charged elementary particle, possesses a <a href="/wiki/Electron_magnetic_moment" title="Electron magnetic moment">nonzero magnetic moment</a>. One of the triumphs of the theory of <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a> is its accurate prediction of the electron <a href="/wiki/Land%C3%A9_g-factor" title="Landé g-factor"><span class="texhtml mvar" style="font-style:italic;">g</span>-factor</a>, which has been experimentally determined to have the value <span class="nowrap"><span data-sort-value="2999799768069563908♠"></span>−2.002<span style="margin-left:.25em;">319</span><span style="margin-left:.25em;">304</span><span style="margin-left:.25em;">360</span><span style="margin-left:.25em;">92</span>(36)</span>, with the digits in parentheses denoting <a href="/wiki/Measurement_uncertainty" title="Measurement uncertainty">measurement uncertainty</a> in the last two digits at one <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a>.<sup id="cite_ref-physconst-ge_16-0" class="reference"><a href="#cite_note-physconst-ge-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> The value of 2 arises from the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a>, a fundamental equation connecting the electron's spin with its electromagnetic properties; and the <a href="/wiki/Anomalous_magnetic_dipole_moment" title="Anomalous magnetic dipole moment">deviation</a> from <span class="nowrap"><span data-sort-value="2999800000000000000♠"></span>−2</span> arises from the electron's interaction with the surrounding quantum fields, including its own electromagnetic field and <a href="/wiki/Virtual_particles" class="mw-redirect" title="Virtual particles">virtual particles</a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>Composite particles also possess magnetic moments associated with their spin. In particular, the <a href="/wiki/Neutron" title="Neutron">neutron</a> possesses a non-zero magnetic moment despite being electrically neutral. This fact was an early indication that the neutron is not an elementary particle. In fact, it is made up of <a href="/wiki/Quarks" class="mw-redirect" title="Quarks">quarks</a>, which are electrically charged particles. The <a href="/wiki/Neutron_magnetic_moment" class="mw-redirect" title="Neutron magnetic moment">magnetic moment of the neutron</a> comes from the spins of the individual quarks and their orbital motions. </p><p><a href="/wiki/Neutrino" title="Neutrino">Neutrinos</a> are both elementary and electrically neutral. The minimally extended <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> that takes into account non-zero neutrino masses predicts neutrino magnetic moments of:<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\nu }\approx 3\times 10^{-19}\mu _{\text{B}}{\frac {m_{\nu }c^{2}}{\text{eV}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>≈</mo> <mn>3</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>19</mn> </mrow> </msup> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mtext>eV</mtext> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\nu }\approx 3\times 10^{-19}\mu _{\text{B}}{\frac {m_{\nu }c^{2}}{\text{eV}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/406114230a2f078e446c4433aa39f1fdfa5e3954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.572ex; height:5.843ex;" alt="{\displaystyle \mu _{\nu }\approx 3\times 10^{-19}\mu _{\text{B}}{\frac {m_{\nu }c^{2}}{\text{eV}}},}"></span></dd></dl> <p>where the <span class="texhtml"><i>μ</i><sub>ν</sub></span> are the neutrino magnetic moments, <span class="texhtml"><i>m</i><sub>ν</sub></span> are the neutrino masses, and <span class="texhtml"><i>μ</i><sub>B</sub></span> is the <a href="/wiki/Bohr_magneton" title="Bohr magneton">Bohr magneton</a>. New physics above the electroweak scale could, however, lead to significantly higher neutrino magnetic moments. It can be shown in a model-independent way that neutrino magnetic moments larger than about 10<sup>−14</sup> <span class="texhtml"><i>μ</i><sub>B</sub></span> are "unnatural" because they would also lead to large radiative contributions to the neutrino mass. Since the neutrino masses are known to be at most about <span class="nowrap"><span data-sort-value="7000100000000000000♠"></span>1 eV/<i>c</i><sup>2</sup></span>, <a href="/wiki/Fine-tuning_(physics)" title="Fine-tuning (physics)">fine-tuning</a> would be necessary in order to prevent large contributions to the neutrino mass via radiative corrections.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> The measurement of neutrino magnetic moments is an active area of research. Experimental results have put the neutrino magnetic moment at less than <span class="nowrap"><span data-sort-value="6990120000000000000♠"></span>1.2<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−10</sup></span> times the electron's magnetic moment. </p><p>On the other hand, elementary particles with spin but without electric charge, such as the <a href="/wiki/Photon" title="Photon">photon</a> and <a href="/wiki/Z_boson" class="mw-redirect" title="Z boson">Z boson</a>, do not have a magnetic moment. </p> <div class="mw-heading mw-heading2"><h2 id="Curie_temperature_and_loss_of_alignment">Curie temperature and loss of alignment</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=11" title="Edit section: Curie temperature and loss of alignment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In ordinary materials, the magnetic dipole moments of individual atoms produce magnetic fields that cancel one another, because each dipole points in a random direction, with the overall average being very near zero. <a href="/wiki/Ferromagnet" class="mw-redirect" title="Ferromagnet">Ferromagnetic</a> materials below their <a href="/wiki/Curie_temperature" title="Curie temperature">Curie temperature</a>, however, exhibit <a href="/wiki/Magnetic_domain" title="Magnetic domain">magnetic domains</a> in which the atomic dipole moments spontaneously align locally, producing a macroscopic, non-zero magnetic field from the domain. These are the ordinary "magnets" with which we are all familiar. </p><p>In <a href="/wiki/Paramagnetic" class="mw-redirect" title="Paramagnetic">paramagnetic</a> materials, the magnetic dipole moments of individual atoms will partially align with an externally applied magnetic field. In <a href="/wiki/Diamagnetic" class="mw-redirect" title="Diamagnetic">diamagnetic</a> materials, on the other hand, the magnetic dipole moments of individual atoms align oppositely to any externally applied magnetic field, even if it requires energy to do so. </p><p>The study of the behavior of such "<a href="/wiki/Spin_model" title="Spin model">spin models</a>" is a thriving area of research in <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed matter physics</a>. For instance, the <a href="/wiki/Ising_model" title="Ising model">Ising model</a> describes spins (dipoles) that have only two possible states, up and down, whereas in the <a href="/wiki/Heisenberg_model_(quantum)" class="mw-redirect" title="Heisenberg model (quantum)">Heisenberg model</a> the spin vector is allowed to point in any direction. These models have many interesting properties, which have led to interesting results in the theory of <a href="/wiki/Phase_transition" title="Phase transition">phase transitions</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Direction">Direction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=12" title="Edit section: Direction"><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">Further information: <a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">Angular momentum operator</a></div> <div class="mw-heading mw-heading3"><h3 id="Spin_projection_quantum_number_and_multiplicity">Spin projection quantum number and multiplicity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=13" title="Edit section: Spin projection quantum number and multiplicity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In classical mechanics, the angular momentum of a particle possesses not only a magnitude (how fast the body is rotating), but also a direction (either up or down on the <a href="/wiki/Axis_of_rotation" class="mw-redirect" title="Axis of rotation">axis of rotation</a> of the particle). Quantum-mechanical spin also contains information about direction, but in a more subtle form. Quantum mechanics states that the <a href="/wiki/Spatial_vector" class="mw-redirect" title="Spatial vector">component</a> of angular momentum for a spin-<i>s</i> particle measured along any direction can only take on the values<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{i}=\hbar s_{i},\quad s_{i}\in \{-s,-(s-1),\dots ,s-1,s\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mi>s</mi> <mo>,</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>s</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>s</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{i}=\hbar s_{i},\quad s_{i}\in \{-s,-(s-1),\dots ,s-1,s\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f1f9654d17506b1b00baf7190f167e6351c7fe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.618ex; height:2.843ex;" alt="{\displaystyle S_{i}=\hbar s_{i},\quad s_{i}\in \{-s,-(s-1),\dots ,s-1,s\},}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">S<sub>i</sub></span> is the spin component along the <span class="texhtml mvar" style="font-style:italic;">i</span>-th axis (either <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">y</span>, or <span class="texhtml mvar" style="font-style:italic;">z</span>), <span class="texhtml mvar" style="font-style:italic;">s<sub>i</sub></span> is the spin projection quantum number along the <span class="texhtml mvar" style="font-style:italic;">i</span>-th axis, and <span class="texhtml mvar" style="font-style:italic;">s</span> is the principal spin quantum number (discussed in the previous section). Conventionally the direction chosen is the <span class="texhtml mvar" style="font-style:italic;">z</span> axis: </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 S_{z}=\hbar s_{z},\quad s_{z}\in \{-s,-(s-1),\dots ,s-1,s\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mi>s</mi> <mo>,</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>s</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>s</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{z}=\hbar s_{z},\quad s_{z}\in \{-s,-(s-1),\dots ,s-1,s\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86facde6219c8d3f40199ac054cd7687829c9a88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.224ex; height:2.843ex;" alt="{\displaystyle S_{z}=\hbar s_{z},\quad s_{z}\in \{-s,-(s-1),\dots ,s-1,s\},}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">S<sub>z</sub></span> is the spin component along the <span class="texhtml mvar" style="font-style:italic;">z</span> axis, <span class="texhtml mvar" style="font-style:italic;">s<sub>z</sub></span> is the spin projection quantum number along the <span class="texhtml mvar" style="font-style:italic;">z</span> axis. </p><p>One can see that there are <span class="texhtml">2<i>s</i> + 1</span> possible values of <span class="texhtml mvar" style="font-style:italic;">s<sub>z</sub></span>. The number "<span class="texhtml">2<i>s</i> + 1</span>" is the <a href="/wiki/Multiplicity_(chemistry)" title="Multiplicity (chemistry)">multiplicity</a> of the spin system. For example, there are only two possible values for a <a href="/wiki/Spin-1/2" title="Spin-1/2">spin-<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></a> particle: <span class="texhtml"><i>s<sub>z</sub></i> = +<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> and <span class="texhtml"><i>s<sub>z</sub></i> = −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>. These correspond to <a href="/wiki/Quantum_state" title="Quantum state">quantum states</a> in which the spin component is pointing in the +<i>z</i> or −<i>z</i> directions respectively, and are often referred to as "spin up" and "spin down". For a spin-<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> particle, like a <a href="/wiki/Delta_baryon" title="Delta baryon">delta baryon</a>, the possible values are +<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, +<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Vector">Vector</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=14" title="Edit section: Vector"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a given <a href="/wiki/Quantum_state" title="Quantum state">quantum state</a>, one could think of a spin 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="{\textstyle \langle S\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>S</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \langle S\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b052829953620eeccd940ba9fa71b24a3deb0cbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.309ex; height:2.843ex;" alt="{\textstyle \langle S\rangle }"></span> whose components are the <a href="/wiki/Expectation_value_(quantum_physics)" class="mw-redirect" title="Expectation value (quantum physics)">expectation values</a> of the spin components along each axis, 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="{\textstyle \langle S\rangle =[\langle S_{x}\rangle ,\langle S_{y}\rangle ,\langle S_{z}\rangle ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>S</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \langle S\rangle =[\langle S_{x}\rangle ,\langle S_{y}\rangle ,\langle S_{z}\rangle ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a52cf750b43ecfaa01df6ca13a322d6561695f15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.695ex; height:3.009ex;" alt="{\textstyle \langle S\rangle =[\langle S_{x}\rangle ,\langle S_{y}\rangle ,\langle S_{z}\rangle ]}"></span>. This vector then would describe the "direction" in which the spin is pointing, corresponding to the classical concept of the <a href="/wiki/Axis_of_rotation" class="mw-redirect" title="Axis of rotation">axis of rotation</a>. It turns out that the spin vector is not very useful in actual quantum-mechanical calculations, because it cannot be measured directly: <span class="texhtml mvar" style="font-style:italic;">s<sub>x</sub></span>, <span class="texhtml mvar" style="font-style:italic;">s<sub>y</sub></span> and <span class="texhtml mvar" style="font-style:italic;">s<sub>z</sub></span> cannot possess simultaneous definite values, because of a quantum <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty relation</a> between them. However, for statistically large collections of particles that have been placed in the same pure quantum state, such as through the use of a <a href="/wiki/Stern%E2%80%93Gerlach_apparatus" class="mw-redirect" title="Stern–Gerlach apparatus">Stern–Gerlach apparatus</a>, the spin vector does have a well-defined experimental meaning: It specifies the direction in ordinary space in which a subsequent detector must be oriented in order to achieve the maximum possible probability (100%) of detecting every particle in the collection. For spin-<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> particles, this probability drops off smoothly as the angle between the spin vector and the detector increases, until at an angle of 180°—that is, for detectors oriented in the opposite direction to the spin vector—the expectation of detecting particles from the collection reaches a minimum of 0%. </p><p>As a qualitative concept, the spin vector is often handy because it is easy to picture classically. For instance, quantum-mechanical spin can exhibit phenomena analogous to classical <a href="/wiki/Gyroscope" title="Gyroscope">gyroscopic effects</a>. For example, one can exert a kind of "<a href="/wiki/Torque" title="Torque">torque</a>" on an electron by putting it in a <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> (the field acts upon the electron's intrinsic <a href="/wiki/Magnetic_dipole_moment" class="mw-redirect" title="Magnetic dipole moment">magnetic dipole moment</a>—see the following section). The result is that the spin vector undergoes <a href="/wiki/Precession" title="Precession">precession</a>, just like a classical gyroscope. This phenomenon is known as <a href="/wiki/Electron_spin_resonance" class="mw-redirect" title="Electron spin resonance">electron spin resonance</a> (ESR). The equivalent behaviour of protons in atomic nuclei is used in <a href="/wiki/Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">nuclear magnetic resonance</a> (NMR) spectroscopy and imaging. </p><p>Mathematically, quantum-mechanical spin states are described by vector-like objects known as <a href="/wiki/Spinor" title="Spinor">spinors</a>. There are subtle differences between the behavior of spinors and vectors under <a href="/wiki/Coordinate_rotation" class="mw-redirect" title="Coordinate rotation">coordinate rotations</a>. For example, rotating a spin-<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> particle by 360° does not bring it back to the same quantum state, but to the state with the opposite quantum <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a>; this is detectable, in