Operator statistic

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class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Cuprins</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mută în bara laterală</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ascunde</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Început</div> </a> </li> <li id="toc-Definiție" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definiție"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definiție</span> </div> </a> <ul id="toc-Definiție-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proprietăți" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proprietăți"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Proprietăți</span> </div> </a> <ul id="toc-Proprietăți-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Colectiv_statistic_pur" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Colectiv_statistic_pur"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Colectiv statistic pur</span> </div> </a> <ul id="toc-Colectiv_statistic_pur-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Colectiv_statistic_mixt" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Colectiv_statistic_mixt"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Colectiv statistic mixt</span> </div> </a> <ul id="toc-Colectiv_statistic_mixt-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Evoluția_în_timp_a_operatorului_statistic" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Evoluția_în_timp_a_operatorului_statistic"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Evoluția în timp a operatorului statistic</span> </div> </a> <ul id="toc-Evoluția_în_timp_a_operatorului_statistic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Statistică_clasică_și_statistică_cuantică" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Statistică_clasică_și_statistică_cuantică"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Statistică clasică și statistică cuantică</span> </div> </a> <ul id="toc-Statistică_clasică_și_statistică_cuantică-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografie"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Bibliografie</span> </div> </a> <ul id="toc-Bibliografie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vezi_și" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vezi_și"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Vezi și</span> </div> </a> <ul id="toc-Vezi_și-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Legături_externe" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Legături_externe"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Legături externe</span> </div> </a> <ul id="toc-Legături_externe-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="Cuprins" 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="Comută cuprinsul" > <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">Comută cuprinsul</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">Operator statistic</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 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Disponibil în 25 limbi" > <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-25" 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">25 limbi</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B5%D9%81%D9%88%D9%81%D8%A9_%D8%A7%D9%84%D9%83%D8%AB%D8%A7%D9%81%D8%A9" title="مصفوفة الكثافة – arabă" lang="ar" hreflang="ar" data-title="مصفوفة الكثافة" data-language-autonym="العربية" data-language-local-name="arabă" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Matriu_densitat" title="Matriu densitat – catalană" lang="ca" hreflang="ca" data-title="Matriu densitat" data-language-autonym="Català" data-language-local-name="catalană" 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/Oper%C3%A1tor_hustoty" title="Operátor hustoty – cehă" lang="cs" hreflang="cs" data-title="Operátor hustoty" data-language-autonym="Čeština" data-language-local-name="cehă" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Dichteoperator" title="Dichteoperator – germană" lang="de" hreflang="de" data-title="Dichteoperator" data-language-autonym="Deutsch" data-language-local-name="germană" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Density_matrix" title="Density matrix – engleză" lang="en" hreflang="en" data-title="Density matrix" data-language-autonym="English" data-language-local-name="engleză" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Estado_mixto" title="Estado mixto – spaniolă" lang="es" hreflang="es" data-title="Estado mixto" data-language-autonym="Español" data-language-local-name="spaniolă" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Matriz_dentsitate" title="Matriz dentsitate – bască" lang="eu" hreflang="eu" data-title="Matriz dentsitate" data-language-autonym="Euskara" data-language-local-name="bască" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%A7%D8%AA%D8%B1%DB%8C%D8%B3_%DA%86%DA%AF%D8%A7%D9%84%DB%8C" title="ماتریس چگالی – persană" lang="fa" hreflang="fa" data-title="ماتریس چگالی" data-language-autonym="فارسی" data-language-local-name="persană" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Tiheysmatriisi" title="Tiheysmatriisi – finlandeză" lang="fi" hreflang="fi" data-title="Tiheysmatriisi" data-language-autonym="Suomi" data-language-local-name="finlandeză" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Matrice_densit%C3%A9" title="Matrice densité – franceză" lang="fr" hreflang="fr" data-title="Matrice densité" data-language-autonym="Français" data-language-local-name="franceză" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%95%D7%A4%D7%A8%D7%98%D7%95%D7%A8_%D7%94%D7%A6%D7%A4%D7%99%D7%A4%D7%95%D7%AA" title="אופרטור הצפיפות – ebraică" lang="he" hreflang="he" data-title="אופרטור הצפיפות" data-language-autonym="עברית" data-language-local-name="ebraică" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/S%C5%B1r%C5%B1s%C3%A9gm%C3%A1trix" title="Sűrűségmátrix – maghiară" lang="hu" hreflang="hu" data-title="Sűrűségmátrix" data-language-autonym="Magyar" data-language-local-name="maghiară" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BD%D5%A1%D5%BC%D5%A8_%D5%BE%D5%AB%D5%B3%D5%A1%D5%AF" title="Խառը վիճակ – armeană" lang="hy" hreflang="hy" data-title="Խառը վիճակ" data-language-autonym="Հայերեն" data-language-local-name="armeană" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Matriks_densitas" title="Matriks densitas – indoneziană" lang="id" hreflang="id" data-title="Matriks densitas" data-language-autonym="Bahasa Indonesia" data-language-local-name="indoneziană" 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/Operatore_densit%C3%A0" title="Operatore densità – italiană" lang="it" hreflang="it" data-title="Operatore densità" data-language-autonym="Italiano" data-language-local-name="italiană" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%AF%86%E5%BA%A6%E8%A1%8C%E5%88%97" title="密度行列 – japoneză" lang="ja" hreflang="ja" data-title="密度行列" data-language-autonym="日本語" data-language-local-name="japoneză" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B0%80%EB%8F%84_%ED%96%89%EB%A0%AC" title="밀도 행렬 – coreeană" lang="ko" hreflang="ko" data-title="밀도 행렬" data-language-autonym="한국어" data-language-local-name="coreeană" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Dichtheidsmatrix" title="Dichtheidsmatrix – neerlandeză" lang="nl" hreflang="nl" data-title="Dichtheidsmatrix" data-language-autonym="Nederlands" data-language-local-name="neerlandeză" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%A1%E0%A9%88%E0%A9%B1%E0%A8%A8%E0%A8%B8%E0%A8%9F%E0%A9%80_%E0%A8%AE%E0%A9%88%E0%A8%9F%E0%A9%8D%E0%A8%B0%E0%A8%BF%E0%A8%95%E0%A8%B8" title="ਡੈੱਨਸਟੀ ਮੈਟ੍ਰਿਕਸ – punjabi" lang="pa" hreflang="pa" data-title="ਡੈੱਨਸਟੀ ਮੈਟ੍ਰਿਕਸ" 