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<title>Kubo-Formel</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Kubo-Formel</h1>

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Die Kubo-Formel (nach <a href="Ry%C5%8Dgo_Kubo" title="Ryōgo Kubo">Ryōgo Kubo</a>)<a href="#cite_note-Kubo_I-1">[1]</a><a href="#cite_note-Kubo_II-2">[2]</a>

ist ein Resultat der <a href="Quantenstatistik" title="Quantenstatistik">Quantenstatistik</a>. Sie gibt die <a href="Lineare_Antwortfunktion" title="Lineare Antwortfunktion">Lineare Antwortfunktion</a> einer messbaren Größe (<a href="Observable" title="Observable">Observable</a>) in zeitabhängiger <a href="St%C3%B6rungstheorie_(Quantenmechanik)" title="Störungstheorie (Quantenmechanik)">Störungstheorie</a> bei endlicher <a href="Temperatur" title="Temperatur">Temperatur</a> als thermischen <a href="Erwartungswert" title="Erwartungswert">Erwartungswert</a> <a href="Hermitescher_Operator" title="Hermitescher Operator">hermitescher Operatoren</a> im <a href="Wechselwirkungsbild" title="Wechselwirkungsbild">Wechselwirkungsbild</a> an.

Zu den zahlreichen Anwendungen der Kubo-Formel gehört die Berechnung <a href="Magnetische_Suszeptibilit%C3%A4t" title="Magnetische Suszeptibilität">magnetischer</a> und <a href="Elektrische_Suszeptibilit%C3%A4t" title="Elektrische Suszeptibilität">elektrischer Suszeptibilitäten</a> und abstrakter Verallgemeinerungen davon als Folge einer zeitabhängigen Störung des <a href="Hamiltonoperator" title="Hamiltonoperator">Hamiltonoperators</a> des Systems.

<div class="mw-heading mw-heading2"><h2 id="Details">Details</h2></div>

Die Kubo-Formel führt auf eine Beziehung zwischen

<ul><li>dem quantenstatistischen Erwartungswert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle A\rangle _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \langle A\rangle _{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eeb76a76aa3dc0d1b8af8ebfbc6bde9979fb3dca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.607ex; height:2.843ex;" alt="{\displaystyle \langle A\rangle _{0}}" loading="lazy"> einer Observable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> in einem ungestörten System mit <a href="Hamilton-Operator" class="mw-redirect" title="Hamilton-Operator">Hamilton-Operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle H_{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43910602a221b7a4c373791f94793e3008622070.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{0}}" loading="lazy"> zu einer Zeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02d3006c4190b1939b04d9b9bb21006fb4e6fa4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}" loading="lazy"> und</li>

<li>dem Erwartungswert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle A(t)\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \langle A(t)\rangle }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7e743cd79c098508f6b4e8d50bbf9652a2715ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.201ex; height:2.843ex;" alt="{\displaystyle \langle A(t)\rangle }" loading="lazy"> derselben Observable nach Einführung einer kleinen Störung des Systems in Form eines Störoperators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(t)}">

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<annotation encoding="application/x-tex">{\displaystyle V(t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/383b47023708ac9df0e198448491048ec7acb2cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.436ex; height:2.843ex;" alt="{\displaystyle V(t)}" loading="lazy"> zu einer Zeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \langle A(t)\rangle -\langle A\rangle _{0}=-\mathrm {i} \int _{t_{0}}^{t}\mathrm {d} t'\langle [A(t),V(t')]\rangle }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0de318f6b6c8c45cc9cc2d7d2e512b0e314559c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.733ex; height:6.509ex;" alt="{\displaystyle \langle A(t)\rangle -\langle A\rangle _{0}=-\mathrm {i} \int _{t_{0}}^{t}\mathrm {d} t'\langle [A(t),V(t')]\rangle }" loading="lazy"></dd></dl>

Dabei bezeichnen

<ul><li>spitze Klammern den quantenstatistischen Erwartungswert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle A\rangle =\operatorname {Tr} [\rho A]}">