principle, with <a href="/wiki/Interference_(wave_propagation)" class="mw-redirect" title="Interference (wave propagation)">interference</a> experiments. To return the particle to its exact original state, one needs a 720° rotation. (The <a href="/wiki/Plate_trick" title="Plate trick">plate trick</a> and <a href="/wiki/M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a> give non-quantum analogies.) A spin-zero particle can only have a single quantum state, even after torque is applied. Rotating a spin-2 particle 180° can bring it back to the same quantum state, and a spin-4 particle should be rotated 90° to bring it back to the same quantum state. The spin-2 particle can be analogous to a straight stick that looks the same even after it is rotated 180°, and a spin-0 particle can be imagined as sphere, which looks the same after whatever angle it is turned through. </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_formulation">Mathematical formulation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=15" title="Edit section: Mathematical formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Operator">Operator</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=16" title="Edit section: Operator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Spin obeys <a href="/wiki/Commutation_relations" class="mw-redirect" title="Commutation relations">commutation relations</a><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> analogous to those of the <a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">orbital angular momentum</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 \left[{\hat {S}}_{j},{\hat {S}}_{k}\right]=i\hbar \varepsilon _{jkl}{\hat {S}}_{l},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> <mi>l</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\hat {S}}_{j},{\hat {S}}_{k}\right]=i\hbar \varepsilon _{jkl}{\hat {S}}_{l},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/075f18d0a435c840e760e9a979522c36dc20c527" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.862ex; height:4.843ex;" alt="{\displaystyle \left[{\hat {S}}_{j},{\hat {S}}_{k}\right]=i\hbar \varepsilon _{jkl}{\hat {S}}_{l},}"></span> where <span class="texhtml mvar" style="font-style:italic;">ε<sub>jkl</sub></span> is the <a href="/wiki/Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a>. It follows (as with <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>) that the <a href="/wiki/Eigenvectors" class="mw-redirect" title="Eigenvectors">eigenvectors</a> 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 {\hat {S}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {S}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f525fc62601674c7d99f4926642ca9eac29703a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:3.343ex;" alt="{\displaystyle {\hat {S}}^{2}}"></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 {\hat {S}}_{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>S</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 {S}}_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac67cb8315ba14707385264d2f64e1e65a965f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.574ex; height:3.176ex;" alt="{\displaystyle {\hat {S}}_{z}}"></span> (expressed as <a href="/wiki/Bra-ket_notation" class="mw-redirect" title="Bra-ket notation">kets</a> in the total <span class="texhtml mvar" style="font-style:italic;">S</span> <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a>) are<sup id="cite_ref-griffiths_2-2" class="reference"><a href="#cite_note-griffiths-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 166">: 166 </span></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 {\begin{aligned}{\hat {S}}^{2}|s,m_{s}\rangle &=\hbar ^{2}s(s+1)|s,m_{s}\rangle ,\\{\hat {S}}_{z}|s,m_{s}\rangle &=\hbar m_{s}|s,m_{s}\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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\hat {S}}^{2}|s,m_{s}\rangle &=\hbar ^{2}s(s+1)|s,m_{s}\rangle ,\\{\hat {S}}_{z}|s,m_{s}\rangle &=\hbar m_{s}|s,m_{s}\rangle .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83a281a47cd5089eaa10b4d4d5337ee7e11dfc56" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.932ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}{\hat {S}}^{2}|s,m_{s}\rangle &=\hbar ^{2}s(s+1)|s,m_{s}\rangle ,\\{\hat {S}}_{z}|s,m_{s}\rangle &=\hbar m_{s}|s,m_{s}\rangle .\end{aligned}}}"></span> </p><p>The spin <a href="/wiki/Creation_and_annihilation_operators" title="Creation and annihilation operators">raising and lowering operators</a> acting on these eigenvectors give <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 {S}}_{\pm }|s,m_{s}\rangle =\hbar {\sqrt {s(s+1)-m_{s}(m_{s}\pm 1)}}|s,m_{s}\pm 1\rangle ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>s</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>±</mo> <mn>1</mn> <mo stretchy="false">)</mo> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>±</mo> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {S}}_{\pm }|s,m_{s}\rangle =\hbar {\sqrt {s(s+1)-m_{s}(m_{s}\pm 1)}}|s,m_{s}\pm 1\rangle ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a230042419dede741677a5a0a83a8b5eb0eebe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:50.634ex; height:4.843ex;" alt="{\displaystyle {\hat {S}}_{\pm }|s,m_{s}\rangle =\hbar {\sqrt {s(s+1)-m_{s}(m_{s}\pm 1)}}|s,m_{s}\pm 1\rangle ,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {S}}_{\pm }={\hat {S}}_{x}\pm i{\hat {S}}_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>±</mo> <mi>i</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {S}}_{\pm }={\hat {S}}_{x}\pm i{\hat {S}}_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/815e5a7df3a40b3d5b8d325e121c19e07bfbf38f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.192ex; height:3.509ex;" alt="{\displaystyle {\hat {S}}_{\pm }={\hat {S}}_{x}\pm i{\hat {S}}_{y}}"></span>.<sup id="cite_ref-griffiths_2-3" class="reference"><a href="#cite_note-griffiths-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 166">: 166 </span></sup> </p><p>But unlike orbital angular momentum, the eigenvectors are not <a href="/wiki/Spherical_harmonics" title="Spherical harmonics">spherical harmonics</a>. They are not functions of <span class="texhtml mvar" style="font-style:italic;">θ</span> and <span class="texhtml mvar" style="font-style:italic;">φ</span>. There is also no reason to exclude half-integer values of <span class="texhtml mvar" style="font-style:italic;">s</span> and <span class="texhtml mvar" style="font-style:italic;">m<sub>s</sub></span>. </p><p>All quantum-mechanical particles possess an intrinsic spin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> (though this value may be equal to zero). The projection of the spin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> on any axis is quantized in units of the <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a>, such that the state function of the particle is, say, not <span class="mwe-math-element"><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 =\psi (\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =\psi (\mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109564f25c267002405f7d3b4daa0cfc8bd9c5ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.036ex; height:2.843ex;" alt="{\displaystyle \psi =\psi (\mathbf {r} )}"></span>, but <span class="mwe-math-element"><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 =\psi (\mathbf {r} ,s_{z})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =\psi (\mathbf {r} ,s_{z})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c179cbddd6b81a3c6f94737251c3985bee6f6cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.162ex; height:2.843ex;" alt="{\displaystyle \psi =\psi (\mathbf {r} ,s_{z})}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fe17b6a1d1ce8d16e8753353f2b8b575ae4381d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.092ex; height:2.009ex;" alt="{\displaystyle s_{z}}"></span> can take only the values of the following discrete set: <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 s_{z}\in \{-s\hbar ,-(s-1)\hbar ,\dots ,+(s-1)\hbar ,+s\hbar \}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mi>s</mi> <mi class="MJX-variant">ℏ</mi> <mo>,</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi class="MJX-variant">ℏ</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi class="MJX-variant">ℏ</mi> <mo>,</mo> <mo>+</mo> <mi>s</mi> <mi class="MJX-variant">ℏ</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{z}\in \{-s\hbar ,-(s-1)\hbar ,\dots ,+(s-1)\hbar ,+s\hbar \}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15111e5f9669c3ddcfbd71677511ac4bc1f39717" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.595ex; height:2.843ex;" alt="{\displaystyle s_{z}\in \{-s\hbar ,-(s-1)\hbar ,\dots ,+(s-1)\hbar ,+s\hbar \}.}"></span> </p><p>One distinguishes <a href="/wiki/Boson" title="Boson">bosons</a> (integer spin) and <a href="/wiki/Fermion" title="Fermion">fermions</a> (half-integer spin). The total angular momentum conserved in interaction processes is then the sum of the orbital angular momentum and the spin. </p> <div class="mw-heading mw-heading3"><h3 id="Pauli_matrices">Pauli matrices</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=17" title="Edit section: Pauli matrices"><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/Pauli_matrices" title="Pauli matrices">Pauli matrices</a></div> <p>The <a href="/wiki/Operator_(physics)#Operators_in_quantum_mechanics" title="Operator (physics)">quantum-mechanical operators</a> associated with spin-<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> <a href="/wiki/Observables" class="mw-redirect" title="Observables">observables</a> are <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 {S} }}={\frac {\hbar }{2}}{\boldsymbol {\sigma }},}"> <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">S</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {S} }}={\frac {\hbar }{2}}{\boldsymbol {\sigma }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1216f2e7db390f94edab53f22253780815fb9f9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.968ex; height:5.343ex;" alt="{\displaystyle {\hat {\mathbf {S} }}={\frac {\hbar }{2}}{\boldsymbol {\sigma }},}"></span> where in Cartesian components <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 S_{x}={\frac {\hbar }{2}}\sigma _{x},\quad S_{y}={\frac {\hbar }{2}}\sigma _{y},\quad S_{z}={\frac {\hbar }{2}}\sigma _{z}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mn>2</mn> </mfrac> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mn>2</mn> </mfrac> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mn>2</mn> </mfrac> </mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{x}={\frac {\hbar }{2}}\sigma _{x},\quad S_{y}={\frac {\hbar }{2}}\sigma _{y},\quad S_{z}={\frac {\hbar }{2}}\sigma _{z}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea23101e82f018cfefd54a349dc6791123b9f504" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.787ex; height:5.343ex;" alt="{\displaystyle S_{x}={\frac {\hbar }{2}}\sigma _{x},\quad S_{y}={\frac {\hbar }{2}}\sigma _{y},\quad S_{z}={\frac {\hbar }{2}}\sigma _{z}.}"></span> </p><p>For the special case of spin-<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> particles, <span class="texhtml mvar" style="font-style:italic;">σ<sub>x</sub></span>, <span class="texhtml mvar" style="font-style:italic;">σ<sub>y</sub></span> and <span class="texhtml mvar" style="font-style:italic;">σ<sub>z</sub></span> are the three <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</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 \sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad \sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad \sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/826b86c552f848c4cd463e3fdc39bdd993d9bb13" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.578ex; height:6.176ex;" alt="{\displaystyle \sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad \sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad \sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Pauli_exclusion_principle">Pauli exclusion principle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=18" title="Edit section: Pauli exclusion principle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a> states that the <a href="/wiki/Wavefunction" class="mw-redirect" title="Wavefunction">wavefunction</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 \psi (\mathbf {r} _{1},\sigma _{1},\dots ,\mathbf {r} _{N},\sigma _{N})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {r} _{1},\sigma _{1},\dots ,\mathbf {r} _{N},\sigma _{N})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d20f2e18a7f12cdd4996c71d7823eb228a28dfe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.919ex; height:2.843ex;" alt="{\displaystyle \psi (\mathbf {r} _{1},\sigma _{1},\dots ,\mathbf {r} _{N},\sigma _{N})}"></span> for a system of <span class="texhtml mvar" style="font-style:italic;">N</span> identical particles having spin <span class="texhtml mvar" style="font-style:italic;">s</span> must change upon interchanges of any two of the <span class="texhtml mvar" style="font-style:italic;">N</span> particles 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 (\dots ,\mathbf {r} _{i},\sigma _{i},\dots ,\mathbf {r} _{j},\sigma _{j},\dots )=(-1)^{2s}\psi (\dots ,\mathbf {r} _{j},\sigma _{j},\dots ,\mathbf {r} _{i},\sigma _{i},\dots ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>s</mi> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (\dots ,\mathbf {r} _{i},\sigma _{i},\dots ,\mathbf {r} _{j},\sigma _{j},\dots )=(-1)^{2s}\psi (\dots ,\mathbf {r} _{j},\sigma _{j},\dots ,\mathbf {r} _{i},\sigma _{i},\dots ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4efd5274f27e2901ca2541a7764aa6391452090" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:63.846ex; height:3.343ex;" alt="{\displaystyle \psi (\dots ,\mathbf {r} _{i},\sigma _{i},\dots ,\mathbf {r} _{j},\sigma _{j},\dots )=(-1)^{2s}\psi (\dots ,\mathbf {r} _{j},\sigma _{j},\dots ,\mathbf {r} _{i},\sigma _{i},\dots ).}"></span> </p><p>Thus, for <a href="/wiki/Boson" title="Boson">bosons</a> the prefactor <span class="texhtml">(−1)<sup>2<i>s</i></sup></span> will reduce to +1, for <a href="/wiki/Fermion" title="Fermion">fermions</a> to −1. This permutation postulate for <span class="texhtml mvar" style="font-style:italic;">N</span>-particle state functions has most important consequences in daily life, e.g. the <a href="/wiki/Periodic_table" title="Periodic table">periodic table</a> of the chemical elements. </p> <div class="mw-heading mw-heading3"><h3 id="Rotations">Rotations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=19" title="Edit section: Rotations"><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/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></div> <p>As described above, quantum mechanics states that <a href="/wiki/Spatial_vector" class="mw-redirect" title="Spatial vector">components</a> of angular momentum measured along any direction can only take a number of discrete values. The most convenient quantum-mechanical description of particle's spin is therefore with a set of complex numbers corresponding to amplitudes of finding a given value of projection of its intrinsic angular momentum on a given axis. For instance, for a spin-<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> particle, we would need two numbers <span class="texhtml"><i>a</i><sub>±1/2</sub></span>, giving amplitudes of finding it with projection of angular momentum equal to <span class="texhtml">+<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>ħ</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> and <span class="texhtml">−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>ħ</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>, satisfying the requirement <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 |a_{+1/2}|^{2}+|a_{-1/2}|^{2}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a_{+1/2}|^{2}+|a_{-1/2}|^{2}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed3dcdc078282eaf7ca6ecc6809960b0a5f823a3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.857ex; height:3.676ex;" alt="{\displaystyle |a_{+1/2}|^{2}+|a_{-1/2}|^{2}=1.