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Display information for equation id:math.2562.12 on revision:2562

* Page found: Das Photonengas im Strahlungshohlraum (eq math.2562.12)

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\begin{align}

& \bar{N}=2\sum\limits_{{\bar{q}}}^{{}}{{}}\left\langle {{N}_{q}} \right\rangle  \\

& \Rightarrow \bar{N}=\frac{8\pi V}{{{\left( 2\pi  \right)}^{3}}{{c}^{3}}}\int_{0}^{\infty }{{}}d\omega {{\omega }^{2}}\left\langle N(\omega ) \right\rangle =\frac{8\pi V}{{{c}^{3}}}\int_{0}^{\infty }{{}}d\nu {{\nu }^{2}}\left\langle {{N}_{\nu }} \right\rangle  \\

& \bar{N}:=\int_{0}^{\infty }{{}}d\nu D\left( \nu  \right)\left\langle {{N}_{\nu }} \right\rangle  \\

& \Rightarrow D\left( \nu  \right)=\frac{8\pi V}{{{c}^{3}}}{{\nu }^{2}} \\

& \bar{N}=\frac{8\pi V}{{{c}^{3}}}\int_{0}^{\infty }{{}}d\nu {{\nu }^{2}}\left\langle {{N}_{\nu }} \right\rangle =\int_{0}^{\infty }{{}}d\nu D\left( \nu  \right)\left\langle {{N}_{\nu }} \right\rangle  \\

& U=\int_{0}^{\infty }{{}}d\nu D\left( \nu  \right)h\nu \left\langle {{N}_{\nu }} \right\rangle  \\

\end{align}

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N¯=2∑q¯⟨Nq⟩⇒N¯=8πV(2π)3c3∫0∞dωω2⟨N(ω)⟩=8πVc3∫0∞dνν2⟨Nν⟩N¯:=∫0∞dνD(ν)⟨Nν⟩⇒D(ν)=8πVc3ν2N¯=8πVc3∫0∞dνν2⟨Nν⟩=∫0∞dνD(ν)⟨Nν⟩U=∫0∞dνD(ν)hν⟨Nν⟩
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data-mjx-texclass="ORD"><mn>8</mn><mi>π</mi><mi>V</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mn>2</mn><mi>π</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><msup><mi>c</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mrow></mrow></mfrac></mrow><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>d</mi><mi>ω</mi><msup><mi>ω</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><mi>N</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo data-mjx-texclass="CLOSE">⟩</mo></mrow><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>8</mn><mi>π</mi><mi>V</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><msup><mi>c</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mrow></mfrac></mrow><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>d</mi><mi>ν</mi><msup><mi>ν</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><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"><mover><mi>N</mi><mo>¯</mo></mover><mo stretchy="false">:=</mo><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>d</mi><mi>ν</mi><mi>D</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ν</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><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"><mo stretchy="false">⇒</mo><mi>D</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ν</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>8</mn><mi>π</mi><mi>V</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><msup><mi>c</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mrow></mfrac></mrow><msup><mi>ν</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mover><mi>N</mi><mo>¯</mo></mover><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>8</mn><mi>π</mi><mi>V</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><msup><mi>c</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mrow></mfrac></mrow><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>d</mi><mi>ν</mi><msup><mi>ν</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><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><mo stretchy="false">=</mo><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>d</mi><mi>ν</mi><mi>D</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ν</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><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>U</mi><mo stretchy="false">=</mo><msubsup><mo stretchy="false">∫</mo><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>d</mi><mi>ν</mi><mi>D</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ν</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><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></mtr></mtable></mstyle></mrow></math>

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