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<annotation encoding="application/x-tex">{\displaystyle \langle A\rangle =\operatorname {Tr} [\rho A]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a2d39c6b57d8011d88a2c0554104f048e90a48dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.479ex; height:2.843ex;" alt="{\displaystyle \langle A\rangle =\operatorname {Tr} [\rho A]}" loading="lazy"> mit der <a href="Dichtematrix" class="mw-redirect" title="Dichtematrix">Dichtematrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"></li>

<li>eckige Klammern den <a href="Kommutator_(Mathematik)" title="Kommutator (Mathematik)">Kommutator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [A,V]=AV-VA}">

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<annotation encoding="application/x-tex">{\displaystyle [A,V]=AV-VA}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d8d27742b14b6960aac77189da91ce01ea77052.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.857ex; height:2.843ex;" alt="{\displaystyle [A,V]=AV-VA}" loading="lazy"></li>

<li>ein <a href="Subskript" class="mw-redirect" title="Subskript">Subskript</a> Null das ungestörte System</li>

<li>i die <a href="Imagin%C3%A4re_Einheit" class="mw-redirect" title="Imaginäre Einheit">imaginäre Einheit</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Herleitung_und_Formulierung">Herleitung und Formulierung</h2></div>

Ein Quantensystem habe den zeitunabhängigen <a href="Hamiltonoperator" title="Hamiltonoperator">Hamiltonoperator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {H}}_{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9747fcac2bfac1d234c2dccd5cb0266995d03b2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.118ex; height:3.176ex;" alt="{\displaystyle {\hat {H}}_{0}}" loading="lazy"> mit den als diskret angenommenen Energiewerten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle E_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad6b82f2a00af6c9efd4c16d4e99329605645c0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.934ex; height:2.509ex;" alt="{\displaystyle E_{n}}" loading="lazy">. Der quantenmechanische und thermische Erwartungswert einer physikalischen Größe mit dem <a href="Hermitescher_Operator#Definitionen" title="Hermitescher Operator">hermiteschen Operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {A}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {A}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f595a6c73d1183d6a1b2ac21fe47ac28c1483821.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.843ex;" alt="{\displaystyle {\hat {A}}}" loading="lazy"> ist dann:

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<annotation encoding="application/x-tex">{\displaystyle \langle {\hat {A}}\rangle ={1 \over Z_{0}}\operatorname {Tr} [{\hat {\rho }}_{0}{\hat {A}}]={1 \over Z_{0}}\sum _{n}\langle n|{\hat {A}}|n\rangle {\mathrm {e} }^{-\beta E_{n}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d87f267c9d125e7f94770b33483b02095697aa4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.286ex; height:6.343ex;" alt="{\displaystyle \langle {\hat {A}}\rangle ={1 \over Z_{0}}\operatorname {Tr} [{\hat {\rho }}_{0}{\hat {A}}]={1 \over Z_{0}}\sum _{n}\langle n|{\hat {A}}|n\rangle {\mathrm {e} }^{-\beta E_{n}}}" loading="lazy"></dd></dl>

<msub>

<mover>

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\rho }}_{0}=e^{-\beta {\hat {H}}_{0}}=\sum _{n}|n\rangle \langle n|\mathrm {e} ^{-\beta E_{n}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b01613542a89e87d4dcc8e3516adb06bba7ed17a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.751ex; height:5.843ex;" alt="{\displaystyle {\hat {\rho }}_{0}=e^{-\beta {\hat {H}}_{0}}=\sum _{n}|n\rangle \langle n|\mathrm {e} ^{-\beta E_{n}}}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{0}=\operatorname {Tr} [\rho _{0}]}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle Z_{0}=\operatorname {Tr} [\rho _{0}]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fec092068572a8b2c1babab221887cb0e9826a0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.88ex; height:2.843ex;" alt="{\displaystyle Z_{0}=\operatorname {Tr} [\rho _{0}]}" loading="lazy"> die <a href="Zustandssumme" title="Zustandssumme">Zustandssumme</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

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<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> die reziproke absolute Temperatur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/(k_{\text{B}}T)}">

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<annotation encoding="application/x-tex">{\displaystyle 1/(k_{\text{B}}T)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8afc1888e27f18b8839d55d668d735e66c821454.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.378ex; height:2.843ex;" alt="{\displaystyle 1/(k_{\text{B}}T)}" loading="lazy"> mit der <a href="Boltzmann-Konstante" title="Boltzmann-Konstante">Boltzmann-Konstanten</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{\text{B}}}">