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/Macierz_g%C4%99sto%C5%9Bci" title="Macierz gęstości – poloneză" lang="pl" hreflang="pl" data-title="Macierz gęstości" data-language-autonym="Polski" data-language-local-name="poloneză" 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/Matriz_densidade" title="Matriz densidade – portugheză" lang="pt" hreflang="pt" data-title="Matriz densidade" data-language-autonym="Português" data-language-local-name="portugheză" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_%D0%BF%D0%BB%D0%BE%D1%82%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Матрица плотности – rusă" lang="ru" hreflang="ru" data-title="Матрица плотности" data-language-autonym="Русский" data-language-local-name="rusă" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/T%C3%A4thetsmatris" title="Täthetsmatris – suedeză" lang="sv" hreflang="sv" data-title="Täthetsmatris" data-language-autonym="Svenska" data-language-local-name="suedeză" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%8F_%D0%B3%D1%83%D1%81%D1%82%D0%B8%D0%BD%D0%B8" title="Матриця густини – ucraineană" lang="uk" hreflang="uk" data-title="Матриця густини" data-language-autonym="Українська" data-language-local-name="ucraineană" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%AF%86%E5%BA%A6%E7%9F%A9%E9%99%A3" title="密度矩陣 – chineză" lang="zh" hreflang="zh" data-title="密度矩陣" data-language-autonym="中文" data-language-local-name="chineză" 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/Q831774#sitelinks-wikipedia" title="Modifică legăturile interlinguale" class="wbc-editpage">Modifică legăturile</a></span></div> </div> </div> </div> </header> <div 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</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="ro" dir="ltr"><p><b>Operatorul statistic</b>, numit și <b>operator densitate</b> sau <b>matrice densitate</b>, este instrumentul matematic al mecanicii statistice cuantice. El reunește într-o tratare unitară o dublă statistică: descrierea statistică a stărilor microscopice ale unui sistem (proprie <a href="/wiki/Mecanic%C4%83_cuantic%C4%83" title="Mecanică cuantică">mecanicii cuantice</a>) și statistica rezultată din cunoașterea incompletă a acestor stări (existentă și în <a href="/wiki/Mecanic%C4%83_statistic%C4%83" title="Mecanică statistică">mecanica statistică</a> clasică). </p><p>Conform mecanicii cuantice, starea dinamică a unui sistem atomic este descrisă de <a href="/wiki/Mecanic%C4%83_cuantic%C4%83#Funcție_de_stare_și_spațiu_Hilbert" title="Mecanică cuantică">funcția de stare</a>. Această descriere are <a href="/wiki/Mecanic%C4%83_cuantic%C4%83#Interpretare_statistică" title="Mecanică cuantică">caracter statistic</a>: funcția de stare se referă la un <i>colectiv statistic</i> alcătuit dintr-un număr mare de exemplare identice ale sistemului, care evoluează în timp conform <a href="/wiki/Mecanic%C4%83_cuantic%C4%83#Formularea_Schrödinger" title="Mecanică cuantică">ecuației lui Schrödinger</a>, pornind de la o anumită stare inițială. Starea inițială a colectivului statistic este „preparată” prin indicarea precisă a valorilor unui număr de observabile compatibile – mărimi fizice reprezentate prin <a href="/wiki/Mecanic%C4%83_cuantic%C4%83#Observabile_și_operatori_hermitici" title="Mecanică cuantică">operatori</a> care comută, doi câte doi. Dacă acest sistem de observabile este <i>complet</i>, funcția de stare este unic determinată. </p><p>Dacă însă starea inițială a sistemului nu este complet determinată (cum se întâmplă în cazul operatorilor cu <a href="/wiki/Mecanic%C4%83_cuantic%C4%83#Valori_proprii_și_vectori_proprii" title="Mecanică cuantică">valori proprii degenerate</a>), există mai multe funcții de stare compatibile. În acest caz este necesară o tratare statistică analoagă celei din mecanica statistică clasică: trebuie luate în considerare toate stările inițiale posibile, atașând fiecăreia o probabilitate, care se determină prin măsurări repetate asupra sistemului incomplet de observabile utilizat. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definiție"><span id="Defini.C8.9Bie"></span>Definiție</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_statistic&veaction=edit&section=1" title="Modifică secțiunea: Definiție" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_statistic&action=edit&section=1" title="Edit section's source code: Definiție"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>În cazul general când informația este incompletă, sistemul se poate afla în oricare dintre stările descrise de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> funcții de stare normate </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 \left\{\psi ^{(1)},\psi ^{(2)},\ldots \psi ^{(\nu )}\right\},\quad \langle \psi ^{(\alpha )},\psi ^{(\alpha )}\rangle =1\,,\quad (\alpha =1,2,\ldots \nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mo>…</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mo>}</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>α</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\psi ^{(1)},\psi ^{(2)},\ldots \psi ^{(\nu )}\right\},\quad \langle \psi ^{(\alpha )},\psi ^{(\alpha )}\rangle =1\,,\quad (\alpha =1,2,\ldots \nu )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e244e13b441e319b45d7ce79075e58c216f14887" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:57.744ex; height:4.843ex;" alt="{\displaystyle \left\{\psi ^{(1)},\psi ^{(2)},\ldots \psi ^{(\nu )}\right\},\quad \langle \psi ^{(\alpha )},\psi ^{(\alpha )}\rangle =1\,,\quad (\alpha =1,2,\ldots \nu )}"></span></dd></dl> <p>și probabilitățile asociate lor </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 \left\{p^{(1)},p^{(2)},\ldots p^{(\nu )}\right\},\quad p^{(\alpha )}\geq 0\,,\quad (\alpha =1,2,\ldots \nu )\,,\quad \sum _{\alpha =1}^{\nu }\,p^{(\alpha )}=1\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mo>…</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mo>}</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>≥</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>α</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </munderover> <mspace width="thinmathspace" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{p^{(1)},p^{(2)},\ldots p^{(\nu )}\right\},\quad p^{(\alpha )}\geq 0\,,\quad (\alpha =1,2,\ldots \nu )\,,\quad \sum _{\alpha =1}^{\nu }\,p^{(\alpha )}=1\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c875906d03b7d01faeb7bfe1f6e9a5dac5a9bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:66.35ex; height:6.843ex;" alt="{\displaystyle \left\{p^{(1)},p^{(2)},\ldots p^{(\nu )}\right\},\quad p^{(\alpha )}\geq 0\,,\quad (\alpha =1,2,\ldots \nu )\,,\quad \sum _{\alpha =1}^{\nu }\,p^{(\alpha )}=1\,.}"></span></dd></dl> <p>Considerând modul în care se compun probabilitățile în această dublă statistică, <a href="/wiki/Mecanic%C4%83_cuantic%C4%83#Principiul_descompunerii_spectrale" title="Mecanică cuantică">valoarea medie</a> a unei observabile reprezentate de operatorul hermitic <font style="vertical-align:10%;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dd68c84d3681911d63970cb9a87d34cae3e6ede" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.407ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} \,}"></span></font> pe colectivul statistic astfel definit este </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 \langle A\rangle =\sum _{\alpha =1}^{\nu }\,p^{(\alpha )}\,\langle \psi ^{(\alpha )},\mathbf {A} \psi ^{(\alpha )}\rangle \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </munderover> <mspace width="thinmathspace" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle A\rangle =\sum _{\alpha =1}^{\nu }\,p^{(\alpha )}\,\langle \psi ^{(\alpha )},\mathbf {A} \psi ^{(\alpha )}\rangle \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8185b713c8eccccf7829b804c0e9cf6df562e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.95ex; height:6.843ex;" alt="{\displaystyle \langle A\rangle =\sum _{\alpha =1}^{\nu }\,p^{(\alpha )}\,\langle \psi ^{(\alpha )},\mathbf {A} \psi ^{(\alpha )}\rangle \,.