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Display information for equation id:math.2504.19 on revision:2504
* Page found: Das ideale Gas (eq math.2504.19)
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\begin{align}
& \Rightarrow {{e}^{-\Psi }}=Y=\sum\limits_{N=0}^{\infty }{{}}\frac{1}{N!}\exp \left( -\alpha N \right)\frac{1}{{{\hbar }^{3N}}}\int_{V}^{{}}{{}}{{d}^{3}}{{q}_{1}}...\int_{V}^{{}}{{}}{{d}^{3}}{{q}_{N}}\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}{{p}_{1}}...\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}{{p}_{N}}\exp \left( -\frac{\beta }{2m}\sum\limits_{i}^{{}}{{}}{{p}_{i}}^{2} \right) \\
& =\sum\limits_{N=0}^{\infty }{{}}\frac{1}{N!}\exp \left( -\alpha N \right){{V}^{N}}{{\left[ \frac{1}{{{\hbar }^{3}}}\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}p\exp \left( -\frac{\beta {{p}^{2}}}{2m} \right) \right]}^{N}} \\
& \left[ \frac{1}{{{\hbar }^{3}}}\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}p\exp \left( -\frac{\beta {{p}^{2}}}{2m} \right) \right]=\frac{1}{{{\hbar }^{3}}}{{\left( \frac{2\pi m}{\beta } \right)}^{\frac{3}{2}}}\equiv \Phi \left( \beta \right) \\
& \Rightarrow {{e}^{-\Psi }}=Y=\sum\limits_{N=0}^{\infty }{{}}\frac{1}{N!}\exp \left( -\alpha N \right){{V}^{N}}{{\left[ \frac{1}{{{\hbar }^{3}}}\int_{{{R}^{3}}}^{{}}{{}}{{d}^{3}}p\exp \left( -\frac{\beta {{p}^{2}}}{2m} \right) \right]}^{N}}=\sum\limits_{N=0}^{\infty }{{}}\frac{1}{N!}{{\left[ V\Phi \left( \beta \right){{e}^{-\alpha }} \right]}^{N}} \\
& =\exp \left[ V\Phi \left( \beta \right){{e}^{-\alpha }} \right] \\
& \Rightarrow {{e}^{-\Psi }}=Y=\exp \left[ V\Phi \left( \beta \right){{e}^{-\alpha }} \right] \\
\end{align}
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