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<annotation encoding="application/x-tex">{\displaystyle k_{\text{B}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9582c7c795d2def2c061f0dfa3a6f0fb3dd2de44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.607ex; height:2.509ex;" alt="{\displaystyle k_{\text{B}}}" loading="lazy"> und der Temperatur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> ist. Im letzten Gleichheitszeichen wurde dabei nach den ungestörten Energieeigenzuständen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |n\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |n\rangle }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89c45bda6ac852da785c06476f2d3e3b6cba812a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.946ex; height:2.843ex;" alt="{\displaystyle |n\rangle }" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}_{0}|n\rangle =E_{n}|n\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {H}}_{0}|n\rangle =E_{n}|n\rangle }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94ff659001a6610e0d74faf9bbc6102df747af50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.042ex; height:3.343ex;" alt="{\displaystyle {\hat {H}}_{0}|n\rangle =E_{n}|n\rangle }" loading="lazy"> entwickelt und deren Vollständigkeit ausgenutzt.

Wenn zur Zeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=t_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle t=t_{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be6d7492e2d48bf34fdd5dffa189b188c140820c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.832ex; height:2.343ex;" alt="{\displaystyle t=t_{0}}" loading="lazy"> eine externe Störung eingeschaltet wird, verlässt das System das thermische Gleichgewicht. Die Störung wird durch einen zeitabhängigen Zusatz zum Hamiltonoperator beschrieben:

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<annotation encoding="application/x-tex">{\displaystyle {\hat {H}}(t)={\hat {H}}_{0}+{\hat {V}}(t)\Theta (t-t_{0}),}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89d4e4a343d60283a2f2c2be7eab0ed7494222ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.044ex; height:3.343ex;" alt="{\displaystyle {\hat {H}}(t)={\hat {H}}_{0}+{\hat {V}}(t)\Theta (t-t_{0}),}" loading="lazy"></dd></dl>

Dabei bezeichnet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (t)}">

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<annotation encoding="application/x-tex">{\displaystyle \Theta (t)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/415d180b65fdc75e95cb50e5786e6c56989dd416.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.457ex; height:2.843ex;" alt="{\displaystyle \Theta (t)}" loading="lazy"> die <a href="Heaviside-Funktion" title="Heaviside-Funktion">Heaviside-Funktion</a>, die für nichtnegative Werte von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t-t_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle t-t_{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffcdb16b021fb595c51ce737b33b82ec05c9f014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.574ex; height:2.343ex;" alt="{\displaystyle t-t_{0}}" loading="lazy"> den Wert Eins annimmt und für alle anderen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t-t_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle t-t_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffcdb16b021fb595c51ce737b33b82ec05c9f014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.574ex; height:2.343ex;" alt="{\displaystyle t-t_{0}}" loading="lazy"> den Wert Null. Damit wird dem instantanen „Einschaltprozess“ zum Zeitpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02d3006c4190b1939b04d9b9bb21006fb4e6fa4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}" loading="lazy"> Rechnung getragen. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {V}}(t)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {V}}(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/294ac05837b4472c7c1a64c75ec720aafe04efaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.436ex; height:3.343ex;" alt="{\displaystyle {\hat {V}}(t)}" loading="lazy"> ist ein für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> definierter hermitescher Operator, sodass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}(t)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {H}}(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/97f6b5b382f48bb86a9cc6e989f6f7dd4cad2094.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.713ex; height:3.343ex;" alt="{\displaystyle {\hat {H}}(t)}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> ein vollständiges Orthonormalsystem von Eigenfunktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |n(t)\rangle }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |n(t)\rangle }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/828615cd4bf848e2d003c9c886a239484f3487e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.595ex; height:2.843ex;" alt="{\displaystyle |n(t)\rangle }" loading="lazy"> und Eigenwerten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}(t)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{n}(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a2569e64043323bc78ce8307125916a63c29cf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.583ex; height:2.843ex;" alt="{\displaystyle E_{n}(t)}" loading="lazy"> besitzt.

