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classical_canonical_partition_function [2016/03/09 11:40]
nikolaj
classical_canonical_partition_function [2016/03/09 11:41] (current)
nikolaj
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 >​$E\,​\dfrac{{\mathrm e}^{-\beta\,​E}}{Z(\beta)}\mapsto\sum_{n=0}^\infty\dfrac{(n(\hbar\omega))\,​{\mathrm e}^{-\beta\,​n\,​(\hbar\omega)}}{\sum_{n=0}^\infty{\mathrm e}^{-\beta\,​n\,​(\hbar\omega)}}=(\hbar\omega)\dfrac{1}{\mathrm e^{\beta\hbar\omega}-1}$ >​$E\,​\dfrac{{\mathrm e}^{-\beta\,​E}}{Z(\beta)}\mapsto\sum_{n=0}^\infty\dfrac{(n(\hbar\omega))\,​{\mathrm e}^{-\beta\,​n\,​(\hbar\omega)}}{\sum_{n=0}^\infty{\mathrm e}^{-\beta\,​n\,​(\hbar\omega)}}=(\hbar\omega)\dfrac{1}{\mathrm e^{\beta\hbar\omega}-1}$
->Roughly 
-> 
->$\int {\mathrm d}E\cdot p_E \, \mapsto \, \sum_{n=0}^\infty\int{\mathrm d}\,​(n\hbar\omega)\,​p_\omega(n)\,​\mapsto\,​\sum_{n=0}^\infty\int{\mathrm d}\,​(n\hbar|k|)\,​|k|^2\,​p_k(n)$ 
  
 The common derivation of **Plancks law** makes the above step and then neglects spatial inhomogenies and just introduces a characteristic length $L$. Then we can use the characteristic speed (of light) $c$ to get a characteristic frequency $c/L$. Thus we can intorduce powers of $\omega$ via the unitless expression $\omega\left/​\right.\dfrac{c}{L}$. If $D$ is quadratic (the case of photon gas), we get The common derivation of **Plancks law** makes the above step and then neglects spatial inhomogenies and just introduces a characteristic length $L$. Then we can use the characteristic speed (of light) $c$ to get a characteristic frequency $c/L$. Thus we can intorduce powers of $\omega$ via the unitless expression $\omega\left/​\right.\dfrac{c}{L}$. If $D$ is quadratic (the case of photon gas), we get
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