}"></span></dd></dl> <p>Introducând în spațiul Hilbert o <a href="/wiki/Mecanic%C4%83_cuantic%C4%83#Valori_proprii_și_vectori_proprii" title="Mecanică cuantică">bază ortonormată</a> oarecare <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{u_{1},u_{2},\ldots u_{i},\dots \right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{u_{1},u_{2},\ldots u_{i},\dots \right\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6332b3234560b73bbce56141e22d1036094ef55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.191ex; height:2.843ex;" alt="{\displaystyle \left\{u_{1},u_{2},\ldots u_{i},\dots \right\},}"></span> funcțiile de stare <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{\psi ^{(\alpha )}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\psi ^{(\alpha )}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2677148bec9cf5573e1d7b21ca173dc4c7952a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.177ex; height:4.843ex;" alt="{\displaystyle \left\{\psi ^{(\alpha )}\right\}}"></span> vor fi reprezentate prin coeficienții dezvoltării </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{(\alpha )}=\sum _{i}\,c_{i}^{(\alpha )}u_{i}\,,\quad \sum _{i}{|c_{i}^{(\alpha )}|}^{2}=1\,,\quad (\alpha =1,2,\ldots \nu )\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mspace width="thinmathspace" /> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>α</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ^{(\alpha )}=\sum _{i}\,c_{i}^{(\alpha )}u_{i}\,,\quad \sum _{i}{|c_{i}^{(\alpha )}|}^{2}=1\,,\quad (\alpha =1,2,\ldots \nu )\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/156671609fd808596b15df30adf7cab942418690" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:54.577ex; height:6.176ex;" alt="{\displaystyle \psi ^{(\alpha )}=\sum _{i}\,c_{i}^{(\alpha )}u_{i}\,,\quad \sum _{i}{|c_{i}^{(\alpha )}|}^{2}=1\,,\quad (\alpha =1,2,\ldots \nu )\,,}"></span></dd></dl> <p>iar operatorul <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dd68c84d3681911d63970cb9a87d34cae3e6ede" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.407ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} \,}"></span> prin elementele de matrice </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 A_{ij}=\langle u_{i},\mathbf {A} u_{j}\rangle \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{ij}=\langle u_{i},\mathbf {A} u_{j}\rangle \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c166941d9e7079df3d93dec07b91a261fd472b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.584ex; height:3.009ex;" alt="{\displaystyle A_{ij}=\langle u_{i},\mathbf {A} u_{j}\rangle \,.}"></span></dd></dl> <p>Cu aceste notații, valoarea medie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle A\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle A\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/471a7c588a72faea10a4d02f6134e670467962a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.552ex; height:2.843ex;" alt="{\displaystyle \langle A\rangle }"></span> devine </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 \langle A\rangle =\sum _{i,j}\,A_{ij}\,\rho _{ji}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <mspace width="thinmathspace" /> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle A\rangle =\sum _{i,j}\,A_{ij}\,\rho _{ji}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b923b24a9fb6a2bbf7f6fc71b8957d47ec5ed7e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.1ex; height:5.843ex;" alt="{\displaystyle \langle A\rangle =\sum _{i,j}\,A_{ij}\,\rho _{ji}\,,}"></span></dd></dl> <p>unde </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 \rho _{ij}=\sum _{\alpha =1}^{\nu }\,p^{(\alpha )}\,c_{i}^{(\alpha )}c_{j}^{(\alpha )*}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </munderover> <mspace width="thinmathspace" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace" /> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>∗</mo> </mrow> </msubsup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{ij}=\sum _{\alpha =1}^{\nu }\,p^{(\alpha )}\,c_{i}^{(\alpha )}c_{j}^{(\alpha )*}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e8dbfb138994dc6ca8414ad493f0b14aba38eb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.023ex; height:6.843ex;" alt="{\displaystyle \rho _{ij}=\sum _{\alpha =1}^{\nu }\,p^{(\alpha )}\,c_{i}^{(\alpha )}c_{j}^{(\alpha )*}\,.}"></span></dd></dl> <p>Matricea de elemente <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{\rho _{ij}\right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\rho _{ij}\right\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/725b47b329d4eb8577a880e06e645c58ec0b0b5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.038ex; height:3.009ex;" alt="{\displaystyle \left\{\rho _{ij}\right\},}"></span> care sintetizează informația (stări posibile și probabilități) asupra colectivului statistic considerat, rezultată dintr-o măsurare simultană (în cazul general incompletă), se numește <i>matricea densitate</i>. Ea este reprezentarea, în baza ortonormată aleasă în spațiul Hilbert, a unui operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\rho }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ρ</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\rho }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/988a152862242c46b08273b3233a0a4a05f45e8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.069ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {\rho }},}"></span> numit <i>operatorul densitate</i> sau <i>operatorul statistic</i>. În notație operatorială, expresia pentru valoarea medie este </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 \langle A\rangle =tr\,\left(\mathbf {A} {\boldsymbol {\rho }}\right)=tr\,\left({\boldsymbol {\rho }}\mathbf {A} \right)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>t</mi> <mi>r</mi> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ρ</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>t</mi> <mi>r</mi> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ρ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle A\rangle =tr\,\left(\mathbf {A} {\boldsymbol {\rho }}\right)=tr\,\left({\boldsymbol {\rho }}\mathbf {A} \right)\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c124e284aff086584ace2940af158563b536da5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.998ex; height:2.843ex;" alt="{\displaystyle \langle A\rangle =tr\,\left(\mathbf {A} {\boldsymbol {\rho }}\right)=tr\,\left({\boldsymbol {\rho }}\mathbf {A} \right)\,,}"></span></dd></dl> <p>unde simbolul <font style="vertical-align:10%;"><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 tr}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle tr}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e049e3abf8645c35b94e206366b2ee4e0a5da76b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.888ex; height:2.009ex;" alt="{\displaystyle tr}"></span></font> denotă <a href="/wiki/Urm%C4%83_(algebr%C4%83)" title="Urmă (algebră)">urma</a> operatorului pe care îl precede. </p> <div class="mw-heading mw-heading2"><h2 id="Proprietăți"><span id="Propriet.C4.83.C8.9Bi"></span>Proprietăți</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_statistic&veaction=edit&section=2" title="Modifică secțiunea: Proprietăți" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_statistic&action=edit&section=2" title="Edit section's source code: Proprietăți"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Din definiția precedentă rezultă câteva proprietăți importante ale operatorului statistic: <sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <ul><li>Operatorul statistic este hermitic:</li></ul> <dl><dd><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 {\rho }}^{\boldsymbol {+}}={\boldsymbol {\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ρ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold">+</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ρ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\rho }}^{\boldsymbol {+}}={\boldsymbol {\rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dd45ffc045f0d36dfe26443f9e84c178ebc1a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.645ex; height:3.009ex;" alt="{\displaystyle {\boldsymbol {\rho }}^{\boldsymbol {+}}={\boldsymbol {\rho }}}"></span></dd></dl></dd></dl> <dl><dd>deci valorile sale proprii sunt numere reale.