Aus der Zeitentwicklung der <a href="Dichtematrix" class="mw-redirect" title="Dichtematrix">Dichtematrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\rho }}(t)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\rho }}(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/375488fa20a1c4f902f489eef1d68acdd4e4ae67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.025ex; height:2.843ex;" alt="{\displaystyle {\hat {\rho }}(t)}" loading="lazy">

<mover>

</mover>

</mrow>

</mrow>

</munder>

</mrow>

</mrow>

<msup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\rho }}(t)=\sum _{n}|n(t)\rangle \langle n(t)|\mathrm {e} ^{-\beta E_{n}(t)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72d6f85e88cf90332586f16833371a27d70b86e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.591ex; height:5.509ex;" alt="{\displaystyle {\hat {\rho }}(t)=\sum _{n}|n(t)\rangle \langle n(t)|\mathrm {e} ^{-\beta E_{n}(t)}}" loading="lazy"></dd></dl>

folgt unter der Annahme, dass zu jedem Zeitpunkt der quantenstatistische Gleichgewichtsformalismus gültig bleibt<a href="#cite_note-3">[3]</a>

, der thermische Erwartungswert der Operatoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {A}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {A}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f595a6c73d1183d6a1b2ac21fe47ac28c1483821.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.843ex;" alt="{\displaystyle {\hat {A}}}" loading="lazy">:

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<msup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle {\hat {A}}\rangle ={\frac {1}{Z(t)}}\operatorname {Tr} [{\hat {\rho }}(t){\hat {A}}]={\frac {1}{Z(t)}}\sum _{n}\langle n(t)|{\hat {A}}|n(t)\rangle \mathrm {e} ^{-\beta E_{n}(t)},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c41569952bf782434da6d0cac1a7575da0e85dc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:55.074ex; height:6.343ex;" alt="{\displaystyle \langle {\hat {A}}\rangle ={\frac {1}{Z(t)}}\operatorname {Tr} [{\hat {\rho }}(t){\hat {A}}]={\frac {1}{Z(t)}}\sum _{n}\langle n(t)|{\hat {A}}|n(t)\rangle \mathrm {e} ^{-\beta E_{n}(t)},}" loading="lazy"></dd></dl>

mit der Zustandssumme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(t)=\operatorname {Tr} [{\hat {\rho }}(t)]}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z(t)=\operatorname {Tr} [{\hat {\rho }}(t)]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b6ed60dc75b09c2355e5235cfeda2e641359e99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.336ex; height:2.843ex;" alt="{\displaystyle Z(t)=\operatorname {Tr} [{\hat {\rho }}(t)]}" loading="lazy">.

Hier wird noch das <a href="Mathematische_Struktur_der_Quantenmechanik" class="mw-redirect" title="Mathematische Struktur der Quantenmechanik">quantenmechanische Schrödingerbild</a> benutzt, allerdings mit zeitabhängigen Hamiltonoperatoren. Es wird aber an dieser Stelle darauf hingewiesen, dass sich im Allgemeinen sowohl die Eigenfunktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |n(t)\rangle }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |n(t)\rangle }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/828615cd4bf848e2d003c9c886a239484f3487e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.595ex; height:2.843ex;" alt="{\displaystyle |n(t)\rangle }" loading="lazy"> als auch die Eigenwerte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}(t)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{n}(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a2569e64043323bc78ce8307125916a63c29cf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.583ex; height:2.843ex;" alt="{\displaystyle E_{n}(t)}" loading="lazy"> des Hamiltonoperators mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> ändern werden. Die Zeitabhängigkeit der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |n(t)\rangle }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |n(t)\rangle }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/828615cd4bf848e2d003c9c886a239484f3487e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.595ex; height:2.843ex;" alt="{\displaystyle |n(t)\rangle }" loading="lazy"> folgt aus der <a href="Schr%C3%B6dingergleichung" title="Schrödingergleichung">Schrödingergleichung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {i} \partial _{t}|n(t)\rangle ={\hat {H}}(t)|n(t)\rangle }">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {i} \partial _{t}|n(t)\rangle ={\hat {H}}(t)|n(t)\rangle }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b560044f4bcf39ea89b9d9bb8a455468d1bb9c65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.708ex; height:3.343ex;" alt="{\displaystyle \mathrm {i} \partial _{t}|n(t)\rangle ={\hat {H}}(t)|n(t)\rangle }" loading="lazy"> (hier und im Folgenden wird die <a href="Reduzierte_Planck-Konstante" class="mw-redirect" title="Reduzierte Planck-Konstante">reduzierte Planck-Konstante</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de68de3a92517953436c93b5a76461d49160cc41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }" loading="lazy"> gleich Eins gesetzt). Da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {V}}(t)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {V}}(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/294ac05837b4472c7c1a64c75ec720aafe04efaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.436ex; height:3.343ex;" alt="{\displaystyle {\hat {V}}(t)}" loading="lazy"> „schwach“ sein soll, liegt es nahe, die niedrigste Ordnung der zeitabhängigen Störungstheorie zu benutzen und zum <a href="Wechselwirkungsbild" title="Wechselwirkungsbild">Wechselwirkungsbild</a> überzugehen (Zustände <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |n\rangle \to |{\hat {n}}\rangle }">

