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

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TeX (original user input):

\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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MathML (117.122 KB / 11.026 KB) :

e - Ψ = Y = N = 0 1 N ! exp ( - α N ) 1 3 N V d 3 q 1 V d 3 q N R 3 d 3 p 1 R 3 d 3 p N exp ( - β 2 m i p i 2 ) = N = 0 1 N ! exp ( - α N ) V N [ 1 3 R 3 d 3 p exp ( - β p 2 2 m ) ] N [ 1 3 R 3 d 3 p exp ( - β p 2 2 m ) ] = 1 3 ( 2 π m β ) 3 2 Φ ( β ) e - Ψ = Y = N = 0 1 N ! exp ( - α N ) V N [ 1 3 R 3 d 3 p exp ( - β p 2 2 m ) ] N = N = 0 1 N ! [ V Φ ( β ) e - α ] N = exp [ V Φ ( β ) e - α ] e - Ψ = Y = exp [ V Φ ( β ) e - α ] absent absent superscript 𝑒 Ψ 𝑌 superscript subscript 𝑁 0 1 𝑁 𝛼 𝑁 1 superscript Planck-constant-over-2-pi 3 𝑁 subscript 𝑉 superscript 𝑑 3 subscript 𝑞 1 subscript 𝑉 superscript 𝑑 3 subscript 𝑞 𝑁 subscript superscript 𝑅 3 superscript 𝑑 3 subscript 𝑝 1 subscript superscript 𝑅 3 superscript 𝑑 3 subscript 𝑝 𝑁 𝛽 2 𝑚 subscript 𝑖 superscript subscript 𝑝 𝑖 2 absent absent superscript subscript 𝑁 0 1 𝑁 𝛼 𝑁 superscript 𝑉 𝑁 superscript delimited-[] 1 superscript Planck-constant-over-2-pi 3 subscript superscript 𝑅 3 superscript 𝑑 3 𝑝 𝛽 superscript 𝑝 2 2 𝑚 𝑁 absent delimited-[] 1 superscript Planck-constant-over-2-pi 3 subscript superscript 𝑅 3 superscript 𝑑 3 𝑝 𝛽 superscript 𝑝 2 2 𝑚 1 superscript Planck-constant-over-2-pi 3 superscript 2 𝜋 𝑚 𝛽 3 2 Φ 𝛽 absent absent superscript 𝑒 Ψ 𝑌 superscript subscript 𝑁 0 1 𝑁 𝛼 𝑁 superscript 𝑉 𝑁 superscript delimited-[] 1 superscript Planck-constant-over-2-pi 3 subscript superscript 𝑅 3 superscript 𝑑 3 𝑝 𝛽 superscript 𝑝 2 2 𝑚 𝑁 superscript subscript 𝑁 0 1 𝑁 superscript delimited-[] 𝑉 Φ 𝛽 superscript 𝑒 𝛼 𝑁 absent absent 𝑉 Φ 𝛽 superscript 𝑒 𝛼 absent absent superscript 𝑒 Ψ 𝑌 𝑉 Φ 𝛽 superscript 𝑒 𝛼 {\displaystyle{\displaystyle\begin{aligned} \par&\displaystyle\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)% \\ \par&\displaystyle=\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}}\\ \par&\displaystyle\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)% \\ \par&\displaystyle\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}}\\ \par&\displaystyle=\exp\left[V\Phi\left(\beta\right){{e}^{-\alpha}}\right]\\ \par&\displaystyle\Rightarrow{{e}^{-\Psi}}=Y=\exp\left[V\Phi\left(\beta\right)% {{e}^{-\alpha}}\right]\\ \par\end{aligned}}}

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eΨ=Y=N=01N!exp(αN)13NVd3q1...Vd3qNR3d3p1...R3d3pNexp(β2mipi2)=N=01N!exp(αN)VN[13R3d3pexp(βp22m)]N[13R3d3pexp(βp22m)]=13(2πmβ)32Φ(β)eΨ=Y=N=01N!exp(αN)VN[13R3d3pexp(βp22m)]N=N=01N![VΦ(β)eα]N=exp[VΦ(β)eα]eΨ=Y=exp[VΦ(β)eα]

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