</dd></dl> <ul><li>Urma operatorului statistic este egală cu 1:</li></ul> <dl><dd><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 tr\,{\boldsymbol {\rho }}=1\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mi>r</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ρ</mi> </mrow> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle tr\,{\boldsymbol {\rho }}=1\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce526e87e5809a9323de3ebed5d18091bbfafc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.993ex; height:2.509ex;" alt="{\displaystyle tr\,{\boldsymbol {\rho }}=1\,.}"></span></dd></dl></dd></dl> <ul><li>Valorile proprii ale operatorului statistic nu pot fi mai mici decât 0 sau mai mari decât 1:</li></ul> <dl><dd><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 0\leq {\rho }_{i}\leq 1\,,\quad (i=0,1,\ldots n)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≤</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>i</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq {\rho }_{i}\leq 1\,,\quad (i=0,1,\ldots n)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcdb1b6a0d4f49198637b004e210b65d6652a6c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.909ex; height:2.843ex;" alt="{\displaystyle 0\leq {\rho }_{i}\leq 1\,,\quad (i=0,1,\ldots n)\,.}"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Colectiv_statistic_pur">Colectiv statistic pur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_statistic&veaction=edit&section=3" title="Modifică secțiunea: Colectiv statistic pur" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_statistic&action=edit&section=3" title="Edit section's source code: Colectiv statistic pur"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dacă starea sistemului e determinată de o singură funcție <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 ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc4043b55bade492740e58cba74198873db1464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.16ex; height:2.509ex;" alt="{\displaystyle \psi ,}"></span> cu probabilitate asociată p = 1, rezultă din discuția precedentă că funcția de stare e funcție proprie a operatorului statistic, corespunzătoare valorii proprii 1, celelalte valori proprii fiind 0: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\rho }}\,\psi =\psi \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ρ</mi> </mrow> <mspace width="thinmathspace" /> <mi>ψ</mi> <mo>=</mo> <mi>ψ</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\rho }}\,\psi =\psi \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/971a9b2a713853283a0c8926e050e2e176a2392e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.968ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\rho }}\,\psi =\psi \,.}"></span></dd></dl> <p>În baza care diagonalizează matricea densitate, aceasta va avea forma </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 \rho \,=\,{\begin{pmatrix}1&0&\cdots &0\\0&0&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &0\end{pmatrix}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <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> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \,=\,{\begin{pmatrix}1&0&\cdots &0\\0&0&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &0\end{pmatrix}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e55f415a9b56573ff9c448207bb4473cde9a8e71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:24.361ex; height:14.176ex;" alt="{\displaystyle \rho \,=\,{\begin{pmatrix}1&0&\cdots &0\\0&0&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &0\end{pmatrix}}\,.}"></span></dd></dl> <p>În această situație particulară se vorbește despre un colectiv statistic <i>pur</i>, în care, conform unui <a href="/wiki/Mecanic%C4%83_cuantic%C4%83#Interpretare_statistică" title="Mecanică cuantică">principiu general</a> al mecanicii cuantice, statistica observabilelor este complet determinată de funcția de stare. </p> <div class="mw-heading mw-heading2"><h2 id="Colectiv_statistic_mixt">Colectiv statistic mixt</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_statistic&veaction=edit&section=4" title="Modifică secțiunea: Colectiv statistic mixt" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_statistic&action=edit&section=4" title="Edit section's source code: Colectiv statistic mixt"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cazul general este cel al unui colectiv statistic <i>mixt</i>, în care operatorul statistic are mai multe valori proprii diferite de zero. În baza de vectori proprii, matricea densitate este diagonală și poate fi descompusă în forma </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 \rho \,=\,{\begin{pmatrix}\rho _{1}&0&\cdots &0\\0&\rho _{2}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &\rho _{n}\end{pmatrix}}\,=\,\rho _{1}{\begin{pmatrix}1&0&\cdots &0\\0&0&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &0\end{pmatrix}}\,+\,\rho _{2}{\begin{pmatrix}0&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &0\end{pmatrix}}\,+\,\cdots \,\rho _{n}{\begin{pmatrix}0&0&\cdots &0\\0&0&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{pmatrix}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <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> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mo>⋯</mo> <mspace width="thinmathspace" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \,=\,{\begin{pmatrix}\rho _{1}&0&\cdots &0\\0&\rho _{2}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &\rho _{n}\end{pmatrix}}\,=\,\rho _{1}{\begin{pmatrix}1&0&\cdots &0\\0&0&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &0\end{pmatrix}}\,+\,\rho _{2}{\begin{pmatrix}0&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &0\end{pmatrix}}\,+\,\cdots \,\rho _{n}{\begin{pmatrix}0&0&\cdots &0\\0&0&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{pmatrix}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/855f28bddcc3cf4c01df2bc4155302ce3ff98b5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:104.095ex; height:14.509ex;" alt="{\displaystyle \rho \,=\,{\begin{pmatrix}\rho _{1}&0&\cdots &0\\0&\rho _{2}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &\rho _{n}\end{pmatrix}}\,=\,\rho _{1}{\begin{pmatrix}1&0&\cdots &0\\0&0&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &0\end{pmatrix}}\,+\,\rho _{2}{\begin{pmatrix}0&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &0\end{pmatrix}}\,+\,\cdots \,\rho _{n}{\begin{pmatrix}0&0&\cdots &0\\0&0&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{pmatrix}}\,.}"></span></dd></dl> <p>Cazul mixt poate fi deci considerat ca un <i>amestec</i> de cazuri pure, cu ponderi statistice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{\rho _{1},\rho _{2},\ldots \rho _{n}\right\}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\rho _{1},\rho _{2},\ldots \rho _{n}\right\}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/290bd3eb10e03c721b922bb754f158a54cefb6b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.857ex; height:2.843ex;" alt="{\displaystyle \left\{\rho _{1},\rho _{2},\ldots \rho _{n}\right\}\,.