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Display information for equation id:math.2563.25 on revision:2563

* Page found: Das Photonengas im Strahlungshohlraum (eq math.2563.25)

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TeX (original user input): 

\begin{align}

& p=kT{{\left( \frac{\partial }{\partial V}\ln Z \right)}_{T}}=-kT\sum\limits_{\nu }^{{}}{{}}\frac{\frac{h}{kT}\left( \frac{\partial \nu }{\partial V} \right){{e}^{-\frac{h\nu }{kT}}}}{\left( 1-{{e}^{-\frac{h\nu }{kT}}} \right)}=kT\sum\limits_{\nu }^{{}}{{}}\frac{\frac{h}{kT}\frac{\nu }{3V}{{e}^{-\frac{h\nu }{kT}}}}{\left( 1-{{e}^{-\frac{h\nu }{kT}}} \right)}=\frac{1}{3V}\sum\limits_{\nu }^{{}}{{}}\frac{h\nu {{e}^{-\frac{h\nu }{kT}}}}{\left( 1-{{e}^{-\frac{h\nu }{kT}}} \right)}= \\

& \Rightarrow p=\frac{1}{3V}\sum\limits_{\nu }^{{}}{{}}h\nu \left\langle {{N}_{\nu }} \right\rangle  \\

& p=\frac{1}{3}\frac{U}{V} \\

\end{align}

TeX (checked): 

{\begin{aligned}&p=kT{{\left({\frac {\partial }{\partial V}}\ln Z\right)}_{T}}=-kT\sum \limits _{\nu }^{}{}{\frac {{\frac {h}{kT}}\left({\frac {\partial \nu }{\partial V}}\right){{e}^{-{\frac {h\nu }{kT}}}}}{\left(1-{{e}^{-{\frac {h\nu }{kT}}}}\right)}}=kT\sum \limits _{\nu }^{}{}{\frac {{\frac {h}{kT}}{\frac {\nu }{3V}}{{e}^{-{\frac {h\nu }{kT}}}}}{\left(1-{{e}^{-{\frac {h\nu }{kT}}}}\right)}}={\frac {1}{3V}}\sum \limits _{\nu }^{}{}{\frac {h\nu {{e}^{-{\frac {h\nu }{kT}}}}}{\left(1-{{e}^{-{\frac {h\nu }{kT}}}}\right)}}=\\&\Rightarrow p={\frac {1}{3V}}\sum \limits _{\nu }^{}{}h\nu \left\langle {{N}_{\nu }}\right\rangle \\&p={\frac {1}{3}}{\frac {U}{V}}\\\end{aligned}}

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p=kT(∂∂Vln⁡Z)T=−kT∑νhkT(∂ν∂V)e−hνkT(1−e−hνkT)=kT∑νhkTν3Ve−hνkT(1−e−hνkT)=13V∑νhνe−hνkT(1−e−hνkT)=⇒p=13V∑νhν⟨Nν⟩p=13UV
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data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>h</mi><mi>ν</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>T</mi></mrow></mrow></mfrac></mrow></mrow></mrow></msup></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mn>1</mn><mo stretchy="false">−</mo><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>h</mi><mi>ν</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>T</mi></mrow></mrow></mfrac></mrow></mrow></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mfrac></mrow><mo stretchy="false">=</mo></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><mi>p</mi><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>3</mn><mi>V</mi></mrow></mrow></mfrac></mrow><munderover><mo form="prefix" movablelimits="false" stretchy="false">∑</mo><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><mi>h</mi><mi>ν</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>N</mi><mrow data-mjx-texclass="ORD"><mi>ν</mi></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mi>p</mi><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>U</mi></mrow><mrow data-mjx-texclass="ORD"><mi>V</mi></mrow></mfrac></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

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  • Z
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  • ν
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  • Nν
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