}"></span> </p><p>For a generic particle with spin <span class="texhtml mvar" style="font-style:italic;">s</span>, we would need <span class="texhtml">2<i>s</i> + 1</span> such parameters. Since these numbers depend on the choice of the axis, they transform into each other non-trivially when this axis is rotated. It is clear that the transformation law must be linear, so we can represent it by associating a matrix with each rotation, and the product of two transformation matrices corresponding to rotations A and B must be equal (up to phase) to the matrix representing rotation AB. Further, rotations preserve the quantum-mechanical inner product, and so should our transformation matrices: <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 _{m=-j}^{j}a_{m}^{*}b_{m}=\sum _{m=-j}^{j}\left(\sum _{n=-j}^{j}U_{nm}a_{n}\right)^{*}\left(\sum _{k=-j}^{j}U_{km}b_{k}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munderover> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munderover> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munderover> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{m=-j}^{j}a_{m}^{*}b_{m}=\sum _{m=-j}^{j}\left(\sum _{n=-j}^{j}U_{nm}a_{n}\right)^{*}\left(\sum _{k=-j}^{j}U_{km}b_{k}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/509e7c5baac38634ca8177647c41abcdd7bb7e8c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:50.631ex; height:7.843ex;" alt="{\displaystyle \sum _{m=-j}^{j}a_{m}^{*}b_{m}=\sum _{m=-j}^{j}\left(\sum _{n=-j}^{j}U_{nm}a_{n}\right)^{*}\left(\sum _{k=-j}^{j}U_{km}b_{k}\right),}"></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 \sum _{n=-j}^{j}\sum _{k=-j}^{j}U_{np}^{*}U_{kq}=\delta _{pq}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munderover> <msubsup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>q</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>q</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=-j}^{j}\sum _{k=-j}^{j}U_{np}^{*}U_{kq}=\delta _{pq}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb6b91db4ffe6f2d9fe883cbe60842093e70741" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.744ex; height:7.676ex;" alt="{\displaystyle \sum _{n=-j}^{j}\sum _{k=-j}^{j}U_{np}^{*}U_{kq}=\delta _{pq}.}"></span> </p><p>Mathematically speaking, these matrices furnish a unitary <a href="/wiki/Projective_representation" title="Projective representation">projective representation</a> of the <a href="/wiki/Rotation_group_SO(3)" class="mw-redirect" title="Rotation group SO(3)">rotation group SO(3)</a>. Each such representation corresponds to a representation of the covering group of SO(3), which is <a href="/wiki/SU(2)" class="mw-redirect" title="SU(2)">SU(2)</a>.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> There is one <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional irreducible representation of SU(2) for each dimension, though this representation is <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional real for odd <span class="texhtml mvar" style="font-style:italic;">n</span> and <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional complex for even <span class="texhtml mvar" style="font-style:italic;">n</span> (hence of real dimension <span class="texhtml">2<i>n</i></span>). For a rotation by angle <span class="texhtml mvar" style="font-style:italic;">θ</span> in the plane with normal 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="{\textstyle {\hat {\boldsymbol {\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\hat {\boldsymbol {\theta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9393802d3a3215abe32cd313c42592c00c681fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.457ex; height:2.843ex;" alt="{\textstyle {\hat {\boldsymbol {\theta }}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=e^{-{\frac {i}{\hbar }}{\boldsymbol {\theta }}\cdot \mathbf {S} },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">θ</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=e^{-{\frac {i}{\hbar }}{\boldsymbol {\theta }}\cdot \mathbf {S} },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6fbb7ecb77e3c6817e0131eb05b372573935ce6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.14ex; height:3.843ex;" alt="{\displaystyle U=e^{-{\frac {i}{\hbar }}{\boldsymbol {\theta }}\cdot \mathbf {S} },}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\boldsymbol {\theta }}=\theta {\hat {\boldsymbol {\theta }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">θ</mi> </mrow> <mo>=</mo> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\boldsymbol {\theta }}=\theta {\hat {\boldsymbol {\theta }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2852b607b922421b60cf2cfee4b6b268a21d41a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.952ex; height:2.843ex;" alt="{\textstyle {\boldsymbol {\theta }}=\theta {\hat {\boldsymbol {\theta }}}}"></span>, and <span class="texhtml"><b>S</b></span> is the vector of <a href="#Operator">spin operators</a>. </p> <style data-mw-deduplicate="TemplateStyles:r1174254338">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style><div class="math_proof" style=""><strong>Proof</strong> <p>Working in the coordinate system where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\hat {\theta }}={\hat {z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\hat {\theta }}={\hat {z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90fefa0e1dd360cf67f7a0a280b336a049bc9c3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.751ex; height:2.843ex;" alt="{\textstyle {\hat {\theta }}={\hat {z}}}"></span>, we would like to show that <span class="texhtml mvar" style="font-style:italic;">S<sub>x</sub></span> and <span class="texhtml mvar" style="font-style:italic;">S<sub>y</sub></span> are rotated into each other by the angle <span class="texhtml mvar" style="font-style:italic;">θ</span>. Starting with <span class="texhtml mvar" style="font-style:italic;">S<sub>x</sub></span>. Using units where <span class="texhtml"><i>ħ</i> = 1</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 {\begin{aligned}S_{x}\rightarrow U^{\dagger }S_{x}U&=e^{i\theta S_{z}}S_{x}e^{-i\theta S_{z}}\\&=S_{x}+(i\theta )\left[S_{z},S_{x}\right]+\left({\frac {1}{2!}}\right)(i\theta )^{2}\left[S_{z},\left[S_{z},S_{x}\right]\right]+\left({\frac {1}{3!}}\right)(i\theta )^{3}\left[S_{z},\left[S_{z},\left[S_{z},S_{x}\right]\right]\right]+\cdots \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> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>U</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> </msup> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>θ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow> <mo>[</mo> <mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>[</mo> <mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <mrow> <mo>[</mo> <mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mo>[</mo> <mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <mrow> <mo>[</mo> <mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <mrow> <mo>[</mo> <mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S_{x}\rightarrow U^{\dagger }S_{x}U&=e^{i\theta S_{z}}S_{x}e^{-i\theta S_{z}}\\&=S_{x}+(i\theta )\left[S_{z},S_{x}\right]+\left({\frac {1}{2!}}\right)(i\theta )^{2}\left[S_{z},\left[S_{z},S_{x}\right]\right]+\left({\frac {1}{3!}}\right)(i\theta )^{3}\left[S_{z},\left[S_{z},\left[S_{z},S_{x}\right]\right]\right]+\cdots \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb75f835c80f9c5b2b2bf4702263a3de3e474185" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:97.528ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}S_{x}\rightarrow U^{\dagger }S_{x}U&=e^{i\theta S_{z}}S_{x}e^{-i\theta S_{z}}\\&=S_{x}+(i\theta )\left[S_{z},S_{x}\right]+\left({\frac {1}{2!}}\right)(i\theta )^{2}\left[S_{z},\left[S_{z},S_{x}\right]\right]+\left({\frac {1}{3!}}\right)(i\theta )^{3}\left[S_{z},\left[S_{z},\left[S_{z},S_{x}\right]\right]\right]+\cdots \end{aligned}}}"></span> </p><p>Using the <a href="#Operator">spin operator commutation relations</a>, we see that the commutators evaluate to <span class="texhtml mvar" style="font-style:italic;">i S<sub>y</sub></span> for the odd terms in the series, and to <span class="texhtml mvar" style="font-style:italic;">S<sub>x</sub></span> for all of the even terms. Thus: <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}U^{\dagger }S_{x}U&=S_{x}\left[1-{\frac {\theta ^{2}}{2!}}+\cdots \right]-S_{y}\left[\theta -{\frac {\theta ^{3}}{3!}}\cdots \right]\\&=S_{x}\cos \theta -S_{y}\sin \theta ,\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> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>U</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow> <mo>[</mo> <mrow> <mi>θ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>⋯</mo> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U^{\dagger }S_{x}U&=S_{x}\left[1-{\frac {\theta ^{2}}{2!}}+\cdots \right]-S_{y}\left[\theta -{\frac {\theta ^{3}}{3!}}\cdots \right]\\&=S_{x}\cos \theta -S_{y}\sin \theta ,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f749d3d323d7a74fda88685fd1c576b36370d916" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:47.199ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}U^{\dagger }S_{x}U&=S_{x}\left[1-{\frac {\theta ^{2}}{2!}}+\cdots \right]-S_{y}\left[\theta -{\frac {\theta ^{3}}{3!}}\cdots \right]\\&=S_{x}\cos \theta -S_{y}\sin \theta ,\end{aligned}}}"></span> as expected. Note that since we only relied on the spin operator commutation relations, this proof holds for any dimension (i.e., for any principal spin quantum number <span class="texhtml mvar" style="font-style:italic;">s</span>)<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 164">: 164 </span></sup> </p> </div> <p>A generic rotation in 3-dimensional space can be built by compounding operators of this type using <a href="/wiki/Euler_angles" title="Euler angles">Euler angles</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 {\mathcal {R}}(\alpha ,\beta ,\gamma )=e^{-i\alpha S_{x}}e^{-i\beta S_{y}}e^{-i\gamma S_{z}}.}"> <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">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>α</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>β</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>γ</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}(\alpha ,\beta ,\gamma )=e^{-i\alpha S_{x}}e^{-i\beta S_{y}}e^{-i\gamma S_{z}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2af68c2977b62c7dab0777ac5cc542ca9f499d5c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.613ex; height:3.176ex;" alt="{\displaystyle {\mathcal {R}}(\alpha ,\beta ,\gamma )=e^{-i\alpha S_{x}}e^{-i\beta S_{y}}e^{-i\gamma S_{z}}.}"></span> </p><p>An irreducible representation of this group of operators is furnished by the <a href="/wiki/Wigner_D-matrix" title="Wigner D-matrix">Wigner D-matrix</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 D_{m'm}^{s}(\alpha ,\beta ,\gamma )\equiv \langle sm'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|sm\rangle =e^{-im'\alpha }d_{m'm}^{s}(\beta )e^{-im\gamma },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>m</mi> <mo>′</mo> </msup> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>s</mi> <msup> <mi>m</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mi>m</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <msup> <mi>m</mi> <mo>′</mo> </msup> <mi>α</mi> </mrow> </msup> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>m</mi> <mo>′</mo> </msup> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>m</mi> <mi>γ</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{m'm}^{s}(\alpha ,\beta ,\gamma )\equiv \langle sm'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|sm\rangle =e^{-im'\alpha }d_{m'm}^{s}(\beta )e^{-im\gamma },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30f34b18f935a58c820d47bf69da56d00e323154" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:60.04ex; height:3.676ex;" alt="{\displaystyle D_{m'm}^{s}(\alpha ,\beta ,\gamma )\equiv \langle sm'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|sm\rangle =e^{-im'\alpha }d_{m'm}^{s}(\beta )e^{-im\gamma },}"></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 d_{m'm}^{s}(\beta )=\langle sm'|e^{-i\beta s_{y}}|sm\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>m</mi> <mo>′</mo> </msup> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>s</mi> <msup> <mi>m</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>β</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mi>m</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{m'm}^{s}(\beta )=\langle sm'|e^{-i\beta s_{y}}|sm\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5280ab88093cc7349aee97339763b2e79e0ba4ab" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.85ex; height:3.509ex;" alt="{\displaystyle d_{m'm}^{s}(\beta )=\langle sm'|e^{-i\beta s_{y}}|sm\rangle }"></span> is <a href="/wiki/Wigner_D-matrix#Wigner_(small)_d-matrix" title="Wigner D-matrix">Wigner's small d-matrix</a>. Note that for <span class="texhtml"><i>γ</i> = 2π</span> and <span class="texhtml"><i>α</i> = <i>β</i> = 0</span>; i.e., a full rotation about the <span class="texhtml mvar" style="font-style:italic;">z</span> axis, the Wigner D-matrix elements become <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{m'm}^{s}(0,0,2\pi )=d_{m'm}^{s}(0)e^{-im2\pi }=\delta _{m'm}(-1)^{2m}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>m</mi> <mo>′</mo> </msup> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>m</mi> <mo>′</mo> </msup> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>m</mi> <mn>2</mn> <mi>π</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>m</mi> <mo>′</mo> </msup> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>m</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{m'm}^{s}(0,0,2\pi )=d_{m'm}^{s}(0)e^{-im2\pi }=\delta _{m'm}(-1)^{2m}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a9de354e53fc72d397982f1b90b6f61a5ddf3b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:47.271ex; height:3.509ex;" alt="{\displaystyle D_{m'm}^{s}(0,0,2\pi )=d_{m'm}^{s}(0)e^{-im2\pi }=\delta _{m'm}(-1)^{2m}.}"></span> </p><p>Recalling that a generic spin state can be written as a superposition of states with definite <span class="texhtml mvar" style="font-style:italic;">m</span>, we see that if <span class="texhtml mvar" style="font-style:italic;">s</span> is an integer, the values of <span class="texhtml mvar" style="font-style:italic;">m</span> are all integers, and this matrix corresponds to the identity operator. However, if <span class="texhtml mvar" style="font-style:italic;">s</span> is a half-integer, the values of <span class="texhtml mvar" style="font-style:italic;">m</span> are also all half-integers, giving <span class="texhtml">(−1)<sup>2<i>m</i></sup> = −1</span> for all <span class="texhtml mvar" style="font-style:italic;">m</span>, and hence upon rotation by 2<span class="texhtml mvar" style="font-style:italic;">π</span> the state picks up a minus sign. This fact is a crucial element of the proof of the <a href="/wiki/Spin%E2%80%93statistics_theorem" title="Spin–statistics theorem">spin–statistics theorem</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Lorentz_transformations">Lorentz transformations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=20" title="Edit section: Lorentz transformations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Spin_(physics)" title="Special:EditPage/Spin (physics)">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">May 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>We could try the same approach to determine the behavior of spin under general <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformations</a>, but we would immediately discover a major obstacle. Unlike SO(3), the group of Lorentz transformations <a href="/wiki/SO(3,1)" class="mw-redirect" title="SO(3,1)">SO(3,1)</a> is <a href="/wiki/Compact_group" title="Compact group">non-compact</a> and therefore does not have any faithful, unitary, finite-dimensional representations. </p><p>In case of spin-<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> particles, it is possible to find a construction that includes both a finite-dimensional representation and a scalar product that is preserved by this representation. We associate a 4-component <a href="/wiki/Dirac_spinor" title="Dirac spinor">Dirac spinor</a> <span class="texhtml mvar" style="font-style:italic;">ψ</span> with each particle. These spinors transform under Lorentz transformations according to the law <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 '=\exp {\left({\tfrac {1}{8}}\omega _{\mu \nu }[\gamma _{\mu },\gamma _{\nu }]\right)}\psi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mstyle> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">[</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mi>ψ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi '=\exp {\left({\tfrac {1}{8}}\omega _{\mu \nu }[\gamma _{\mu },\gamma _{\nu }]\right)}\psi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e11ff7900e4d19601a8520c61497413e2b47e9f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.433ex; height:4.843ex;" alt="{\displaystyle \psi '=\exp {\left({\tfrac {1}{8}}\omega _{\mu \nu }[\gamma _{\mu },\gamma _{\nu }]\right)}\psi ,}"></span> where <span class="texhtml mvar" style="font-style:italic;">γ<sub>ν</sub></span> are <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a>, and <span class="texhtml mvar" style="font-style:italic;">ω<sub>μν</sub></span> is an antisymmetric 4 × 4 matrix parametrizing the transformation. It can be shown that the scalar 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 \langle \psi |\phi \rangle ={\bar {\psi }}\phi =\psi ^{\dagger }\gamma _{0}\phi }"> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mi>ϕ</mi> <mo>=</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \psi |\phi \rangle ={\bar {\psi }}\phi =\psi ^{\dagger }\gamma _{0}\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14c91fd04bfaba7dc42a9a14c9d1ebb61ec13789" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.652ex; height:3.176ex;" alt="{\displaystyle \langle \psi |\phi \rangle ={\bar {\psi }}\phi =\psi ^{\dagger }\gamma _{0}\phi }"></span> is preserved. It is not, however, positive-definite, so the representation is not unitary. </p> <div class="mw-heading mw-heading3"><h3 id="Measurement_of_spin_along_the_x,_y,_or_z_axes"><span id="Measurement_of_spin_along_the_x.2C_y.2C_or_z_axes"></span>Measurement of spin along the <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">y</span>, or <span class="texhtml mvar" style="font-style:italic;">z</span> axes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=21" title="Edit section: Measurement of spin along the x, y, or z axes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Each of the (<a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian</a>) Pauli matrices of spin-<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> particles has two <a href="/wiki/Eigenvalues" class="mw-redirect" title="Eigenvalues">eigenvalues</a>, +1 and −1. The corresponding <a href="/wiki/Normalisable_wavefunction" class="mw-redirect" title="Normalisable wavefunction">normalized</a> <a href="/wiki/Eigenvectors" class="mw-redirect" title="Eigenvectors">eigenvectors</a> are <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{array}{lclc}\psi _{x+}=\left|{\frac {1}{2}},{\frac {+1}{2}}\right\rangle _{x}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{1}\end{pmatrix}},&\psi _{x-}=\left|{\frac {1}{2}},{\frac {-1}{2}}\right\rangle _{x}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{-1}\end{pmatrix}},\\\psi _{y+}=\left|{\frac {1}{2}},{\frac {+1}{2}}\right\rangle _{y}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{i}\end{pmatrix}},&\psi _{y-}=\left|{\frac {1}{2}},{\frac {-1}{2}}\right\rangle _{y}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{-i}\end{pmatrix}},\\\psi _{z+}=\left|{\frac {1}{2}},{\frac {+1}{2}}\right\rangle _{z}=&{\begin{pmatrix}1\\0\end{pmatrix}},&\psi _{z-}=\left|{\frac {1}{2}},{\frac {-1}{2}}\right\rangle _{z}=&{\begin{pmatrix}0\\1\end{pmatrix}}.