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |n\rangle \to |{\hat {n}}\rangle }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1f68bf0e58fb5767f30b746a2b732af85a73383.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.506ex; height:2.843ex;" alt="{\displaystyle |n\rangle \to |{\hat {n}}\rangle }" loading="lazy">). Das Ergebnis ist:

</mrow>

<msup>

</mrow>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</msup>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |n(t)\rangle =:\mathrm {e} ^{-\mathrm {i} {\hat {H}}_{0}t_{0}}|{\hat {n}}(t)\rangle =\mathrm {e} ^{-\mathrm {i} {\hat {H}}_{0}t}{\hat {U}}(t,t_{0})|{\hat {n}}(t_{0})\rangle }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff38acbcaa37729be3f3c769409af4e06264c20d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.644ex; height:3.676ex;" alt="{\displaystyle |n(t)\rangle =:\mathrm {e} ^{-\mathrm {i} {\hat {H}}_{0}t_{0}}|{\hat {n}}(t)\rangle =\mathrm {e} ^{-\mathrm {i} {\hat {H}}_{0}t}{\hat {U}}(t,t_{0})|{\hat {n}}(t_{0})\rangle }" loading="lazy">, wobei per Definition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\hat {n}}(t_{0})\rangle =\mathrm {e} ^{+\mathrm {i} {\hat {H}}_{0}t_{0}}|n(t_{0})\rangle }">

</mrow>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |{\hat {n}}(t_{0})\rangle =\mathrm {e} ^{+\mathrm {i} {\hat {H}}_{0}t_{0}}|n(t_{0})\rangle }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b7dd001319dc7c437ce018a37ee8ab772fa3ed5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.114ex; height:3.676ex;" alt="{\displaystyle |{\hat {n}}(t_{0})\rangle =\mathrm {e} ^{+\mathrm {i} {\hat {H}}_{0}t_{0}}|n(t_{0})\rangle }" loading="lazy"> ist.</dd></dl>

In linearer Ordnung in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {V}}(t)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {V}}(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/294ac05837b4472c7c1a64c75ec720aafe04efaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.436ex; height:3.343ex;" alt="{\displaystyle {\hat {V}}(t)}" loading="lazy"> gilt:

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

<msup>

</msup>

<mover>

</mover>

</mrow>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {U}}(t,t_{0})=1-\mathrm {i} \int _{t_{0}}^{t}\mathrm {d} t'{\hat {V}}(t')}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68beef856cee3088a03cd4cd10052c9b53b971d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.109ex; height:6.509ex;" alt="{\displaystyle {\hat {U}}(t,t_{0})=1-\mathrm {i} \int _{t_{0}}^{t}\mathrm {d} t'{\hat {V}}(t')}" loading="lazy">.</dd></dl>

Auf diese Weise erhält man für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {A}}(t)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {A}}(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8162d2ff06f25fa27d2cb08c270eed3a5fe125df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.425ex; height:3.343ex;" alt="{\displaystyle {\hat {A}}(t)}" loading="lazy"> in linearer Ordnung das Endresultat (in dieser Ordnung sind ferner alle oben angesprochenen Probleme beseitigt, weil bei Störungsrechnungen erster Ordnung nur die Eigenfunktionen nullter Ordnung benötigt werden):