}"></span> Se poate arăta că un caz pur nu poate fi considerat, la rândul său, ca un amestec de cazuri cu statistică mai simplă. <sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Concluzia este că statistica din cazul pur nu poate fi atribuită unei cunoașteri incomplete a sistemului atomic considerat: operatorul statistic având o unică valoare proprie egală cu 1, respectiv funcția proprie corespunzătoare acestei valori proprii, reprezintă informația maximă despre starea sistemului, în contextul mecanicii cuantice. </p><p>Operatorul statistic își dovedește utilitatea mai ales în cazul sistemelor compuse dintr-un număr mare de particule, de exemplu când se cercetează proprietățile corpurilor macroscopice pe baza structurii lor atomice; dar el este folosit și în sisteme cu un număr mai redus de componente, atunci când condițiile experimentale nu înlesnesc înregistrarea de informații complete. <sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Evoluția_în_timp_a_operatorului_statistic"><span id="Evolu.C8.9Bia_.C3.AEn_timp_a_operatorului_statistic"></span>Evoluția în timp a operatorului statistic</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_statistic&veaction=edit&section=5" title="Modifică secțiunea: Evoluția în timp a operatorului statistic" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_statistic&action=edit&section=5" title="Edit section's source code: Evoluția în timp a operatorului statistic"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Considerentele precedente se referă la starea unui sistem atomic la un anumit moment, însă această stare evoluează în timp. În cazul pur, evoluția temporală a funcției de stare se face conform <a href="/wiki/Mecanic%C4%83_cuantic%C4%83#Formularea_Schrödinger" title="Mecanică cuantică">ecuației lui Schrödinger</a>. Utilizând coeficienții funcției de stare, respectiv elementele de matrice ale hamiltonianului, într-o bază ortonormată oarecare, se obține ecuația corespunzătoare pentru matricea densitate, care în formă operatorială este <sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <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 i\hbar {\frac {d{\boldsymbol {\rho }}}{dt}}=\left[H,{\boldsymbol {\rho }}\right]\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ρ</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mi>H</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ρ</mi> </mrow> </mrow> <mo>]</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {d{\boldsymbol {\rho }}}{dt}}=\left[H,{\boldsymbol {\rho }}\right]\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b637ddc2bed95973b9a5e18182f768fed4aa4947" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.917ex; height:5.509ex;" alt="{\displaystyle i\hbar {\frac {d{\boldsymbol {\rho }}}{dt}}=\left[H,{\boldsymbol {\rho }}\right]\,.}"></span></dd></dl> <p>Întrucât operatorul statistic pentru un amestec este o combinație liniară de operatori statistici pentru cazuri pure, această ecuație rămâne valabilă și în cazul mixt. </p> <div class="mw-heading mw-heading2"><h2 id="Statistică_clasică_și_statistică_cuantică"><span id="Statistic.C4.83_clasic.C4.83_.C8.99i_statistic.C4.83_cuantic.C4.83"></span>Statistică clasică și statistică cuantică</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_statistic&veaction=edit&section=6" title="Modifică secțiunea: Statistică clasică și statistică cuantică" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_statistic&action=edit&section=6" title="Edit section's source code: Statistică clasică și statistică cuantică"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se vede că <i>operatorul statistic</i> în spațiul Hilbert joacă, în mecanica statistică cuantică, un rol similar cu <i><a href="/wiki/Mecanic%C4%83_statistic%C4%83#Colectiv_statistic" title="Mecanică statistică">densitatea de probabilitate</a></i> în spațiul fazelor din mecanica statistică clasică (de unde și denumirea alternativă de operator sau matrice <i>densitate</i>). Tabelul de mai jos, care rezumă paralelismul dintre cele două teorii, este totodată o exemplificare a <a href="/w/index.php?title=Principiul_de_coresponden%C8%9B%C4%83&action=edit&redlink=1" class="new" title="Principiul de corespondență — pagină inexistentă">principiului de corespondență</a>. <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> </p> <table class="wikitable" style="margin-left:1.0em;text-align:center;"> <caption> </caption> <tbody><tr> <td style="background:#f2f2f2;"><b>Statistică clasică</b> </td> <td style="background:#f2f2f2;"><b>Statistică cuantică</b> </td></tr> <tr> <td>Densitatea de probabilitate <font style="vertical-align:10%;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}"> <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">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span></font> în spațiul fazelor </td> <td>Operatorul statistic <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 {\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ρ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa514c733a0add8e7c3af8bf4f930fa918d16ff8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.423ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {\rho }}}"></span> în spațiul Hilbert </td></tr> <tr> <td>Condiția de normare<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int {\mathcal {P}}\,dp\,dq=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>q</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\mathcal {P}}\,dp\,dq=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70ac35879b8af7ee3caa55ef00d316d1ea6af51d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.99ex; height:5.676ex;" alt="{\displaystyle \int {\mathcal {P}}\,dp\,dq=1}"></span> </td> <td>Condiția de normare<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle tr\,{\boldsymbol {\rho }}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mi>r</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ρ</mi> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle tr\,{\boldsymbol {\rho }}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f32b365a5a2ac3887e75638e6117fdb2c74f335a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.959ex; height:2.509ex;" alt="{\displaystyle tr\,{\boldsymbol {\rho }}=1}"></span> </td></tr> <tr> <td>Valoarea medie a mărimii <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5397b34cab7d96daac496a937b0c0fa076dff7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.666ex; height:2.509ex;" alt="{\displaystyle f\,}"></span><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle f\right\rangle =\int f{\mathcal {P}}\,dp\,dq}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mi>f</mi> <mo>⟩</mo> </mrow> <mo>=</mo> <mo>∫</mo> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle f\right\rangle =\int f{\mathcal {P}}\,dp\,dq}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87014d7c553a39bd94b3b7318bea38bb0e6074c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.194ex; height:5.676ex;" alt="{\displaystyle \left\langle f\right\rangle =\int f{\mathcal {P}}\,dp\,dq}"></span> </td> <td>Valoarea medie a observabilei <font style="vertical-align:10%;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6aaf5ce10d6add44b973e28fb3d95f37abf3721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.13ex; height:2.176ex;" alt="{\displaystyle A\,}"></span></font><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle A\rangle =tr\,\left(\mathbf {A} {\boldsymbol {\rho }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>t</mi> <mi>r</mi> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ρ</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle A\rangle =tr\,\left(\mathbf {A} {\boldsymbol {\rho }}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d9a673269f58c8a51ca98d415c66b6d065c894c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.565ex; height:2.843ex;" alt="{\displaystyle \langle A\rangle =tr\,\left(\mathbf {A} {\boldsymbol {\rho }}\right)}"></span> </td></tr> <tr> <td>Distribuția canonică la temperatura <font style="vertical-align:10%;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/476a8389064c06ab89963a2467aef525838da0cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.023ex; height:2.176ex;" alt="{\displaystyle T\,}"></span></font><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}={\frac {1}{Z}}\,e^{-\beta H}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>Z</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mi>H</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}={\frac {1}{Z}}\,e^{-\beta H}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/972bf07cba52fff6b036f85f704043aabc0f3d87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.701ex; height:5.176ex;" alt="{\displaystyle {\mathcal {P}}={\frac {1}{Z}}\,e^{-\beta H}}"></span><br />Funcția de partiție<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=\int e^{-\beta H}\,dp\,dq}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mo>∫</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mi>H</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=\int e^{-\beta H}\,dp\,dq}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b76db0fd36fee6d89581eebeb37ac98ac97be4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.8ex; height:5.676ex;" alt="{\displaystyle Z=\int e^{-\beta H}\,dp\,dq}"></span><br /><font style="vertical-align:10%;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span></font> este hamiltonianul iar <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 \beta =1/kT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =1/kT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcbbd7dfc2c3bf360822cda0b3c7227f468e476f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.603ex; height:2.843ex;" alt="{\displaystyle \beta =1/kT}"></span> </td> <td>Distribuția canonică la temperatura <font style="vertical-align:10%;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/476a8389064c06ab89963a2467aef525838da0cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.023ex; height:2.176ex;" alt="{\displaystyle T\,}"></span></font><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\rho }}={\frac {1}{Z}}\,e^{-\beta {\mathcal {H}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ρ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>Z</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\rho }}={\frac {1}{Z}}\,e^{-\beta {\mathcal {H}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/690acd97852f25df075261daed4a19974ed7fcb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.349ex; height:5.176ex;" alt="{\displaystyle {\boldsymbol {\rho }}={\frac {1}{Z}}\,e^{-\beta {\mathcal {H}}}}"></span><br />Funcția de partiție<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=tr\left(e^{-\beta {\mathcal {H}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=tr\left(e^{-\beta {\mathcal {H}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56b110b39bcf726adcc1f9b450cf48a8ae993c27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.109ex; height:3.343ex;" alt="{\displaystyle Z=tr\left(e^{-\beta {\mathcal {H}}}\right)}"></span><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19ef4c7b923a5125ac91aa491838a95ee15b804f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}"></span> este operatorul hamiltonian iar <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 \beta =1/kT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =1/kT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcbbd7dfc2c3bf360822cda0b3c7227f468e476f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.603ex; height:2.843ex;" alt="{\displaystyle \beta =1/kT}"></span> </td></tr> <tr> <td>Expresia generală a entropiei<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ S=-k\int {\mathcal {P}}\,ln{\mathcal {P}}\,dp\,dq}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>S</mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>l</mi> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ S=-k\int {\mathcal {P}}\,ln{\mathcal {P}}\,dp\,dq}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bda7f77a6d3d95424c76feefea514369d036a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.493ex; height:5.676ex;" alt="{\displaystyle \ S=-k\int {\mathcal {P}}\,ln{\mathcal {P}}\,dp\,dq}"></span> </td> <td>Expresia generală a entropiei<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ S=-k\,tr\left({\boldsymbol {\rho }}\,ln{\boldsymbol {\rho }}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>S</mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mspace width="thinmathspace" /> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ρ</mi> </mrow> <mspace width="thinmathspace" /> <mi>l</mi> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ρ</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ S=-k\,tr\left({\boldsymbol {\rho }}\,ln{\boldsymbol {\rho }}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56474be701ea1f127970d1e42df9ac5990f88b04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.99ex; height:2.843ex;" alt="{\displaystyle \ S=-k\,tr\left({\boldsymbol {\rho }}\,ln{\boldsymbol {\rho }}\right)}"></span> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_statistic&veaction=edit&section=7" title="Modifică secțiunea: Note" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_statistic&action=edit&section=7" title="Edit section's source code: Note"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><b><a href="#cite_ref-1">^</a></b> <span class="reference-text">Țițeica, p. 416.</span> </li> <li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> <span class="reference-text">Țițeica, p. 419.</span> </li> <li id="cite_note-3"><b><a href="#cite_ref-3">^</a></b> <span class="reference-text">Țițeica, p. 421.</span> </li> <li id="cite_note-4"><b><a href="#cite_ref-4">^</a></b> <span class="reference-text">Țițeica, p. 420.</span> </li> <li id="cite_note-5"><b><a href="#cite_ref-5">^</a></b> <span class="reference-text">Messiah, pp. 285–286.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografie">Bibliografie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_statistic&veaction=edit&section=8" title="Modifică secțiunea: Bibliografie" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_statistic&action=edit&section=8" title="Edit section's source code: Bibliografie"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Lev_Landau" title="Lev Landau">L.D. Landau</a>, <a href="/wiki/Evgheni_Lif%C8%99i%C8%9B" title="Evgheni Lifșiț">E.M. Lifshitz</a>: <i>Statistical Physics</i>, Pergamon Press, Oxford, 1980, pp. 14–20. <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/0080230385" class="internal mw-magiclink-isbn">ISBN 0-08-023038-5</a>.</li> <li>Landau, L.D., Lifshitz, E.M.: <i>Quantum mechanics: non-relativistic theory</i>, Pergamon Press, Oxford, 1991, pp. 38–41. <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/0080291406" class="internal mw-magiclink-isbn">ISBN 0-08-029140-6</a></li> <li>Messiah, Albert: <i>Mécanique quantique</i>, Dunod, Paris, 1962, Tome 1, pp. 279–286.</li> <li><a href="/wiki/%C8%98erban_%C8%9Ai%C8%9Beica" title="Șerban Țițeica">Țițeica, Șerban</a>: <i>Mecanica cuantică</i>, Editura Academiei Republicii Socialiste România, București, 1984, pp. 414–421.