\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left center left center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>+</mo> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>−</mo> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>+</mo> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>−</mo> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>+</mo> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>−</mo> </mrow> </msub> <mo>=</mo> <msub> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lclc}\psi _{x+}=\left|{\frac {1}{2}},{\frac {+1}{2}}\right\rangle _{x}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{1}\end{pmatrix}},&\psi _{x-}=\left|{\frac {1}{2}},{\frac {-1}{2}}\right\rangle _{x}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{-1}\end{pmatrix}},\\\psi _{y+}=\left|{\frac {1}{2}},{\frac {+1}{2}}\right\rangle _{y}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{i}\end{pmatrix}},&\psi _{y-}=\left|{\frac {1}{2}},{\frac {-1}{2}}\right\rangle _{y}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{-i}\end{pmatrix}},\\\psi _{z+}=\left|{\frac {1}{2}},{\frac {+1}{2}}\right\rangle _{z}=&{\begin{pmatrix}1\\0\end{pmatrix}},&\psi _{z-}=\left|{\frac {1}{2}},{\frac {-1}{2}}\right\rangle _{z}=&{\begin{pmatrix}0\\1\end{pmatrix}}.\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/039abb68de98761481bdf1c0a7834581c003a830" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.338ex; width:63.548ex; height:19.843ex;" alt="{\displaystyle {\begin{array}{lclc}\psi _{x+}=\left|{\frac {1}{2}},{\frac {+1}{2}}\right\rangle _{x}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{1}\end{pmatrix}},&\psi _{x-}=\left|{\frac {1}{2}},{\frac {-1}{2}}\right\rangle _{x}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{-1}\end{pmatrix}},\\\psi _{y+}=\left|{\frac {1}{2}},{\frac {+1}{2}}\right\rangle _{y}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{i}\end{pmatrix}},&\psi _{y-}=\left|{\frac {1}{2}},{\frac {-1}{2}}\right\rangle _{y}=\displaystyle {\frac {1}{\sqrt {2}}}\!\!\!\!\!&{\begin{pmatrix}{1}\\{-i}\end{pmatrix}},\\\psi _{z+}=\left|{\frac {1}{2}},{\frac {+1}{2}}\right\rangle _{z}=&{\begin{pmatrix}1\\0\end{pmatrix}},&\psi _{z-}=\left|{\frac {1}{2}},{\frac {-1}{2}}\right\rangle _{z}=&{\begin{pmatrix}0\\1\end{pmatrix}}.\end{array}}}"></span> </p><p>(Because any eigenvector multiplied by a constant is still an eigenvector, there is ambiguity about the overall sign. In this article, the convention is chosen to make the first element imaginary and negative if there is a sign ambiguity. The present convention is used by software such as <a href="/wiki/SymPy" title="SymPy">SymPy</a>; while many physics textbooks, such as Sakurai and Griffiths, prefer to make it real and positive.) </p><p>By the <a href="/wiki/Postulates_of_quantum_mechanics#Postulates_of_quantum_mechanics" class="mw-redirect" title="Postulates of quantum mechanics">postulates of quantum mechanics</a>, an experiment designed to measure the electron spin on the <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">y</span>, or <span class="texhtml mvar" style="font-style:italic;">z</span> axis can only yield an eigenvalue of the corresponding spin operator (<span class="texhtml mvar" style="font-style:italic;">S<sub>x</sub></span>, <span class="texhtml mvar" style="font-style:italic;">S<sub>y</sub></span> or <span class="texhtml mvar" style="font-style:italic;">S<sub>z</sub></span>) on that axis, i.e. <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>ħ</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> or <span class="texhtml">–<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>ħ</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>. The <a href="/wiki/Quantum_state" title="Quantum state">quantum state</a> of a particle (with respect to spin), can be represented by a two-component <a href="/wiki/Spinor" title="Spinor">spinor</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 \psi ={\begin{pmatrix}a+bi\\c+di\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mi>i</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ={\begin{pmatrix}a+bi\\c+di\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d04a01c26db12a7b5fcbf518df94326dcdc39a5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.301ex; height:6.176ex;" alt="{\displaystyle \psi ={\begin{pmatrix}a+bi\\c+di\end{pmatrix}}.}"></span> </p><p>When the spin of this particle is measured with respect to a given axis (in this example, the <span class="texhtml mvar" style="font-style:italic;">x</span> axis), the probability that its spin will be measured as <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>ħ</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> is just <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\big |}\langle \psi _{x+}|\psi \rangle {\big |}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>+</mo> </mrow> </msub> <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-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\big |}\langle \psi _{x+}|\psi \rangle {\big |}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af7168738404eb1a61accc04c9e36ff1a753f47e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.281ex; height:3.676ex;" alt="{\displaystyle {\big |}\langle \psi _{x+}|\psi \rangle {\big |}^{2}}"></span>. Correspondingly, the probability that its spin will be measured as <span class="texhtml">–<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>ħ</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> is just <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\big |}\langle \psi _{x-}|\psi \rangle {\big |}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>−</mo> </mrow> </msub> <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-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\big |}\langle \psi _{x-}|\psi \rangle {\big |}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3a440d4bc31361fed6e2cbde5dc61815df9711e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.281ex; height:3.676ex;" alt="{\displaystyle {\big |}\langle \psi _{x-}|\psi \rangle {\big |}^{2}}"></span>. Following the measurement, the spin state of the particle <a href="/wiki/Wavefunction_collapse" class="mw-redirect" title="Wavefunction collapse">collapses</a> into the corresponding eigenstate. As a result, if the particle's spin along a given axis has been measured to have a given eigenvalue, all measurements will yield the same eigenvalue (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 {\big |}\langle \psi _{x+}|\psi _{x+}\rangle {\big |}^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>+</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>+</mo> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\big |}\langle \psi _{x+}|\psi _{x+}\rangle {\big |}^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29f316d82747234fe1278b41cd84dab1d5e4c7f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.993ex; height:3.676ex;" alt="{\displaystyle {\big |}\langle \psi _{x+}|\psi _{x+}\rangle {\big |}^{2}=1}"></span>, etc.), provided that no measurements of the spin are made along other axes. </p> <div class="mw-heading mw-heading3"><h3 id="Measurement_of_spin_along_an_arbitrary_axis">Measurement of spin along an arbitrary axis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=22" title="Edit section: Measurement of spin along an arbitrary axis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Spin_(physics)" title="Special:EditPage/Spin (physics)">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">May 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>The operator to measure spin along an arbitrary axis direction is easily obtained from the Pauli spin matrices. Let <span class="texhtml"><i>u</i> = (<i>u<sub>x</sub></i>, <i>u<sub>y</sub></i>, <i>u<sub>z</sub></i>)</span> be an arbitrary unit vector. Then the operator for spin in this direction is simply <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 S_{u}={\frac {\hbar }{2}}(u_{x}\sigma _{x}+u_{y}\sigma _{y}+u_{z}\sigma _{z}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{u}={\frac {\hbar }{2}}(u_{x}\sigma _{x}+u_{y}\sigma _{y}+u_{z}\sigma _{z}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb1183035de365800e059aed81b68d6b7bcc061" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.393ex; height:5.343ex;" alt="{\displaystyle S_{u}={\frac {\hbar }{2}}(u_{x}\sigma _{x}+u_{y}\sigma _{y}+u_{z}\sigma _{z}).}"></span> </p><p>The operator <span class="texhtml mvar" style="font-style:italic;">S<sub>u</sub></span> has eigenvalues of <span class="texhtml">±<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>ħ</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>, just like the usual spin matrices. This method of finding the operator for spin in an arbitrary direction generalizes to higher spin states, one takes the <a href="/wiki/Dot_product" title="Dot product">dot product</a> of the direction with a vector of the three operators for the three <span class="texhtml mvar" style="font-style:italic;">x</span>-, <span class="texhtml mvar" style="font-style:italic;">y</span>-, <span class="texhtml mvar" style="font-style:italic;">z</span>-axis directions. </p><p>A normalized spinor for spin-<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> in the <span class="texhtml">(<i>u<sub>x</sub></i>, <i>u<sub>y</sub></i>, <i>u<sub>z</sub></i>)</span> direction (which works for all spin states except spin down, where it will give <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">0</span><span class="sr-only">/</span><span class="den">0</span></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 {\frac {1}{\sqrt {2+2u_{z}}}}{\begin{pmatrix}1+u_{z}\\u_{x}+iu_{y}\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> <mo>+</mo> <mn>2</mn> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {2+2u_{z}}}}{\begin{pmatrix}1+u_{z}\\u_{x}+iu_{y}\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2f0f06b2b1152507b3e3a7c241d972236216415" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:23.612ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {2+2u_{z}}}}{\begin{pmatrix}1+u_{z}\\u_{x}+iu_{y}\end{pmatrix}}.}"></span> </p><p>The above spinor is obtained in the usual way by diagonalizing the <span class="texhtml mvar" style="font-style:italic;">σ<sub>u</sub></span> matrix and finding the eigenstates corresponding to the eigenvalues. In quantum mechanics, vectors are termed "normalized" when multiplied by a normalizing factor, which results in the vector having a length of unity. </p> <div class="mw-heading mw-heading3"><h3 id="Compatibility_of_spin_measurements">Compatibility of spin measurements</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=23" title="Edit section: Compatibility of spin measurements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Spin_(physics)" title="Special:EditPage/Spin (physics)">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">May 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>Since the Pauli matrices do not <a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">commute</a>, measurements of spin along the different axes are incompatible. This means that if, for example, we know the spin along the <span class="texhtml mvar" style="font-style:italic;">x</span> axis, and we then measure the spin along the <span class="texhtml mvar" style="font-style:italic;">y</span> axis, we have invalidated our previous knowledge of the <span class="texhtml mvar" style="font-style:italic;">x</span> axis spin. This can be seen from the property of the eigenvectors (i.e. eigenstates) of the Pauli matrices 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 {\big |}\langle \psi _{x\pm }|\psi _{y\pm }\rangle {\big |}^{2}={\big |}\langle \psi _{x\pm }|\psi _{z\pm }\rangle {\big |}^{2}={\big |}\langle \psi _{y\pm }|\psi _{z\pm }\rangle {\big |}^{2}={\tfrac {1}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>±</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>±</mo> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>±</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>±</mo> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>±</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>±</mo> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\big |}\langle \psi _{x\pm }|\psi _{y\pm }\rangle {\big |}^{2}={\big |}\langle \psi _{x\pm }|\psi _{z\pm }\rangle {\big |}^{2}={\big |}\langle \psi _{y\pm }|\psi _{z\pm }\rangle {\big |}^{2}={\tfrac {1}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ba727da42d413ebb77f266ccc639e648a88c512" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:49.21ex; height:3.843ex;" alt="{\displaystyle {\big |}\langle \psi _{x\pm }|\psi _{y\pm }\rangle {\big |}^{2}={\big |}\langle \psi _{x\pm }|\psi _{z\pm }\rangle {\big |}^{2}={\big |}\langle \psi _{y\pm }|\psi _{z\pm }\rangle {\big |}^{2}={\tfrac {1}{2}}.