<mtr>

<mtd>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

<msup>

</msup>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</munder>

<msup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</msup>

<mover>

</mover>

</mrow>

<msup>

</msup>

<mover>

</mover>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

<msup>

</msup>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<msup>

</msup>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\langle {\hat {A}}(t)\rangle _{T}&=\langle {\hat {A}}\rangle _{0,T}-\mathrm {i} \int _{t_{0}}^{t}\mathrm {d} t'{1 \over Z_{0}}\ \sum _{n}\mathrm {e} ^{-\beta E_{n}}\langle n(t_{0})|{\hat {A}}(t){\hat {V}}(t')-{\hat {V}}(t'){\hat {A}}(t)|n(t_{0})\rangle \\&=\langle {\hat {A}}\rangle _{0,T}-\mathrm {i} \int _{t_{0}}^{t}\mathrm {d} t'\langle [{\hat {A}}(t),{\hat {V}}(t')]\rangle _{0,T}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4af293a553600104d821a56f343a1fc612287505.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.894ex; margin-bottom: -0.277ex; width:77.175ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}\langle {\hat {A}}(t)\rangle _{T}&=\langle {\hat {A}}\rangle _{0,T}-\mathrm {i} \int _{t_{0}}^{t}\mathrm {d} t'{1 \over Z_{0}}\ \sum _{n}\mathrm {e} ^{-\beta E_{n}}\langle n(t_{0})|{\hat {A}}(t){\hat {V}}(t')-{\hat {V}}(t'){\hat {A}}(t)|n(t_{0})\rangle \\&=\langle {\hat {A}}\rangle _{0,T}-\mathrm {i} \int _{t_{0}}^{t}\mathrm {d} t'\langle [{\hat {A}}(t),{\hat {V}}(t')]\rangle _{0,T}\end{aligned}}}" loading="lazy"></dd></dl>

Hier bedeutet der Ausdruck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \dots \rangle _{0,T}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \dots \rangle _{0,T}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/407409419de11452f0d4be003f8d6eadb9cdf6d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.201ex; height:3.009ex;" alt="{\displaystyle \langle \dots \rangle _{0,T}}" loading="lazy"> einen mit dem Hamiltonoperator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}_{0}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {H}}_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9747fcac2bfac1d234c2dccd5cb0266995d03b2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.118ex; height:3.176ex;" alt="{\displaystyle {\hat {H}}_{0}}" loading="lazy"> berechneten quantenstatistischen Erwartungswert, bei der Temperatur <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy">, während die Ausdrücke darüber, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle n(t_{0})|\dots |n(t_{0})\rangle ,}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle n(t_{0})|\dots |n(t_{0})\rangle ,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82020bc4cad7fe3786df6ad48555182f87141e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.443ex; height:2.843ex;" alt="{\displaystyle \langle n(t_{0})|\dots |n(t_{0})\rangle ,}" loading="lazy"> gewöhnliche quantenmechanische Erwartungswerte sind, welche die Temperatur nicht berücksichtigen. Ferner sind in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {e} ^{-\beta E_{n}}}">

<msup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {e} ^{-\beta E_{n}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63af1b34fc1beda9ca915c5bb0b9f3a13b63efd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.663ex; height:2.676ex;" alt="{\displaystyle \mathrm {e} ^{-\beta E_{n}}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad6b82f2a00af6c9efd4c16d4e99329605645c0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.934ex; height:2.509ex;" alt="{\displaystyle E_{n}}" loading="lazy"> die Eigenwerte von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}(t_{0})}">

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {H}}(t_{0})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ac79de3e377ca281471da4499e9f39a3fe83151.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.767ex; height:3.343ex;" alt="{\displaystyle {\hat {H}}(t_{0})}" loading="lazy"> gemeint.

Da zum Zeitpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02d3006c4190b1939b04d9b9bb21006fb4e6fa4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}" loading="lazy"> die verschiedenen Bilder identisch sind, gilt dasselbe auch für obiges Endresultat.