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Vezi_și"><span id="Vezi_.C8.99i"></span>Vezi și</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_statistic&veaction=edit&section=9" title="Modifică secțiunea: Vezi și" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_statistic&action=edit&section=9" title="Edit section's source code: Vezi și"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Mecanic%C4%83_cuantic%C4%83" title="Mecanică cuantică">Mecanică cuantică</a></li> <li><a href="/wiki/Mecanic%C4%83_statistic%C4%83" title="Mecanică statistică">Mecanică statistică</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Legături_externe"><span id="Leg.C4.83turi_externe"></span>Legături externe</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_statistic&veaction=edit&section=10" title="Modifică secțiunea: Legături externe" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_statistic&action=edit&section=10" title="Edit section's source code: Legături externe"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_9corr.pdf">Density Matrices – University of Vienna</a></li> <li><a rel="nofollow" class="external text" href="http://pages.uoregon.edu/soper/QuantumMechanics/density.pdf">The density operator in quantum mechanics – University of Oregon</a></li> <li><a rel="nofollow" class="external text" href="http://www.cithep.caltech.edu/~fcp/physics/quantumMechanics/densityMatrix/densityMatrix.pdf">Density Matrix Formalism – Caltech</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151128230632/http://www.cithep.caltech.edu/~fcp/physics/quantumMechanics/densityMatrix/densityMatrix.pdf">Arhivat</a> în <time datetime="2015-11-28">28 noiembrie 2015</time>, la <a href="/wiki/Wayback_Machine" class="mw-redirect" title="Wayback Machine">Wayback Machine</a>.</li> <li><a rel="nofollow" class="external text" href="http://ocw.mit.edu/courses/chemistry/5-74-introductory-quantum-mechanics-ii-spring-2009/lecture-notes/MIT5_74s09_lec12.pdf">The density matrix – MIT</a></li> <li><a rel="nofollow" class="external text" href="http://www.physik.fu-berlin.de/en/einrichtungen/ag/ag-eisert/teaching/QMChapter4.pdf">Density operators – Freie Universität Berlin</a></li></ul> <p><br /> </p> <table class="navbox" cellspacing="0" style=""> <tbody><tr> <td style="padding:2px;"> <table cellspacing="0" class="nowraplinks collapsible" style="width:100%;background:transparent;color:inherit;;"> <tbody><tr> <th style=";" colspan="2" class="navbox-title"><div style="float:left; width:6em;text-align:left;"><div class="noprint plainlinks" style="padding:0; font-size:xx-small; color:var(--color-base, #000); white-space:nowrap; ;"><span style=";;border:none;"><a href="/wiki/Format:Fizic%C4%83_cuantic%C4%83" title="Format:Fizică cuantică"><span title="Vizualizare format" style=";;border:none;;">v</span></a> <span style="font-size:80%;">•</span> <a href="/wiki/Discu%C8%9Bie_Format:Fizic%C4%83_cuantic%C4%83" title="Discuție Format:Fizică cuantică"><span title="Discuție format" style=";;border:none;;">d</span></a> <span style="font-size:80%;">•</span> <a class="external text" href="https://ro.wikipedia.org/w/index.php?title=Format:Fizic%C4%83_cuantic%C4%83&action=edit"><span title="Acest format se poate modifica. Folosiți butonul de previzualizare înainte de a salva." style=";;border:none;;">m</span></a></span></div></div><span class="" style="font-size: 110%;">Fizică cuantică</span> </th></tr> <tr style="height:2px;"> <td> </td></tr> <tr> <td class="navbox-group" style=";;">Teorie cuantică veche </td> <td style="text-align:left;border-left:2px solid #fdfdfd;width:100%;padding:0px;;;" class="navbox-list navbox-odd"> <div style="padding:0em 0.25em"> <a href="/wiki/Constanta_Planck" title="Constanta Planck">Constanta Planck</a> • <a href="/wiki/Cuant%C4%83" title="Cuantă">Cuantă</a> • <a href="/wiki/Difrac%C8%9Bia_electronilor" title="Difracția electronilor">Difracția electronilor</a> • <a href="/wiki/Dualismul_und%C4%83-particul%C4%83" title="Dualismul undă-particulă">Dualismul undă-particulă</a> • <a href="/wiki/Formula_lui_Planck" title="Formula lui Planck">Formula lui Planck</a> • <a href="/wiki/Ipoteza_De_Broglie" title="Ipoteza De Broglie">Ipoteza De Broglie</a> • <a href="/wiki/Modelul_atomic_Bohr" title="Modelul atomic Bohr">Modelul atomic Bohr</a> • <a href="/wiki/Num%C4%83r_cuantic" title="Număr cuantic">Număr cuantic</a></div> </td></tr> <tr style="height:2px"> <td> </td></tr> <tr> <td class="navbox-group" style=";;">Mecanică cuantică </td> <td style="text-align:left;border-left:2px solid #fdfdfd;width:100%;padding:0px;;;" class="navbox-list navbox-even"><div style="padding:0em 0.25em"> <a href="/wiki/Ecua%C8%9Bia_lui_Dirac" title="Ecuația lui Dirac">Ecuația lui Dirac</a> • <a href="/wiki/Ecua%C8%9Bia_lui_Schr%C3%B6dinger" title="Ecuația lui Schrödinger">Ecuația lui Schrödinger</a> • <a href="/wiki/Efectul_tunel" title="Efectul tunel">Efectul tunel</a> • <a href="/wiki/Func%C8%9Bie_de_und%C4%83" title="Funcție de undă">Funcție de undă</a> • <a href="/wiki/Hamiltonian_(mecanic%C4%83_cuantic%C4%83)" title="Hamiltonian (mecanică cuantică)">Hamiltonian (mecanică cuantică)</a> • <a href="/wiki/Inseparabilitate_cuantic%C4%83" title="Inseparabilitate cuantică">Inseparabilitate cuantică</a> • <a href="/wiki/Interpretarea_Copenhaga" title="Interpretarea Copenhaga">Interpretarea Copenhaga</a> • <a href="/wiki/Interpret%C4%83rile_mecanicii_cuantice" title="Interpretările mecanicii cuantice">Interpretările mecanicii cuantice</a> • <a href="/wiki/Introducere_%C3%AEn_mecanica_cuantic%C4%83" title="Introducere în mecanica cuantică">Introducere în mecanica cuantică</a> • <a href="/wiki/Mecanic%C4%83_cuantic%C4%83" title="Mecanică cuantică">Mecanică cuantică</a> • <a href="/wiki/Moment_cinetic_(mecanic%C4%83_cuantic%C4%83)" title="Moment cinetic (mecanică cuantică)">Moment cinetic (mecanică cuantică)</a> • <a href="/wiki/Nota%C8%9Bia_bra-ket" title="Notația bra-ket">Notația bra-ket</a> • <a class="mw-selflink selflink">Operator statistic</a> • <a href="/wiki/Oscilatorul_armonic_liniar_(cuantic)" title="Oscilatorul armonic liniar (cuantic)">Oscilatorul armonic liniar</a> • <a href="/wiki/Particule_identice" title="Particule identice">Particule identice</a> • <a href="/wiki/Principiul_de_excluziune" title="Principiul de excluziune">Principiul de excluziune</a> • <a href="/wiki/Principiul_incertitudinii" title="Principiul incertitudinii">Principiul incertitudinii</a> • <a href="/wiki/Reprezentarea_numerelor_de_ocupare" title="Reprezentarea numerelor de ocupare">Reprezentarea numerelor de ocupare</a> • <a href="/wiki/Spin_(fizic%C4%83)" title="Spin (fizică)">Spin (fizică)</a> • <a href="/wiki/Spin_%C2%BD_%C8%99i_matricile_lui_Pauli" title="Spin ½ și matricile lui Pauli">Spin ½ și matricile lui Pauli</a></div> </td></tr> <tr style="height:2px"> <td> </td></tr> <tr> <td class="navbox-group" style=";;">Teorie cuantică relativistă </td> <td style="text-align:left;border-left:2px solid #fdfdfd;width:100%;padding:0px;;;" class="navbox-list navbox-odd"><div style="padding:0em 0.25em"> <a href="/wiki/Ecua%C8%9Bia_Schr%C3%B6dinger_neliniar%C4%83" title="Ecuația Schrödinger neliniară">Ecuația Schrödinger neliniară</a> • <a href="/wiki/Electrodinamic%C4%83_cuantic%C4%83" title="Electrodinamică cuantică">Electrodinamică cuantică</a> • <a href="/wiki/Ruperea_spontan%C4%83_a_simetriei" title="Ruperea spontană a simetriei">Ruperea spontană a simetriei</a> • <a href="/wiki/Teoria_coardelor" title="Teoria coardelor">Teoria coardelor</a></div>  </td></tr> <tr style="height:2px;"> <td> </td></tr> <tr> <td class="navbox-abovebelow" style=";" colspan="2"><span typeof="mw:File"><a href="/wiki/Proiect:Mecanic%C4%83_cuantic%C4%83" title="Proiect:Mecanică cuantică"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Symbol_spin_blue2.svg/15px-Symbol_spin_blue2.svg.png" decoding="async" width="15" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Symbol_spin_blue2.svg/23px-Symbol_spin_blue2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Symbol_spin_blue2.svg/30px-Symbol_spin_blue2.svg.png 2x" data-file-width="89" data-file-height="87" 