}"></span> </p><p>So when <a href="/wiki/Physicist" title="Physicist">physicists</a> measure the spin of a particle along the <span class="texhtml mvar" style="font-style:italic;">x</span> axis as, for example, <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>ħ</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>, the particle's spin state <a href="/wiki/Wavefunction_collapse" class="mw-redirect" title="Wavefunction collapse">collapses</a> into the 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="{\displaystyle |\psi _{x+}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>+</mo> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi _{x+}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/719629d08083f0fc9b1b0e486ebd84a8d5a42b5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.516ex; height:2.843ex;" alt="{\displaystyle |\psi _{x+}\rangle }"></span>. When we then subsequently measure the particle's spin along the <span class="texhtml mvar" style="font-style:italic;">y</span> axis, the spin state will now collapse into either <span class="mwe-math-element"><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 _{y+}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>+</mo> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi _{y+}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3706cfed3c242e08d7860dee6e09a70624698ac0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.393ex; height:3.009ex;" alt="{\displaystyle |\psi _{y+}\rangle }"></span> or <span class="mwe-math-element"><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 _{y-}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>−</mo> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi _{y-}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/583b95f9311740e9634a4c06cdc19d325128ccdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.393ex; height:3.009ex;" alt="{\displaystyle |\psi _{y-}\rangle }"></span>, each with probability <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>. Let us say, in our example, that we measure <span class="texhtml">−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>ħ</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>. When we now return to measure the particle's spin along the <span class="texhtml mvar" style="font-style:italic;">x</span> axis again, the probabilities that we will measure <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>ħ</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> or <span class="texhtml">−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>ħ</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> are each <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> (i.e. they are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\big |}\langle \psi _{x+}|\psi _{y-}\rangle {\big |}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>+</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>−</mo> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\big |}\langle \psi _{x+}|\psi _{y-}\rangle {\big |}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d39ca429e7e4a1aea95c9efc067652609c6f500" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.609ex; height:3.676ex;" alt="{\displaystyle {\big |}\langle \psi _{x+}|\psi _{y-}\rangle {\big |}^{2}}"></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 {\big |}\langle \psi _{x-}|\psi _{y-}\rangle {\big |}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>−</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>−</mo> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\big |}\langle \psi _{x-}|\psi _{y-}\rangle {\big |}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/828a4edc580d27f878fdeaf40dcc8dedc7662a14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.609ex; height:3.676ex;" alt="{\displaystyle {\big |}\langle \psi _{x-}|\psi _{y-}\rangle {\big |}^{2}}"></span> respectively). This implies that the original measurement of the spin along the <span class="texhtml mvar" style="font-style:italic;">x</span> axis is no longer valid, since the spin along the <span class="texhtml mvar" style="font-style:italic;">x</span> axis will now be measured to have either eigenvalue with equal probability. </p> <div class="mw-heading mw-heading3"><h3 id="Higher_spins">Higher spins</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=24" title="Edit section: Higher spins"><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/3D_rotation_group#A_note_on_Lie_algebras" title="3D rotation group">3D rotation group § A note on Lie algebras</a></div> <p>The spin-<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> operator <span class="texhtml"><b>S</b> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>ħ</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><b>σ</b></span> forms the <a href="/wiki/Fundamental_representation" title="Fundamental representation">fundamental representation</a> of <a href="/wiki/Representation_theory_of_SU(2)" title="Representation theory of SU(2)">SU(2)</a>. By taking <a href="/wiki/Kronecker_product" title="Kronecker product">Kronecker products</a> of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting <a href="/wiki/Spin_operator" class="mw-redirect" title="Spin operator">spin operators</a> for higher-spin systems in three spatial dimensions can be calculated for arbitrarily large <span class="texhtml mvar" style="font-style:italic;">s</span> using this <a href="/wiki/Spin_operator" class="mw-redirect" title="Spin operator">spin operator</a> and <a href="/wiki/Ladder_operator#Angular_momentum" title="Ladder operator">ladder operators</a>. For example, taking the Kronecker product of two spin-<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> yields a four-dimensional representation, which is separable into a 3-dimensional spin-1 (<a href="/wiki/Triplet_state" title="Triplet state">triplet states</a>) and a 1-dimensional spin-0 representation (<a href="/wiki/Singlet_state" title="Singlet state">singlet state</a>). </p><p>The resulting irreducible representations yield the following spin matrices and eigenvalues in the z-basis: </p> <div><ol><li>For spin 1 they are <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}S_{x}&={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}},&\left|1,+1\right\rangle _{x}&={\frac {1}{2}}{\begin{pmatrix}1\\{\sqrt {2}}\\1\end{pmatrix}},&\left|1,0\right\rangle _{x}&={\frac {1}{\sqrt {2}}}{\begin{pmatrix}-1\\0\\1\end{pmatrix}},&\left|1,-1\right\rangle _{x}&={\frac {1}{2}}{\begin{pmatrix}1\\{-{\sqrt {2}}}\\1\end{pmatrix}}\\S_{y}&={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&-i&0\\i&0&-i\\0&i&0\end{pmatrix}},&\left|1,+1\right\rangle _{y}&={\frac {1}{2}}{\begin{pmatrix}-1\\-i{\sqrt {2}}\\1\end{pmatrix}},&\left|1,0\right\rangle _{y}&={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\0\\1\end{pmatrix}},&\left|1,-1\right\rangle _{y}&={\frac {1}{2}}{\begin{pmatrix}-1\\i{\sqrt {2}}\\1\end{pmatrix}}\\S_{z}&=\hbar {\begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\end{pmatrix}},&\left|1,+1\right\rangle _{z}&={\begin{pmatrix}1\\0\\0\end{pmatrix}},&\left|1,0\right\rangle _{z}&={\begin{pmatrix}0\\1\\0\end{pmatrix}},&\left|1,-1\right\rangle _{z}&={\begin{pmatrix}0\\0\\1\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> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mrow> <mo>|</mo> <mrow> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mrow> <mo>|</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>0</mn> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mrow> <mo>|</mo> <mrow> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>i</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>i</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mrow> <mo>|</mo> <mrow> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mrow> <mo>|</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>0</mn> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mrow> <mo>|</mo> <mrow> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mrow> <mo>|</mo> <mrow> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mrow> <mo>|</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>0</mn> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> <mtd> <msub> <mrow> <mo>|</mo> <mrow> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S_{x}&={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}},&\left|1,+1\right\rangle _{x}&={\frac {1}{2}}{\begin{pmatrix}1\\{\sqrt {2}}\\1\end{pmatrix}},&\left|1,0\right\rangle _{x}&={\frac {1}{\sqrt {2}}}{\begin{pmatrix}-1\\0\\1\end{pmatrix}},&\left|1,-1\right\rangle _{x}&={\frac {1}{2}}{\begin{pmatrix}1\\{-{\sqrt {2}}}\\1\end{pmatrix}}\\S_{y}&={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&-i&0\\i&0&-i\\0&i&0\end{pmatrix}},&\left|1,+1\right\rangle _{y}&={\frac {1}{2}}{\begin{pmatrix}-1\\-i{\sqrt {2}}\\1\end{pmatrix}},&\left|1,0\right\rangle _{y}&={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\0\\1\end{pmatrix}},&\left|1,-1\right\rangle _{y}&={\frac {1}{2}}{\begin{pmatrix}-1\\i{\sqrt {2}}\\1\end{pmatrix}}\\S_{z}&=\hbar {\begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\end{pmatrix}},&\left|1,+1\right\rangle _{z}&={\begin{pmatrix}1\\0\\0\end{pmatrix}},&\left|1,0\right\rangle _{z}&={\begin{pmatrix}0\\1\\0\end{pmatrix}},&\left|1,-1\right\rangle _{z}&={\begin{pmatrix}0\\0\\1\end{pmatrix}}\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60b57208794d3f3124ab0d5cbc6f3b2fb5816016" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.838ex; width:109.238ex; height:28.843ex;" alt="{\displaystyle {\begin{aligned}S_{x}&={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}},&\left|1,+1\right\rangle _{x}&={\frac {1}{2}}{\begin{pmatrix}1\\{\sqrt {2}}\\1\end{pmatrix}},&\left|1,0\right\rangle _{x}&={\frac {1}{\sqrt {2}}}{\begin{pmatrix}-1\\0\\1\end{pmatrix}},&\left|1,-1\right\rangle _{x}&={\frac {1}{2}}{\begin{pmatrix}1\\{-{\sqrt {2}}}\\1\end{pmatrix}}\\S_{y}&={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&-i&0\\i&0&-i\\0&i&0\end{pmatrix}},&\left|1,+1\right\rangle _{y}&={\frac {1}{2}}{\begin{pmatrix}-1\\-i{\sqrt {2}}\\1\end{pmatrix}},&\left|1,0\right\rangle _{y}&={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\0\\1\end{pmatrix}},&\left|1,-1\right\rangle _{y}&={\frac {1}{2}}{\begin{pmatrix}-1\\i{\sqrt {2}}\\1\end{pmatrix}}\\S_{z}&=\hbar {\begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\end{pmatrix}},&\left|1,+1\right\rangle _{z}&={\begin{pmatrix}1\\0\\0\end{pmatrix}},&\left|1,0\right\rangle _{z}&={\begin{pmatrix}0\\1\\0\end{pmatrix}},&\left|1,-1\right\rangle _{z}&={\begin{pmatrix}0\\0\\1\end{pmatrix}}\\\end{aligned}}}"></span></li><li>For spin <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> they are <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{array}{lclc}S_{x}={\frac {\hbar }{2}}{\begin{pmatrix}0&{\sqrt {3}}&0&0\\{\sqrt {3}}&0&2&0\\0&2&0&{\sqrt {3}}\\0&0&{\sqrt {3}}&0\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {+3}{2}}\right\rangle _{x}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}1\\{\sqrt {3}}\\{\sqrt {3}}\\1\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {+1}{2}}\right\rangle _{x}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}{-{\sqrt {3}}}\\-1\\1\\{\sqrt {3}}\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {-1}{2}}\right\rangle _{x}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}{\sqrt {3}}\\-1\\-1\\{\sqrt {3}}\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {-3}{2}}\right\rangle _{x}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}-1\\{\sqrt {3}}\\{-{\sqrt {3}}}\\1\end{pmatrix}}\\S_{y}={\frac {\hbar }{2}}{\begin{pmatrix}0&-i{\sqrt {3}}&0&0\\i{\sqrt {3}}&0&-2i&0\\0&2i&0&-i{\sqrt {3}}\\0&0&i{\sqrt {3}}&0\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {+3}{2}}\right\rangle _{y}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}{i}\\{-{\sqrt {3}}}\\{-i{\sqrt {3}}}\\1\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {+1}{2}}\right\rangle _{y}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}{-i{\sqrt {3}}}\\1\\{-i}\\{\sqrt {3}}\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {-1}{2}}\right\rangle _{y}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}{i{\sqrt {3}}}\\1\\{i}\\{\sqrt {3}}\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {-3}{2}}\right\rangle _{y}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}{-i}\\{-{\sqrt {3}}}\\{i{\sqrt {3}}}\\1\end{pmatrix}}\\S_{z}={\frac {\hbar }{2}}{\begin{pmatrix}3&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-3\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {+3}{2}}\right\rangle _{z}=\!\!\!&{\begin{pmatrix}1\\0\\0\\0\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {+1}{2}}\right\rangle _{z}=\!\!\!&{\begin{pmatrix}0\\1\\0\\0\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {-1}{2}}\right\rangle _{z}=\!\!\!&{\begin{pmatrix}0\\0\\1\\0\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {-3}{2}}\right\rangle _{z}=\!\!\!&{\begin{pmatrix}0\\0\\0\\1\end{pmatrix}}\\\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left center left center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> 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<mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mtd> <mtd> <msub> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>3</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mrow 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</mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mtd> <mtd> <msub> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>+</mo> <mn>3</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mtd> <mtd> <msub> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>+</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mtd> <mtd> <msub> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mtd> <mtd> <msub> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>3</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lclc}S_{x}={\frac {\hbar }{2}}{\begin{pmatrix}0&{\sqrt {3}}&0&0\\{\sqrt {3}}&0&2&0\\0&2&0&{\sqrt {3}}\\0&0&{\sqrt {3}}&0\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {+3}{2}}\right\rangle _{x}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}1\\{\sqrt {3}}\\{\sqrt {3}}\\1\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {+1}{2}}\right\rangle _{x}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}{-{\sqrt {3}}}\\-1\\1\\{\sqrt {3}}\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {-1}{2}}\right\rangle _{x}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}{\sqrt {3}}\\-1\\-1\\{\sqrt {3}}\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {-3}{2}}\right\rangle _{x}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}-1\\{\sqrt {3}}\\{-{\sqrt {3}}}\\1\end{pmatrix}}\\S_{y}={\frac {\hbar }{2}}{\begin{pmatrix}0&-i{\sqrt {3}}&0&0\\i{\sqrt {3}}&0&-2i&0\\0&2i&0&-i{\sqrt {3}}\\0&0&i{\sqrt {3}}&0\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {+3}{2}}\right\rangle _{y}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}{i}\\{-{\sqrt {3}}}\\{-i{\sqrt {3}}}\\1\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {+1}{2}}\right\rangle _{y}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}{-i{\sqrt {3}}}\\1\\{-i}\\{\sqrt {3}}\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {-1}{2}}\right\rangle _{y}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}{i{\sqrt {3}}}\\1\\{i}\\{\sqrt {3}}\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {-3}{2}}\right\rangle _{y}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}{-i}\\{-{\sqrt {3}}}\\{i{\sqrt {3}}}\\1\end{pmatrix}}\\S_{z}={\frac {\hbar }{2}}{\begin{pmatrix}3&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-3\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {+3}{2}}\right\rangle _{z}=\!\!\!&{\begin{pmatrix}1\\0\\0\\0\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {+1}{2}}\right\rangle _{z}=\!\!\!&{\begin{pmatrix}0\\1\\0\\0\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {-1}{2}}\right\rangle _{z}=\!\!\!&{\begin{pmatrix}0\\0\\1\\0\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {-3}{2}}\right\rangle _{z}=\!\!\!&{\begin{pmatrix}0\\0\\0\\1\end{pmatrix}}\\\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbcb1b7efce769a427fc9118743a42af17177d30" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -19.505ex; width:151.148ex; height:40.176ex;" alt="{\displaystyle {\begin{array}{lclc}S_{x}={\frac {\hbar }{2}}{\begin{pmatrix}0&{\sqrt {3}}&0&0\\{\sqrt {3}}&0&2&0\\0&2&0&{\sqrt {3}}\\0&0&{\sqrt {3}}&0\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {+3}{2}}\right\rangle _{x}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}1\\{\sqrt {3}}\\{\sqrt {3}}\\1\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {+1}{2}}\right\rangle _{x}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}{-{\sqrt {3}}}\\-1\\1\\{\sqrt {3}}\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {-1}{2}}\right\rangle _{x}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}{\sqrt {3}}\\-1\\-1\\{\sqrt {3}}\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {-3}{2}}\right\rangle _{x}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}-1\\{\sqrt {3}}\\{-{\sqrt {3}}}\\1\end{pmatrix}}\\S_{y}={\frac {\hbar }{2}}{\begin{pmatrix}0&-i{\sqrt {3}}&0&0\\i{\sqrt {3}}&0&-2i&0\\0&2i&0&-i{\sqrt {3}}\\0&0&i{\sqrt {3}}&0\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {+3}{2}}\right\rangle _{y}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}{i}\\{-{\sqrt {3}}}\\{-i{\sqrt {3}}}\\1\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {+1}{2}}\right\rangle _{y}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}{-i{\sqrt {3}}}\\1\\{-i}\\{\sqrt {3}}\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {-1}{2}}\right\rangle _{y}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}{i{\sqrt {3}}}\\1\\{i}\\{\sqrt {3}}\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {-3}{2}}\right\rangle _{y}=\!\!\!&{\frac {1}{2{\sqrt {2}}}}{\begin{pmatrix}{-i}\\{-{\sqrt {3}}}\\{i{\sqrt {3}}}\\1\end{pmatrix}}\\S_{z}={\frac {\hbar }{2}}{\begin{pmatrix}3&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-3\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {+3}{2}}\right\rangle _{z}=\!\!\!&{\begin{pmatrix}1\\0\\0\\0\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {+1}{2}}\right\rangle _{z}=\!\!\!&{\begin{pmatrix}0\\1\\0\\0\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {-1}{2}}\right\rangle _{z}=\!\!\!&{\begin{pmatrix}0\\0\\1\\0\end{pmatrix}},\!\!\!&\left|{\frac {3}{2}},{\frac {-3}{2}}\right\rangle _{z}=\!\!\!