Hier wurden <a href="Boson" title="Boson">bosonische</a> Zustände betrachtet. Für <a href="Fermion" title="Fermion">fermionische</a> Zustände ergeben sich zusätzliche Besonderheiten.<a href="#cite_note-Mahan-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise_und_Fußnoten">Einzelnachweise und Fußnoten</h2></div>

<li id="cite_note-Kubo_I-1"><a href="#cite_ref-Kubo_I_1-0">↑</a> Ryogo Kubo: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems</cite>. In: <cite class="lang" lang="en" dir="auto" style="font-style:italic">J. Phys. Soc. Jpn.</cite> 12. Jahrgang, 1957, S. 570–586, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1143/JPSJ.12.570">10.1143/JPSJ.12.570</a> (englisch, <a rel="nofollow" class="external text" href="https://journals.jps.jp/doi/pdf/10.1143/JPSJ.12.570">jps.jp</a> [PDF]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Kubo-Formel&rft.atitle=Statistical-Mechanical+Theory+of+Irreversible+Processes.+I.+General+Theory+and+Simple+Applications+to+Magnetic+and+Conduction+Problems&rft.au=Ryogo%26%2332%3BKubo&rft.btitle=J.+Phys.+Soc.+Jpn.&rft.date=1957&rft.doi=10.1143%2FJPSJ.12.570&rft.genre=book&rft.pages=570-586&rft.volume=12.+Jahrgang" style="display:none">

</li>

<li id="cite_note-Kubo_II-2"><a href="#cite_ref-Kubo_II_2-0">↑</a> Ryogo Kubo, Mario Yokota, Sadao Nakajima: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Statistical-Mechanical Theory of Irreversible Processes. II. Response to Thermal Disturbance</cite>. In: <cite class="lang" lang="en" dir="auto" style="font-style:italic">J. Phys. Soc. Jpn.</cite> 12. Jahrgang, 1957, S. 1203–1211, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1143/JPSJ.12.1203">10.1143/JPSJ.12.1203</a> (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Kubo-Formel&rft.atitle=Statistical-Mechanical+Theory+of+Irreversible+Processes.+II.+Response+to+Thermal+Disturbance&rft.au=Ryogo%26%2332%3BKubo%2C%26%2332%3BMario%26%2332%3BYokota%2C%26%2332%3BSadao%26%2332%3BNakajima&rft.btitle=J.+Phys.+Soc.+Jpn.&rft.date=1957&rft.doi=10.1143%2FJPSJ.12.1203&rft.genre=book&rft.pages=1203-1211&rft.volume=12.+Jahrgang" style="display:none">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Im allgemeinsten Fall der Quantenstatistik kann <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\rho }}(t)=\mathrm {e} ^{-\beta {\hat {H}}(t)}}">

<mover>

</mover>

</mrow>

<msup>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\rho }}(t)=\mathrm {e} ^{-\beta {\hat {H}}(t)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f367d09f0dff1e273d8af4bdd59bd83989f5c5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.94ex; height:3.676ex;" alt="{\displaystyle {\hat {\rho }}(t)=\mathrm {e} ^{-\beta {\hat {H}}(t)}}" loading="lazy"> durch einen beliebigen hermiteschen Operator ersetzt werden, dessen Eigenwerte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c5421b423b193df0b692addd2e4cf025e52bc09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.883ex; height:2.009ex;" alt="{\displaystyle w_{n}}" loading="lazy"> die beiden Bedingungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum w_{n}=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum w_{n}=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14d8e4938b775b30e294044e21104c6ef3944033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.886ex; height:3.843ex;" alt="{\displaystyle \sum w_{n}=1}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum w_{n}^{2}<1}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum w_{n}^{2}<1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/313f0f76b35d2eaa23d2d0e7df502a684651791b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.886ex; height:3.843ex;" alt="{\displaystyle \sum w_{n}^{2}<1}" loading="lazy"> erfüllen.

</li>

<li id="cite_note-Mahan-4"><a href="#cite_ref-Mahan_4-0">↑</a> GD Mahan: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Many-particle physics</cite>. Springer, New York 1981, ISBN 0-306-46338-5 (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Kubo-Formel&rft.au=GD%26%2332%3BMahan&rft.btitle=Many-particle+physics&rft.date=1981&rft.genre=book&rft.isbn=0306463385&rft.place=New+York&rft.pub=Springer" style="display:none">

</li>

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