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Clausius-Rankine">Ciclul Clausius-Rankine</a> • <a href="/wiki/Coeficient_de_transformare_adiabatic%C4%83" title="Coeficient de transformare adiabatică">Coeficient de transformare adiabatică</a> • <a href="/wiki/Constanta_universal%C4%83_a_gazului_ideal" title="Constanta universală a gazului ideal">Constanta universală a gazului ideal</a> • <a href="/wiki/Echilibru_termodinamic" title="Echilibru termodinamic">Echilibru termodinamic</a> • <a href="/wiki/Energie_intern%C4%83" title="Energie internă">Energie internă</a> • <a href="/wiki/Energie_liber%C4%83" title="Energie liberă">Energie liberă</a> • <a href="/wiki/Entalpie" title="Entalpie">Entalpie</a> • <a href="/wiki/Entalpie_liber%C4%83" title="Entalpie liberă">Entalpie liberă</a> • <a href="/wiki/Entropia_radia%C8%9Biei_electromagnetice" title="Entropia radiației electromagnetice">Entropia radiației electromagnetice</a> • <a href="/wiki/Entropia_termodinamic%C4%83_(dup%C4%83_Carath%C3%A9odory)" title="Entropia termodinamică (după Carathéodory)">Entropia termodinamică (după Carathéodory)</a> • <a href="/wiki/Entropie" title="Entropie">Entropie</a> • <a href="/wiki/Entropie_termodinamic%C4%83" title="Entropie termodinamică">Entropie termodinamică</a> • <a href="/wiki/Evaporare" title="Evaporare">Evaporare</a> • <a href="/wiki/Faz%C4%83_(termodinamic%C4%83)" title="Fază (termodinamică)">Fază (termodinamică)</a> • <a href="/wiki/Fierbere" title="Fierbere">Fierbere</a> • <a href="/wiki/Formula_lui_Planck" title="Formula lui Planck">Formula lui Planck</a> • <a href="/wiki/Frac%C8%9Bie_molar%C4%83" title="Fracție molară">Fracție molară</a> • <a href="/wiki/Gaz_ideal" title="Gaz ideal">Gaz ideal</a> • <a href="/wiki/Gaz_perfect" title="Gaz perfect">Gaz perfect</a> • <a href="/wiki/Gaz_real" title="Gaz real">Gaz real</a> • <a href="/wiki/Legea_Boyle-Mariotte" title="Legea Boyle-Mariotte">Legea Boyle-Mariotte</a> • <a href="/wiki/Legea_Dulong-Petit" title="Legea Dulong-Petit">Legea Dulong-Petit</a> • <a href="/wiki/Legea_lui_Avogadro" title="Legea lui Avogadro">Legea lui Avogadro</a> • <a href="/wiki/Legea_lui_Dalton" title="Legea lui Dalton">Legea lui Dalton</a> • <a href="/wiki/Legea_lui_Henry" title="Legea lui Henry">Legea lui Henry</a> • <a href="/wiki/Legea_lui_Raoult" title="Legea lui Raoult">Legea lui Raoult</a> • <a href="/wiki/Legile_de_deplasare_ale_lui_Wien" title="Legile de deplasare ale lui Wien">Legile de deplasare ale lui Wien</a> • <a href="/wiki/Legile_lui_Kirchhoff_(radia%C8%9Bie)" title="Legile lui Kirchhoff (radiație)">Legile lui Kirchhoff (radiație)</a> • <a href="/wiki/Lema_lui_Carath%C3%A9odory_(termodinamic%C4%83)" title="Lema lui Carathéodory (termodinamică)">Lema lui Carathéodory (termodinamică)</a> • <a href="/wiki/M%C4%83rimi_molare_de_exces" title="Mărimi molare de exces">Mărimi molare de exces</a> • <a href="/wiki/Paradoxul_lui_Gibbs_(termodinamic%C4%83)" title="Paradoxul lui Gibbs (termodinamică)">Paradoxul lui Gibbs (termodinamică)</a> • <a href="/wiki/Perpetuum_mobile" title="Perpetuum mobile">Perpetuum mobile</a> • <a href="/wiki/Poten%C8%9Bial_chimic" title="Potențial chimic">Potențial chimic</a> • <a href="/wiki/Poten%C8%9Bial_termodinamic" title="Potențial termodinamic">Potențial termodinamic</a> • <a href="/wiki/Presiune_de_vapori" title="Presiune de vapori">Presiune de vapori</a> • <a href="/wiki/Principiile_termodinamicii" title="Principiile termodinamicii">Principiile termodinamicii</a> • <a href="/wiki/Principiul_al_doilea_al_termodinamicii" title="Principiul al doilea al termodinamicii">Principiul al doilea al termodinamicii</a> • <a href="/wiki/Principiul_al_doilea_al_termodinamicii:_Planck_versus_Carath%C3%A9odory" title="Principiul al doilea al termodinamicii: Planck versus Carathéodory">Principiul al doilea al termodinamicii: Planck versus Carathéodory</a> • <a href="/wiki/Principiul_al_treilea_al_termodinamicii" title="Principiul al treilea al termodinamicii">Principiul al treilea al termodinamicii</a> • <a href="/wiki/Principiul_%C3%AEnt%C3%A2i_al_termodinamicii" title="Principiul întâi al termodinamicii">Principiul întâi al termodinamicii</a> • <a href="/wiki/Principiul_zero_al_termodinamicii" title="Principiul zero al termodinamicii">Principiul zero al termodinamicii</a> • <a href="/wiki/Proces_adiabatic" title="Proces adiabatic">Proces adiabatic</a> • <a href="/wiki/Punct_de_fierbere" title="Punct de fierbere">Punct de fierbere</a> • <a href="/wiki/Punct_de_topire" title="Punct de topire">Punct de topire</a> • <a href="/wiki/Radia%C8%9Bie_termic%C4%83" title="Radiație termică">Radiație termică</a> • <a href="/wiki/Rela%C8%9Bia_lui_Mayer" title="Relația lui Mayer">Relația lui Mayer</a> • <a href="/wiki/Rezonatorul_lui_Planck" title="Rezonatorul lui Planck">Rezonatorul lui Planck</a> • <a href="/wiki/Sistem_termodinamic" title="Sistem termodinamic">Sistem termodinamic</a> • <a href="/wiki/Temperatur%C4%83" title="Temperatură">Temperatură</a> • <a href="/wiki/Termochimie" title="Termochimie">Termochimie</a> • <a href="/wiki/Termodinamic%C4%83" title="Termodinamică">Termodinamică</a> • <a href="/wiki/Transformare_Legendre" title="Transformare Legendre">Transformare Legendre</a> • <a href="/wiki/Transformare_termodinamic%C4%83" title="Transformare termodinamică">Transformare termodinamică</a> • <a href="/wiki/Termodinamic%C4%83_chimic%C4%83" title="Termodinamică chimică">Termodinamică chimică</a> •</div> </td></tr> <tr style="height:2px"> <td> </td></tr> <tr> <td class="navbox-group" style=";;">Mecanică statistică </td> <td style="text-align:left;border-left:2px solid #fdfdfd;width:100%;padding:0px;;;" class="navbox-list navbox-even"><div style="padding:0em 0.25em"> <a href="/wiki/Entropie_statistic%C4%83" title="Entropie statistică">Entropie statistică</a> • <a href="/wiki/Faz%C4%83_(mecanic%C4%83_statistic%C4%83)" title="Fază (mecanică statistică)">Fază (mecanică statistică)</a> • <a href="/wiki/Grad_de_libertate" title="Grad de libertate">Grad de libertate</a> • <a href="/wiki/Mecanic%C4%83_statistic%C4%83" title="Mecanică statistică">Mecanică statistică</a> • <a class="mw-selflink selflink">Operator statistic</a></div> </td></tr> <tr style="height:2px"> <td> </td></tr> <tr> <td class="navbox-group" style=";;">Teorie cinetică </td> <td style="text-align:left;border-left:2px solid #fdfdfd;width:100%;padding:0px;;;" class="navbox-list navbox-odd"><div style="padding:0em 0.25em"> <a href="/wiki/Agita%C8%9Bie_termic%C4%83" class="mw-redirect" title="Agitație termică">Agitație termică</a> • <a href="/wiki/Constanta_Boltzmann" title="Constanta Boltzmann">Constanta Boltzmann</a> • <a href="/wiki/Demonul_lui_Maxwell" title="Demonul lui Maxwell">Demonul lui Maxwell</a> • <a href="/wiki/Num%C4%83rul_lui_Avogadro" title="Numărul lui Avogadro">Numărul lui Avogadro</a> • <a href="/wiki/Teoria_cinetic%C4%83_a_gazelor" title="Teoria cinetică a gazelor">Teoria cinetică a gazelor</a> • <a href="/wiki/Teoria_haosului" title="Teoria haosului">Teoria haosului</a></div> </td></tr></tbody></table> </td></tr></tbody></table><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16513826">    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Adus de la <a dir="ltr" href="https://ro.wikipedia.org/w/index.php?title=Operator_statistic&oldid=16375514">https://ro.wikipedia.org/w/index.php?title=Operator_statistic&oldid=16375514</a></div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:Categorii" title="Special:Categorii">Categorii</a>: <ul><li><a href="/wiki/Categorie:Mecanic%C4%83_cuantic%C4%83" title="Categorie:Mecanică cuantică">Mecanică cuantică</a></li><li><a href="/wiki/Categorie:Fizic%C4%83_statistic%C4%83" title="Categorie:Fizică 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Operator statistic - Wikipedia