&{\begin{pmatrix}0\\0\\0\\1\end{pmatrix}}\\\end{array}}}"></span></li><li>For spin <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> they are <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}{\boldsymbol {S}}_{x}&={\frac {\hbar }{2}}{\begin{pmatrix}0&{\sqrt {5}}&0&0&0&0\\{\sqrt {5}}&0&2{\sqrt {2}}&0&0&0\\0&2{\sqrt {2}}&0&3&0&0\\0&0&3&0&2{\sqrt {2}}&0\\0&0&0&2{\sqrt {2}}&0&{\sqrt {5}}\\0&0&0&0&{\sqrt {5}}&0\end{pmatrix}},\\{\boldsymbol {S}}_{y}&={\frac {\hbar }{2}}{\begin{pmatrix}0&-i{\sqrt {5}}&0&0&0&0\\i{\sqrt {5}}&0&-2i{\sqrt {2}}&0&0&0\\0&2i{\sqrt {2}}&0&-3i&0&0\\0&0&3i&0&-2i{\sqrt {2}}&0\\0&0&0&2i{\sqrt {2}}&0&-i{\sqrt {5}}\\0&0&0&0&i{\sqrt {5}}&0\end{pmatrix}},\\{\boldsymbol {S}}_{z}&={\frac {\hbar }{2}}{\begin{pmatrix}5&0&0&0&0&0\\0&3&0&0&0&0\\0&0&1&0&0&0\\0&0&0&-1&0&0\\0&0&0&0&-3&0\\0&0&0&0&0&-5\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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> <mi>i</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> <mi>i</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>5</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>5</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\boldsymbol {S}}_{x}&={\frac {\hbar }{2}}{\begin{pmatrix}0&{\sqrt {5}}&0&0&0&0\\{\sqrt {5}}&0&2{\sqrt {2}}&0&0&0\\0&2{\sqrt {2}}&0&3&0&0\\0&0&3&0&2{\sqrt {2}}&0\\0&0&0&2{\sqrt {2}}&0&{\sqrt {5}}\\0&0&0&0&{\sqrt {5}}&0\end{pmatrix}},\\{\boldsymbol {S}}_{y}&={\frac {\hbar }{2}}{\begin{pmatrix}0&-i{\sqrt {5}}&0&0&0&0\\i{\sqrt {5}}&0&-2i{\sqrt {2}}&0&0&0\\0&2i{\sqrt {2}}&0&-3i&0&0\\0&0&3i&0&-2i{\sqrt {2}}&0\\0&0&0&2i{\sqrt {2}}&0&-i{\sqrt {5}}\\0&0&0&0&i{\sqrt {5}}&0\end{pmatrix}},\\{\boldsymbol {S}}_{z}&={\frac {\hbar }{2}}{\begin{pmatrix}5&0&0&0&0&0\\0&3&0&0&0&0\\0&0&1&0&0&0\\0&0&0&-1&0&0\\0&0&0&0&-3&0\\0&0&0&0&0&-5\end{pmatrix}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c7a98a1a3545d27e182c41949e47c7d5042e335" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -29.332ex; margin-bottom: -0.172ex; width:59.978ex; height:60.176ex;" alt="{\displaystyle {\begin{aligned}{\boldsymbol {S}}_{x}&={\frac {\hbar }{2}}{\begin{pmatrix}0&{\sqrt {5}}&0&0&0&0\\{\sqrt {5}}&0&2{\sqrt {2}}&0&0&0\\0&2{\sqrt {2}}&0&3&0&0\\0&0&3&0&2{\sqrt {2}}&0\\0&0&0&2{\sqrt {2}}&0&{\sqrt {5}}\\0&0&0&0&{\sqrt {5}}&0\end{pmatrix}},\\{\boldsymbol {S}}_{y}&={\frac {\hbar }{2}}{\begin{pmatrix}0&-i{\sqrt {5}}&0&0&0&0\\i{\sqrt {5}}&0&-2i{\sqrt {2}}&0&0&0\\0&2i{\sqrt {2}}&0&-3i&0&0\\0&0&3i&0&-2i{\sqrt {2}}&0\\0&0&0&2i{\sqrt {2}}&0&-i{\sqrt {5}}\\0&0&0&0&i{\sqrt {5}}&0\end{pmatrix}},\\{\boldsymbol {S}}_{z}&={\frac {\hbar }{2}}{\begin{pmatrix}5&0&0&0&0&0\\0&3&0&0&0&0\\0&0&1&0&0&0\\0&0&0&-1&0&0\\0&0&0&0&-3&0\\0&0&0&0&0&-5\end{pmatrix}}.\end{aligned}}}"></span></li><li>The generalization of these matrices for arbitrary spin <span class="texhtml mvar" style="font-style:italic;">s</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 {\begin{aligned}\left(S_{x}\right)_{ab}&={\frac {\hbar }{2}}\left(\delta _{a,b+1}+\delta _{a+1,b}\right){\sqrt {(s+1)(a+b-1)-ab}},\\\left(S_{y}\right)_{ab}&={\frac {i\hbar }{2}}\left(\delta _{a,b+1}-\delta _{a+1,b}\right){\sqrt {(s+1)(a+b-1)-ab}},\\\left(S_{z}\right)_{ab}&=\hbar (s+1-a)\delta _{a,b}=\hbar (s+1-b)\delta _{a,b},\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> <msub> <mrow> <mo>(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>b</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>a</mi> <mi>b</mi> </msqrt> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow> <mo>(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>b</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>a</mi> <mi>b</mi> </msqrt> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow> <mo>(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left(S_{x}\right)_{ab}&={\frac {\hbar }{2}}\left(\delta _{a,b+1}+\delta _{a+1,b}\right){\sqrt {(s+1)(a+b-1)-ab}},\\\left(S_{y}\right)_{ab}&={\frac {i\hbar }{2}}\left(\delta _{a,b+1}-\delta _{a+1,b}\right){\sqrt {(s+1)(a+b-1)-ab}},\\\left(S_{z}\right)_{ab}&=\hbar (s+1-a)\delta _{a,b}=\hbar (s+1-b)\delta _{a,b},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d08e0836e2e4d5393ad92e94d9d03816076217a5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:55.049ex; height:14.009ex;" alt="{\displaystyle {\begin{aligned}\left(S_{x}\right)_{ab}&={\frac {\hbar }{2}}\left(\delta _{a,b+1}+\delta _{a+1,b}\right){\sqrt {(s+1)(a+b-1)-ab}},\\\left(S_{y}\right)_{ab}&={\frac {i\hbar }{2}}\left(\delta _{a,b+1}-\delta _{a+1,b}\right){\sqrt {(s+1)(a+b-1)-ab}},\\\left(S_{z}\right)_{ab}&=\hbar (s+1-a)\delta _{a,b}=\hbar (s+1-b)\delta _{a,b},\end{aligned}}}"></span> where 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 a,b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}"></span> are integer numbers such 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 1\leq a\leq 2s+1,\quad 1\leq b\leq 2s+1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>a</mi> <mo>≤</mo> <mn>2</mn> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mspace width="1em" /> <mn>1</mn> <mo>≤</mo> <mi>b</mi> <mo>≤</mo> <mn>2</mn> <mi>s</mi> <mo>+</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq a\leq 2s+1,\quad 1\leq b\leq 2s+1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa0edc78c5d5b526f0ca537abc9dff041a2878dc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.461ex; height:2.509ex;" alt="{\displaystyle 1\leq a\leq 2s+1,\quad 1\leq b\leq 2s+1.}"></span></li></ol></div> <p>Also useful in the <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> of multiparticle systems, the general <a href="/wiki/Pauli_group" title="Pauli group">Pauli group</a> <span class="texhtml mvar" style="font-style:italic;">G<sub>n</sub></span> is defined to consist of all <span class="texhtml mvar" style="font-style:italic;">n</span>-fold <a href="/wiki/Tensor" title="Tensor">tensor</a> products of Pauli matrices. </p><p>The analog formula of <a href="/wiki/Pauli_matrices#Exponential_of_a_Pauli_vector" title="Pauli matrices">Euler's formula in terms of the Pauli matrices</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 {\hat {R}}(\theta ,{\hat {\mathbf {n} }})=e^{i{\frac {\theta }{2}}{\hat {\mathbf {n} }}\cdot {\boldsymbol {\sigma }}}=I\cos {\frac {\theta }{2}}+i\left({\hat {\mathbf {n} }}\cdot {\boldsymbol {\sigma }}\right)\sin {\frac {\theta }{2}}}"> <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> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mrow> </msup> <mo>=</mo> <mi>I</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>i</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {R}}(\theta ,{\hat {\mathbf {n} }})=e^{i{\frac {\theta }{2}}{\hat {\mathbf {n} }}\cdot {\boldsymbol {\sigma }}}=I\cos {\frac {\theta }{2}}+i\left({\hat {\mathbf {n} }}\cdot {\boldsymbol {\sigma }}\right)\sin {\frac {\theta }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48aca6de6ef6484576babc9fa50062a7ebd075b8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.684ex; height:5.343ex;" alt="{\displaystyle {\hat {R}}(\theta ,{\hat {\mathbf {n} }})=e^{i{\frac {\theta }{2}}{\hat {\mathbf {n} }}\cdot {\boldsymbol {\sigma }}}=I\cos {\frac {\theta }{2}}+i\left({\hat {\mathbf {n} }}\cdot {\boldsymbol {\sigma }}\right)\sin {\frac {\theta }{2}}}"></span> for higher spins is tractable, but less simple.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Parity">Parity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=25" title="Edit section: Parity"><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/Parity_(physics)" title="Parity (physics)">Parity (physics)</a></div> <p>In tables of the <a href="/wiki/Spin_quantum_number" title="Spin quantum number">spin quantum number</a> <span class="texhtml mvar" style="font-style:italic;">s</span> for nuclei or particles, the spin is often followed by a "+" or "−".<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. (February 2024)">citation needed</span></a></i>]</sup> This refers to the <a href="/wiki/Parity_(physics)" title="Parity (physics)">parity</a> with "+" for even parity (wave function unchanged by spatial inversion) and "−" for odd parity (wave function negated by spatial inversion). For example, see the <a href="/wiki/Isotopes_of_bismuth" title="Isotopes of bismuth">isotopes of bismuth</a>, in which the list of isotopes includes the column <a href="/wiki/Spin_quantum_number#Nuclear_spin" title="Spin quantum number">nuclear spin</a> and parity. For Bi-209, the longest-lived isotope, the entry 9/2– means that the nuclear spin is 9/2 and the parity is odd. </p> <div class="mw-heading mw-heading2"><h2 id="Measuring_spin">Measuring spin</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=26" title="Edit section: Measuring spin"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The nuclear spin of atoms can be determined by sophisticated improvements to the original <a href="/wiki/Stern-Gerlach_experiment" class="mw-redirect" title="Stern-Gerlach experiment">Stern-Gerlach experiment</a>.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> A single-energy (monochromatic) <a href="/wiki/Molecular_beam" title="Molecular beam">molecular beam</a> of atoms in an inhomogeneous magnetic field will split into beams representing each possible spin quantum state. For an atom with electronic spin <span class="texhtml mvar" style="font-style:italic;">S</span> and nuclear spin <span class="texhtml mvar" style="font-style:italic;">I</span>, there are <span class="texhtml">(2<i>S</i> + 1)(2<i>I</i> + 1)</span> spin states. For example, neutral <a href="/wiki/Sodium" title="Sodium">Na</a> atoms, which have <span class="texhtml"><i>S</i> = 1/2</span>, were passed through a series of inhomogeneous magnetic fields that selected one of the two electronic spin states and separated the nuclear spin states, from which four beams were observed. Thus, the nuclear spin for <sup>23</sup>Na atoms was found to be <span class="texhtml"><i>I</i> = 3/2</span>.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p><p>The spin of <a href="/wiki/Pions" class="mw-redirect" title="Pions">pions</a>, a type of elementary particle, was determined by the principle of <a href="/wiki/Detailed_balance" title="Detailed balance">detailed balance</a> applied to those collisions of protons that produced charged pions and <a href="/wiki/Deuterium" title="Deuterium">deuterium</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 p+p\rightarrow \pi _{-}+d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>+</mo> <mi>p</mi> <mo stretchy="false">→</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>+</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p+p\rightarrow \pi _{-}+d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a24edea736ff642aa7ad22585d72e6090616fd53" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:15.775ex; height:2.509ex;" alt="{\displaystyle p+p\rightarrow \pi _{-}+d}"></span> The known spin values for protons and deuterium allows analysis of the collision cross-section to show that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bb6cd2a0370e3a910eb2731c1f57236ef6221c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.836ex; height:2.009ex;" alt="{\displaystyle \pi _{-}}"></span> has spin <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{\pi }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{\pi }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90c0d226a852519f3ff0b0683ae496cb8d5e6fb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.526ex; height:2.509ex;" alt="{\displaystyle s_{\pi }=0}"></span>. A different approach is needed for neutral pions. In that case the decay produced two <a href="/wiki/Gamma_ray" title="Gamma ray">gamma ray</a> photons with spin one: <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 \pi _{0}\rightarrow 2\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">→</mo> <mn>2</mn> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{0}\rightarrow 2\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9917dda28e97c5bcc289f7ad9a449cc5bb3a4355" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.418ex; height:2.676ex;" alt="{\displaystyle \pi _{0}\rightarrow 2\gamma }"></span> This result supplemented with additional analysis leads to the conclusion that the neutral pion also has spin zero.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 66">: 66 </span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=27" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Spin has important theoretical implications and practical applications. Well-established <i>direct</i> applications of spin include: </p> <ul><li><a href="/wiki/Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">Nuclear magnetic resonance</a> (NMR) spectroscopy in chemistry;</li> <li><a href="/wiki/Electron_spin_resonance" class="mw-redirect" title="Electron spin resonance">Electron spin resonance</a> (ESR or EPR) spectroscopy in chemistry and physics;</li> <li><a href="/wiki/Magnetic_resonance_imaging" title="Magnetic resonance imaging">Magnetic resonance imaging</a> (MRI) in medicine, a type of applied NMR, which relies on proton spin density;</li> <li><a href="/wiki/Giant_magnetoresistive_effect" class="mw-redirect" title="Giant magnetoresistive effect">Giant magnetoresistive</a> (GMR) drive-head technology in modern <a href="/wiki/Hard_disk" class="mw-redirect" title="Hard disk">hard disks</a>.</li></ul> <p>Electron spin plays an important role in <a href="/wiki/Magnetism" title="Magnetism">magnetism</a>, with applications for instance in computer memories. The manipulation of <i>nuclear spin</i> by radio-frequency waves (<a href="/wiki/Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">nuclear magnetic resonance</a>) is important in chemical spectroscopy and medical imaging. </p><p><a href="/wiki/Spin%E2%80%93orbit_coupling" class="mw-redirect" title="Spin–orbit coupling">Spin–orbit coupling</a> leads to the <a href="/wiki/Fine_structure" title="Fine structure">fine structure</a> of atomic spectra, which is used in <a href="/wiki/Atomic_clock" title="Atomic clock">atomic clocks</a> and in the modern definition of the <a href="/wiki/Second" title="Second">second</a>. Precise measurements of the <span class="texhtml mvar" style="font-style:italic;">g</span>-factor of the electron have played an important role in the development and verification of <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a>. <i>Photon spin</i> is associated with the <a href="/wiki/Polarization_(waves)" title="Polarization (waves)">polarization</a> of light (<a href="/wiki/Photon_polarization" title="Photon polarization">photon polarization</a>). </p><p>An emerging application of spin is as a binary information carrier in <a href="/wiki/Spin_transistor" title="Spin transistor">spin transistors</a>. The original concept, proposed in 1990, is known as Datta–Das <a href="/wiki/Spin_transistor" title="Spin transistor">spin transistor</a>.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> Electronics based on spin transistors are referred to as <a href="/wiki/Spintronics" title="Spintronics">spintronics</a>. The manipulation of spin in <a href="/wiki/ZnO-based_diluted_magnetic_semiconductors" class="mw-redirect" title="ZnO-based diluted magnetic semiconductors">dilute magnetic semiconductor materials</a>, such as metal-doped <a href="/wiki/Zinc_oxide" title="Zinc oxide">ZnO</a> or <a href="/wiki/Titanium_dioxide" title="Titanium dioxide">TiO<sub>2</sub></a> imparts a further degree of freedom and has the potential to facilitate the fabrication of more efficient electronics.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p><p>There are many <i>indirect</i> applications and manifestations of spin and the associated <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a>, starting with the <a href="/wiki/Periodic_table" title="Periodic table">periodic table</a> of chemistry. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=28" title="Edit section: History"><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/History_of_quantum_mechanics" title="History of quantum mechanics">History of quantum mechanics</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Wolfgang_Pauli_young.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Wolfgang_Pauli_young.jpg/220px-Wolfgang_Pauli_young.jpg" decoding="async" width="220" height="264" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/4/43/Wolfgang_Pauli_young.jpg 1.5x" data-file-width="250" data-file-height="300" /></a><figcaption><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Wolfgang Pauli</a> lecturing</figcaption></figure> <p>Spin was first discovered in the context of the <a href="/wiki/Emission_spectrum" title="Emission spectrum">emission spectrum</a> of <a href="/wiki/Alkali_metal" title="Alkali metal">alkali metals</a>. Starting around 1910, many experiments on different atoms produced a collection of relationships involving <a href="/wiki/Quantum_numbers" class="mw-redirect" title="Quantum numbers">quantum numbers</a> for atomic energy levels partially summarized in <a href="/wiki/Bohr_atom" class="mw-redirect" title="Bohr atom">Bohr's model for the atom</a><sup id="cite_ref-Whittaker_33-0" class="reference"><a href="#cite_note-Whittaker-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 106">: 106 </span></sup> Transitions between levels obeyed <a href="/wiki/Selection_rules" class="mw-redirect" title="Selection rules">selection rules</a> and the rules were known to be correlated with even or odd <a href="/wiki/Atomic_number" title="Atomic number">atomic number</a>. Additional information was known from changes to atomic spectra observed in strong magnetic fields, known as the <a href="/wiki/Zeeman_effect" title="Zeeman effect">Zeeman effect</a>. In 1924, <a href="/wiki/Wolfgang_Ernst_Pauli" class="mw-redirect" title="Wolfgang Ernst Pauli">Wolfgang Pauli</a> used this large collection of empirical observations to propose a new degree of freedom,<sup id="cite_ref-Frohlich_7-2" class="reference"><a href="#cite_note-Frohlich-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> introducing what he called a "two-valuedness not describable classically"<sup id="cite_ref-PauliNobel_34-0" class="reference"><a href="#cite_note-PauliNobel-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> associated with the electron in the outermost <a href="/wiki/Electron_shell" title="Electron shell">shell</a>. </p><p>The physical interpretation of Pauli's "degree of freedom" was initially unknown. <a href="/wiki/Ralph_Kronig" title="Ralph Kronig">Ralph Kronig</a>, one of <a href="/wiki/Alfred_Land%C3%A9" title="Alfred Landé">Alfred Landé</a>'s assistants, suggested in early 1925 that it was produced by the self-rotation of the electron. When Pauli heard about the idea, he criticized it severely, noting that the electron's hypothetical surface would have to be moving faster than the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> in order for it to rotate quickly enough to produce the necessary angular momentum. This would violate the <a href="/wiki/Theory_of_relativity" title="Theory of relativity">theory of relativity</a>. Largely due to Pauli's criticism, Kronig decided not to publish his idea.