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* Page found: Das ideale Gas (eq math.2507.19)

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Das ideale Gas Eq: math.2507.19

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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}

TeX (checked):

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

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\begin{aligned} \Rightarrow e^{- Ψ} = Y = \sum_{N = 0}^{\infty} \frac{1}{N!} \exp (- α N) \frac{1}{ℏ^{3 N}} \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 (- \frac{β}{2 m} \sum_{i}^{} {p_{i}}^{2}) \\ = \sum_{N = 0}^{\infty} \frac{1}{N!} \exp (- α N) V^{N} {[\frac{1}{ℏ^{3}} \int_{R^{3}}^{} d^{3} p \exp (- \frac{β p^{2}}{2 m})]}^{N} \\ [\frac{1}{ℏ^{3}} \int_{R^{3}}^{} d^{3} p \exp (- \frac{β p^{2}}{2 m})] = \frac{1}{ℏ^{3}} {(\frac{2 π m}{β})}^{\frac{3}{2}} \equiv Φ (β) \\ \Rightarrow e^{- Ψ} = Y = \sum_{N = 0}^{\infty} \frac{1}{N!} \exp (- α N) V^{N} {[\frac{1}{ℏ^{3}} \int_{R^{3}}^{} d^{3} p \exp (- \frac{β p^{2}}{2 m})]}^{N} = \sum_{N = 0}^{\infty} \frac{1}{N!} {[V Φ (β) e^{- α}]}^{N} \\ = \exp [V Φ (β) e^{- α}] \\ \Rightarrow e^{- Ψ} = Y = \exp [V Φ (β) e^{- α}] \end{aligned}

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