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Display information for equation id:math.2569.12 on revision:2569

* Page found: Das Photonengas im Strahlungshohlraum (eq math.2569.12)

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Das Photonengas im Strahlungshohlraum Eq: math.2569.12

Hash: d54c52deb59a6f4443e3acc3c8c52d8d

TeX (original user input):

\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}

TeX (checked):

{\begin{aligned}&{\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{aligned}}

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\begin{matrix} \bar{N} = 2 \sum_{\bar{q}}^{} ⟨ N_{q} ⟩ \\ \Rightarrow \bar{N} = \frac{8 π V}{{(2 π)}^{3} c^{3}} \int_{0}^{\infty} d ω ω^{2} ⟨ N (ω) ⟩ = \frac{8 π V}{c^{3}} \int_{0}^{\infty} d ν ν^{2} ⟨ N_{ν} ⟩ \\ \bar{N} := \int_{0}^{\infty} d ν D (ν) ⟨ N_{ν} ⟩ \\ \Rightarrow D (ν) = \frac{8 π V}{c^{3}} ν^{2} \\ \bar{N} = \frac{8 π V}{c^{3}} \int_{0}^{\infty} d ν ν^{2} ⟨ N_{ν} ⟩ = \int_{0}^{\infty} d ν D (ν) ⟨ N_{ν} ⟩ \\ U = \int_{0}^{\infty} d ν D (ν) h ν ⟨ N_{ν} ⟩ \end{matrix}

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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 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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 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Identifiers

$\bar{N}$

$\bar{q}$

$N_{q}$

$\bar{N}$

$π$

$V$

$π$

$c$

$ω$

$ω$

$N$

$ω$

$π$

$V$

$c$

$ν$

$ν$

$N_{ν}$

$\bar{N}$

$ν$

$D$

$ν$

$N_{ν}$

$D$

$ν$

$π$

$V$

$c$

$ν$

$\bar{N}$

$π$

$V$

$c$

$ν$

$ν$

$N_{ν}$

$ν$

$D$

$ν$

$N_{ν}$

$U$

$ν$

$D$

$ν$

$h$

$ν$

$N_{ν}$

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