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p><p>In the autumn of 1925, the same thought came to Dutch physicists <a href="/wiki/George_Uhlenbeck" title="George Uhlenbeck">George Uhlenbeck</a> and <a href="/wiki/Samuel_Goudsmit" title="Samuel Goudsmit">Samuel Goudsmit</a> at <a href="/wiki/Leiden_University" title="Leiden University">Leiden University</a>. Under the advice of <a href="/wiki/Paul_Ehrenfest" title="Paul Ehrenfest">Paul Ehrenfest</a>, they published their results.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> The young physicists immediately regretted the publication: <a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Hendrik Lorentz</a> and <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a> both pointed out problems with the concept of a spinning electron.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p><p>Pauli was especially unconvinced and continued to pursue his two-valued degree of freedom. This allowed him to formulate the <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a>, stating that no two electrons can have the same <a href="/wiki/Quantum_state" title="Quantum state">quantum state</a> in the same quantum system. </p><p>Fortunately, by February 1926, <a href="/wiki/Llewellyn_Thomas" title="Llewellyn Thomas">Llewellyn Thomas</a> managed to resolve a factor-of-two discrepancy between experimental results for the <a href="/wiki/Fine_structure" title="Fine structure">fine structure</a> in the hydrogen spectrum and calculations based on Uhlenbeck and Goudsmit's (and Kronig's unpublished) model.<sup id="cite_ref-griffiths_2-4" class="reference"><a href="#cite_note-griffiths-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 385">: 385 </span></sup> This discrepancy was due to a relativistic effect, the difference between the electron's rotating rest frame and the nuclear rest frame; the effect is now known as <a href="/wiki/Thomas_precession" title="Thomas precession">Thomas precession</a>.<sup id="cite_ref-Frohlich_7-3" class="reference"><a href="#cite_note-Frohlich-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Thomas' result convinced Pauli that electron spin was the correct interpretation of his two-valued degree of freedom, while he continued to insist that the classical rotating charge model is invalid.<sup id="cite_ref-PauliNobel_34-1" class="reference"><a href="#cite_note-PauliNobel-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Giulini_6-2" class="reference"><a href="#cite_note-Giulini-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>In 1927, Pauli formalized the theory of spin using the theory of quantum mechanics invented by <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a> and <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a>. He pioneered the use of <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a> as a <a href="/wiki/Group_representation" title="Group representation">representation</a> of the spin operators and introduced a two-component <a href="/wiki/Spinor" title="Spinor">spinor</a> wave-function. </p><p>Pauli's theory of spin was non-relativistic. In 1928, <a href="/wiki/Paul_Dirac" title="Paul Dirac">Paul Dirac</a> published his relativistic electron equation, using a four-component spinor (known as a "<a href="/wiki/Dirac_spinor" title="Dirac spinor">Dirac spinor</a>") for the electron wave-function. In 1940, Pauli proved the <i><a href="/wiki/Spin%E2%80%93statistics_theorem" title="Spin–statistics theorem">spin–statistics theorem</a></i>, which states that <a href="/wiki/Fermion" title="Fermion">fermions</a> have half-integer spin, and <a href="/wiki/Boson" title="Boson">bosons</a> have integer spin.<sup id="cite_ref-Frohlich_7-4" class="reference"><a href="#cite_note-Frohlich-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>In retrospect, the first direct experimental evidence of the electron spin was the <a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach experiment</a> of 1922. However, the correct explanation of this experiment was only given in 1927.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> The original interpretation assumed the two spots observed in the experiment were due to quantized <a href="/wiki/Orbital_angular_momentum" class="mw-redirect" title="Orbital angular momentum">orbital angular momentum</a>. However, in 1927 Ronald Fraser showed that Sodium atoms are isotropic with no orbital angular momentum and suggested that the observed magnetic properties were due to electron spin.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> In same year, Phipps and Taylor applied the Stern-Gerlach technique to hydrogen atoms; the ground state of hydrogen has zero angular momentum but the measurements again showed two peaks.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> Once the quantum theory became established, it became clear that the original interpretation could not have been correct: the possible values of orbital angular momentum along one axis is always an odd number, unlike the observations. Hydrogen atoms have a single electron with two spin states giving the two spots observed; silver atoms have closed shells which do not contribute to the magnetic moment and only the unmatched outer electron's spin responds to the field. </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=Spin_(physics)&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:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 21em;"> <ul><li><a href="/wiki/Chirality_(physics)" title="Chirality (physics)">Chirality (physics)</a></li> <li><a href="/wiki/Dynamic_nuclear_polarization" title="Dynamic nuclear polarization">Dynamic nuclear polarization</a></li> <li><a href="/wiki/Helicity_(particle_physics)" title="Helicity (particle physics)">Helicity (particle physics)</a></li> <li><a href="/wiki/Holstein%E2%80%93Primakoff_transformation" title="Holstein–Primakoff transformation">Holstein–Primakoff transformation</a></li> <li><a href="/wiki/Kramers%27_theorem" title="Kramers' theorem">Kramers' theorem</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli equation</a></li> <li><a href="/wiki/Pauli%E2%80%93Lubanski_pseudovector" title="Pauli–Lubanski pseudovector">Pauli–Lubanski pseudovector</a></li> <li><a href="/wiki/Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger equation</a></li> <li><a href="/wiki/Representation_theory_of_SU(2)" title="Representation theory of SU(2)">Representation theory of SU(2)</a></li> <li><a href="/wiki/Spin_angular_momentum_of_light" title="Spin angular momentum of light">Spin angular momentum of light</a></li> <li><a href="/wiki/Spin_engineering" title="Spin engineering">Spin engineering</a></li> <li><a href="/wiki/Spin-flip" title="Spin-flip">Spin-flip</a></li> <li><a href="/wiki/Spin_isomers_of_hydrogen" title="Spin isomers of hydrogen">Spin isomers of hydrogen</a></li> <li><a href="/wiki/Spin%E2%80%93orbit_interaction" title="Spin–orbit interaction">Spin–orbit interaction</a></li> <li><a href="/wiki/Spin_tensor" title="Spin tensor">Spin tensor</a></li> <li><a href="/wiki/Spintronics" title="Spintronics">Spintronics</a></li> <li><a href="/wiki/Spin_wave" title="Spin wave">Spin wave</a></li> <li><a href="/wiki/Yrast" title="Yrast">Yrast</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=30" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-merzbacher372-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-merzbacher372_1-0">^</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 id="CITEREFMerzbacher1998" class="citation book cs1"><a href="/wiki/Eugen_Merzbacher" title="Eugen 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John Wiley & Sons. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/quantummechanics00merz_136/page/n385">372</a>–373. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-88702-7" title="Special:BookSources/978-0-471-88702-7"><bdi>978-0-471-88702-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics&rft.pages=372-373&rft.edition=3rd&rft.pub=John+Wiley+%26+Sons&rft.date=1998&rft.isbn=978-0-471-88702-7&rft.aulast=Merzbacher&rft.aufirst=Eugen&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantummechanics00merz_136&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-griffiths-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-griffiths_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-griffiths_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-griffiths_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-griffiths_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-griffiths_2-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffiths2005" class="citation book cs1"><a href="/wiki/David_J._Griffiths" title="David J. 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(1986-06-01). <a rel="nofollow" class="external text" href="https://physics.mcmaster.ca/phys3mm3/notes/whatisspin.pdf">"What is spin?"</a> <span class="cs1-format">(PDF)</span>. <i>American Journal of Physics</i>. <b>54</b> (6): 500–505. <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/1986AmJPh..54..500O">1986AmJPh..54..500O</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.14580">10.1119/1.14580</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9505">0002-9505</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=What+is+spin%3F&rft.volume=54&rft.issue=6&rft.pages=500-505&rft.date=1986-06-01&rft.issn=0002-9505&rft_id=info%3Adoi%2F10.1119%2F1.14580&rft_id=info%3Abibcode%2F1986AmJPh..54..500O&rft.aulast=Ohanian&rft.aufirst=Hans+C.&rft_id=https%3A%2F%2Fphysics.mcmaster.ca%2Fphys3mm3%2Fnotes%2Fwhatisspin.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBliokhPunzmannXiaNori2022" class="citation journal cs1">Bliokh, Konstantin Y.; Punzmann, Horst; Xia, Hua; Nori, Franco; Shats, Michael (2022-01-21). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8782445">"Field theory spin and momentum in water waves"</a>. <i>Science Advances</i>. <b>8</b> (3): eabm1295. <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/2022SciA....8.1295B">2022SciA....8.1295B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fsciadv.abm1295">10.1126/sciadv.abm1295</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2375-2548">2375-2548</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8782445">8782445</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/35061526">35061526</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science+Advances&rft.atitle=Field+theory+spin+and+momentum+in+water+waves&rft.volume=8&rft.issue=3&rft.pages=eabm1295&rft.date=2022-01-21&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8782445%23id-name%3DPMC&rft_id=info%3Abibcode%2F2022SciA....8.1295B&rft_id=info%3Apmid%2F35061526&rft_id=info%3Adoi%2F10.1126%2Fsciadv.abm1295&rft.issn=2375-2548&rft.aulast=Bliokh&rft.aufirst=Konstantin+Y.&rft.au=Punzmann%2C+Horst&rft.au=Xia%2C+Hua&rft.au=Nori%2C+Franco&rft.au=Shats%2C+Michael&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8782445&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-Whittaker_1989_p.87-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Whittaker_1989_p.87_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhittaker1989" class="citation book cs1">Whittaker, Edmund, Sir (1989). <i>A History of the Theories of Aether and Electricity</i>. Vol. 2. Courier Dover Publications. p. 87, 131. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-26126-3" title="Special:BookSources/0-486-26126-3"><bdi>0-486-26126-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+the+Theories+of+Aether+and+Electricity&rft.pages=87%2C+131&rft.pub=Courier+Dover+Publications&rft.date=1989&rft.isbn=0-486-26126-3&rft.aulast=Whittaker&rft.aufirst=Edmund%2C+Sir&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://home.cern/topics/higgs-boson">Information about Higgs Boson</a> in <a href="/wiki/CERN" title="CERN">CERN</a>'s official website.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPauli1940" class="citation journal cs1"><a href="/wiki/Wolfgang_Pauli" title="Wolfgang Pauli">Pauli, Wolfgang</a> (1940). <a rel="nofollow" class="external text" href="http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Pauli_spin_statistics_1940.pdf">"The Connection Between Spin and Statistics"</a> <span class="cs1-format">(PDF)</span>. <i>Phys. Rev</i>. <b>58</b> (8): 716–722. <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/1940PhRv...58..716P">1940PhRv...58..716P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.58.716">10.1103/PhysRev.58.716</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Rev.&rft.atitle=The+Connection+Between+Spin+and+Statistics&rft.volume=58&rft.issue=8&rft.pages=716-722&rft.date=1940&rft_id=info%3Adoi%2F10.1103%2FPhysRev.58.716&rft_id=info%3Abibcode%2F1940PhRv...58..716P&rft.aulast=Pauli&rft.aufirst=Wolfgang&rft_id=http%3A%2F%2Fhermes.ffn.ub.es%2Fluisnavarro%2Fnuevo_maletin%2FPauli_spin_statistics_1940.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Physics of Atoms and Molecules, B. H. Bransden, C. J. Joachain, Longman, 1983, <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/0-582-44401-2" title="Special:BookSources/0-582-44401-2">0-582-44401-2</a>.</span> </li> <li id="cite_note-physconst-ge-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-physconst-ge_16-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://physics.nist.gov/cgi-bin/cuu/Value?gem">"2022 CODATA Value: electron g factor"</a>. <i>The NIST Reference on Constants, Units, and Uncertainty</i>. <a href="/wiki/National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">NIST</a>. May 2024<span class="reference-accessdate">. Retrieved <span class="nowrap">2024-05-18</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+NIST+Reference+on+Constants%2C+Units%2C+and+Uncertainty&rft.atitle=2022+CODATA+Value%3A+electron+g+factor&rft.date=2024-05&rft_id=https%3A%2F%2Fphysics.nist.gov%2Fcgi-bin%2Fcuu%2FValue%3Fgem&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynman,_R._P.1985" class="citation book cs1"><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman, R. P.</a> (1985). 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This correction was worked out for the first time in 1948 by Schwinger as <span class="texhtml"><i>j</i> × <i>j</i></span> divided by <span class="texhtml">2<i>π</i></span>  [<i><a href="/wiki/Sic" title="Sic">sic</a></i>] [where <span class="texhtml mvar" style="font-style:italic;">j</span> is the square root of the <a href="/wiki/Fine-structure_constant" title="Fine-structure constant">fine-structure constant</a>], and was due to an alternative way the electron can go from place to place: Instead of going directly from one point to another, the electron goes along for a while and suddenly emits a photon; then (horrors!) it absorbs its own photon.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Electrons+and+their+interactions&rft.btitle=QED%3A+The+Strange+Theory+of+Light+and+Matter&rft.place=Princeton%2C+New+Jersey&rft.pages=115&rft.pub=Princeton+University+Press&rft.date=1985&rft.isbn=978-0-691-08388-9&rft.au=Feynman%2C+R.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarciano,_W._J.Sanda,_A._I.1977" class="citation journal cs1"><a href="/wiki/William_Marciano" title="William Marciano">Marciano, W. 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New York: Dover Publ. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-26126-3" title="Special:BookSources/978-0-486-26126-3"><bdi>978-0-486-26126-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+history+of+the+theories+of+aether+%26+electricity.+2%3A+The+modern+theories%2C+1900+-+1926&rft.place=New+York&rft.edition=Repr&rft.pub=Dover+Publ&rft.date=1989&rft.isbn=978-0-486-26126-3&rft.aulast=Whittaker&rft.aufirst=Edmund+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-PauliNobel-34"><span class="mw-cite-backlink">^ <a href="#cite_ref-PauliNobel_34-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-PauliNobel_34-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWolfgang_Pauli1946" class="citation web cs1">Wolfgang Pauli (December 13, 1946). <a rel="nofollow" class="external text" href="https://www.nobelprize.org/prizes/physics/1945/pauli/lecture/">"Exclusion Principle and Quantum Mechanics"</a>. <i>Nobel Lecture</i>. <a href="/wiki/Nobel_Prize" title="Nobel Prize">Nobel Prize</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Nobel+Lecture&rft.atitle=Exclusion+Principle+and+Quantum+Mechanics&rft.date=1946-12-13&rft.au=Wolfgang+Pauli&rft_id=https%3A%2F%2Fwww.nobelprize.org%2Fprizes%2Fphysics%2F1945%2Fpauli%2Flecture%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPais1991" class="citation book cs1"><a href="/wiki/Abraham_Pais" title="Abraham Pais">Pais, Abraham</a> (1991). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/nielsbohrstimesi00pais_0"><i>Niels Bohr's Times</i></a></span>. Oxford: Clarendon Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/nielsbohrstimesi00pais_0/page/244">244</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-852049-8" title="Special:BookSources/978-0-19-852049-8"><bdi>978-0-19-852049-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Niels+Bohr%27s+Times&rft.place=Oxford&rft.pages=244&rft.pub=Clarendon+Press&rft.date=1991&rft.isbn=978-0-19-852049-8&rft.aulast=Pais&rft.aufirst=Abraham&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnielsbohrstimesi00pais_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFUhlenbeck,_G.Goudsmit1925" class="citation journal cs1 cs1-prop-foreign-lang-source">Uhlenbeck, G., G.; Goudsmit, S. (November 1925). "Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elektrons". <i>Die Naturwissenschaften</i> (in German). <b>13</b> (47): 953–954. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf01558878">10.1007/bf01558878</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0028-1042">0028-1042</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:32211960">32211960</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Die+Naturwissenschaften&rft.atitle=Ersetzung+der+Hypothese+vom+unmechanischen+Zwang+durch+eine+Forderung+bez%C3%BCglich+des+inneren+Verhaltens+jedes+einzelnen+Elektrons&rft.volume=13&rft.issue=47&rft.pages=953-954&rft.date=1925-11&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A32211960%23id-name%3DS2CID&rft.issn=0028-1042&rft_id=info%3Adoi%2F10.1007%2Fbf01558878&rft.aulast=Uhlenbeck%2C+G.&rft.aufirst=G.&rft.au=Goudsmit%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPais1989" class="citation journal cs1">Pais, Abraham (1989-12-01). <a rel="nofollow" class="external text" href="https://pubs.aip.org/physicstoday/article/42/12/34/405387/George-Uhlenbeck-and-the-Discovery-of-Electron">"George Uhlenbeck and the Discovery of Electron Spin"</a>. <i>Physics Today</i>. <b>42</b> (12): 34–40. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.881186">10.1063/1.881186</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9228">0031-9228</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=George+Uhlenbeck+and+the+Discovery+of+Electron+Spin&rft.volume=42&rft.issue=12&rft.pages=34-40&rft.date=1989-12-01&rft_id=info%3Adoi%2F10.1063%2F1.881186&rft.issn=0031-9228&rft.aulast=Pais&rft.aufirst=Abraham&rft_id=https%3A%2F%2Fpubs.aip.org%2Fphysicstoday%2Farticle%2F42%2F12%2F34%2F405387%2FGeorge-Uhlenbeck-and-the-Discovery-of-Electron&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFB._FriedrichD._Herschbach2003" class="citation journal cs1">B. Friedrich; D. Herschbach (2003). <a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1650229">"Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics"</a>. <i><a href="/wiki/Physics_Today" title="Physics Today">Physics Today</a></i>. <b>56</b> (12): 53. <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/2003PhT....56l..53F">2003PhT....56l..53F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1650229">10.1063/1.1650229</a></span>. <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:17572089">17572089</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=Stern+and+Gerlach%3A+How+a+Bad+Cigar+Helped+Reorient+Atomic+Physics&rft.volume=56&rft.issue=12&rft.pages=53&rft.date=2003&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17572089%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1063%2F1.1650229&rft_id=info%3Abibcode%2F2003PhT....56l..53F&rft.au=B.+Friedrich&rft.au=D.+Herschbach&rft_id=https%3A%2F%2Fdoi.org%2F10.1063%252F1.1650229&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1"><a rel="nofollow" class="external text" href="https://royalsocietypublishing.org/doi/10.1098/rspa.1927.0036">"The effective cross section of the oriented hydrogen atom"</a>. <i>Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character</i>. <b>114</b> (767): 212–221. March 1927. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1927.0036">10.1098/rspa.1927.0036</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0950-1207">0950-1207</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+of+London.+Series+A%2C+Containing+Papers+of+a+Mathematical+and+Physical+Character&rft.atitle=The+effective+cross+section+of+the+oriented+hydrogen+atom&rft.volume=114&rft.issue=767&rft.pages=212-221&rft.date=1927-03&rft_id=info%3Adoi%2F10.1098%2Frspa.1927.0036&rft.issn=0950-1207&rft_id=https%3A%2F%2Froyalsocietypublishing.org%2Fdoi%2F10.1098%2Frspa.1927.0036&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFResnick,_R.Eisberg,_R.1985" class="citation book cs1">Resnick, R.; Eisberg, R. (1985). <a rel="nofollow" class="external text" href="https://archive.org/details/quantumphysicsof00eisb/page/274"><i>Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles</i></a> (2nd ed.). John Wiley & Sons. p. <a rel="nofollow" class="external text" href="https://archive.org/details/quantumphysicsof00eisb/page/274">274</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-87373-0" title="Special:BookSources/978-0-471-87373-0"><bdi>978-0-471-87373-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Physics+of+Atoms%2C+Molecules%2C+Solids%2C+Nuclei+and+Particles&rft.pages=274&rft.edition=2nd&rft.pub=John+Wiley+%26+Sons&rft.date=1985&rft.isbn=978-0-471-87373-0&rft.au=Resnick%2C+R.&rft.au=Eisberg%2C+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumphysicsof00eisb%2Fpage%2F274&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=31" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohen-TannoudjiDiuLaloë2006" class="citation book cs1">Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2006). <i>Quantum Mechanics</i> (2 volume set ed.). John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-56952-7" title="Special:BookSources/978-0-471-56952-7"><bdi>978-0-471-56952-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics&rft.edition=2+volume+set&rft.pub=John+Wiley+%26+Sons&rft.date=2006&rft.isbn=978-0-471-56952-7&rft.aulast=Cohen-Tannoudji&rft.aufirst=Claude&rft.au=Diu%2C+Bernard&rft.au=Lalo%C3%AB%2C+Franck&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCondonShortley1935" class="citation book cs1">Condon, E. U.; Shortley, G. H. (1935). "Especially Chapter 3". <i>The Theory of Atomic Spectra</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-09209-8" title="Special:BookSources/978-0-521-09209-8"><bdi>978-0-521-09209-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Especially+Chapter+3&rft.btitle=The+Theory+of+Atomic+Spectra&rft.pub=Cambridge+University+Press&rft.date=1935&rft.isbn=978-0-521-09209-8&rft.aulast=Condon&rft.aufirst=E.+U.&rft.au=Shortley%2C+G.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHippleSommerThomas1949" class="citation journal cs1">Hipple, J. A.; Sommer, H.; Thomas, H.A. (1949). <a rel="nofollow" class="external text" href="https://www.academia.edu/6483539">"A precise method of determining the faraday by magnetic resonance"</a>. <i>Physical Review</i>. <b>76</b> (12): 1877–1878. <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/1949PhRv...76.1877H">1949PhRv...76.1877H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.76.1877.2">10.1103/PhysRev.76.1877.2</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review&rft.atitle=A+precise+method+of+determining+the+faraday+by+magnetic+resonance&rft.volume=76&rft.issue=12&rft.pages=1877-1878&rft.date=1949&rft_id=info%3Adoi%2F10.1103%2FPhysRev.76.1877.2&rft_id=info%3Abibcode%2F1949PhRv...76.1877H&rft.aulast=Hipple&rft.aufirst=J.+A.&rft.au=Sommer%2C+H.&rft.au=Thomas%2C+H.A.&rft_id=https%3A%2F%2Fwww.academia.edu%2F6483539&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdmonds1957" class="citation book cs1">Edmonds, A. R. (1957). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/angularmomentumi0000edmo"><i>Angular Momentum in Quantum Mechanics</i></a></span>. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-07912-7" title="Special:BookSources/978-0-691-07912-7"><bdi>978-0-691-07912-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Angular+Momentum+in+Quantum+Mechanics&rft.pub=Princeton+University+Press&rft.date=1957&rft.isbn=978-0-691-07912-7&rft.aulast=Edmonds&rft.aufirst=A.+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fangularmomentumi0000edmo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJackson1998" class="citation book cs1">Jackson, John David (1998). <i>Classical Electrodynamics</i> (3rd ed.). John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-30932-1" title="Special:BookSources/978-0-471-30932-1"><bdi>978-0-471-30932-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Electrodynamics&rft.edition=3rd&rft.pub=John+Wiley+%26+Sons&rft.date=1998&rft.isbn=978-0-471-30932-1&rft.aulast=Jackson&rft.aufirst=John+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerwayJewett2004" class="citation book cs1">Serway, Raymond A.; Jewett, John W. (2004). <a rel="nofollow" class="external text" href="https://archive.org/details/physicssciengv2p00serw"><i>Physics for Scientists and Engineers</i></a> (6th ed.). Brooks/Cole. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-534-40842-8" title="Special:BookSources/978-0-534-40842-8"><bdi>978-0-534-40842-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+for+Scientists+and+Engineers&rft.edition=6th&rft.pub=Brooks%2FCole&rft.date=2004&rft.isbn=978-0-534-40842-8&rft.aulast=Serway&rft.aufirst=Raymond+A.&rft.au=Jewett%2C+John+W.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fphysicssciengv2p00serw&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThompson1994" class="citation book cs1">Thompson, William J. (1994). <i>Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems</i>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-55264-2" title="Special:BookSources/978-0-471-55264-2"><bdi>978-0-471-55264-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Angular+Momentum%3A+An+Illustrated+Guide+to+Rotational+Symmetries+for+Physical+Systems&rft.pub=Wiley&rft.date=1994&rft.isbn=978-0-471-55264-2&rft.aulast=Thompson&rft.aufirst=William+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTipler2004" class="citation book cs1">Tipler, Paul (2004). <i>Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics</i> (5th ed.). W. H. Freeman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-0809-4" title="Special:BookSources/978-0-7167-0809-4"><bdi>978-0-7167-0809-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+for+Scientists+and+Engineers%3A+Mechanics%2C+Oscillations+and+Waves%2C+Thermodynamics&rft.edition=5th&rft.pub=W.+H.+Freeman&rft.date=2004&rft.isbn=978-0-7167-0809-4&rft.aulast=Tipler&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpin+%28physics%29" class="Z3988"></span></li> <li>Sin-Itiro Tomonaga, The Story of Spin, 1997</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spin_(physics)&action=edit&section=32" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Spin_(intrinsic_angular_momentum)" class="extiw" title="commons:Category:Spin (intrinsic angular momentum)">Spin (intrinsic angular momentum)</a></span>.</div></div> </div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikiquote-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/13px-Wikiquote-logo.svg.png" decoding="async" width="13" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/20px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/27px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></a></span> Quotations related to <a href="https://en.wikiquote.org/wiki/Special:Search/Spin_(physics)" class="extiw" title="wikiquote:Special:Search/Spin (physics)">Spin (physics)</a> at Wikiquote</li> <li><a rel="nofollow" class="external text" href="http://www.lorentz.leidenuniv.nl/history/spin/goudsmit.html">Goudsmit on the discovery of electron spin.</a></li> <li><i><a href="/wiki/Nature_(journal)" title="Nature (journal)">Nature</a></i>: "<a rel="nofollow" class="external text" href="https://www.nature.com/collections/idgejiafca">Milestones in 'spin' since 1896.</a>"</li> <li><a rel="nofollow" class="external text" href="http://nanohub.org/resources/6025">ECE 495N Lecture 36: Spin</a> Online lecture by S. 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class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Space and time</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/D%27Alembert_operator" title="D'Alembert operator">d'Alembertian</a></li> <li><a href="/wiki/Parity_(physics)" title="Parity (physics)">Parity</a></li> <li><a href="/wiki/Time_evolution" title="Time evolution">Time</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Particles</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/C-symmetry" title="C-symmetry">C-symmetry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operators for operators</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/Anti-symmetric_operator" title="Anti-symmetric operator">Anti-symmetric operator</a></li> <li><a href="/wiki/Ladder_operator" title="Ladder operator">Ladder operator</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Fundamental</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/Momentum_operator" title="Momentum operator">Momentum</a></li> <li><a href="/wiki/Position_operator" title="Position operator">Position</a></li> <li><a href="/wiki/Rotation_operator_(quantum_mechanics)" title="Rotation operator (quantum mechanics)">Rotation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Energy</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/Energy_operator" title="Energy operator">Total energy</a></li> <li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Kinetic_energy" title="Kinetic energy">Kinetic energy</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Angular momentum</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/Clebsch%E2%80%93Gordan_coefficients#Angular_momentum_operators" title="Clebsch–Gordan coefficients">Total</a></li> <li><a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">Orbital</a></li> <li><a href="/wiki/Spin_(particle_physics)#Operator" class="mw-redirect" title="Spin (particle physics)">Spin</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Electromagnetism</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/Transition_dipole_moment" title="Transition dipole moment">Transition dipole moment</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Optics</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/Displacement_operator" title="Displacement operator">Displacement</a></li> <li><a href="/wiki/Hanbury_Brown_and_Twiss_effect" title="Hanbury Brown and Twiss effect">Hanbury Brown and Twiss effect</a></li> <li><a href="/w/index.php?title=Quantum_correlator&action=edit&redlink=1" class="new" title="Quantum correlator (page does not exist)">Quantum correlator</a></li> <li><a href="/wiki/Squeeze_operator" title="Squeeze operator">Squeeze</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Particle physics</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/Casimir_element" title="Casimir element">Casimir invariant</a></li> <li><a href="/wiki/Creation_and_annihilation_operators" title="Creation and annihilation operators">Creation and annihilation</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Quantum_mechanics" style="padding:3px"><table 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 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href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamentals</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Born_rule" title="Born rule">Born rule</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li> <li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)"> Complementarity</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a> <ul><li><a href="/wiki/Ground_state" title="Ground state">Ground state</a></li> <li><a href="/wiki/Excited_state" title="Excited state">Excited state</a></li> <li><a href="/wiki/Degenerate_energy_levels" title="Degenerate energy levels">Degenerate levels</a></li> <li><a href="/wiki/Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li></ul></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a 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="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Quantum_mechanics" title="Category:Quantum mechanics">Category</a></li></ul> </div></td></tr></tbody></table></div> <div 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Spin (physics) - Wikipedia