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grand_canonical_partition_function [2016/03/04 16:49]
nikolaj
grand_canonical_partition_function [2016/03/04 18:10]
nikolaj
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 Important grand canonical partition functions in QM are those for bosons and fermions denoted $\Xi^+$ and $\Xi^-$, respectively. We only deal with one sort of particle, but introduce the index $r$ which runs over different energy levels. Using the identities Important grand canonical partition functions in QM are those for bosons and fermions denoted $\Xi^+$ and $\Xi^-$, respectively. We only deal with one sort of particle, but introduce the index $r$ which runs over different energy levels. Using the identities
  
-$\sum_{N=0}^{N^\text{max}}({\mathrm e}^{\beta\sum_r \mu})^N {\mathrm e}^{-\beta\sum_r N\varepsilon_r}$+$\sum_{N=0}^{N^\text{max}}({\mathrm e}^{\beta\sum_r \mu})^N ​{\cdot} ​{\mathrm e}^{-\beta\sum_r N\varepsilon_r}$
  
-$\sum_{N=0}^{N^\text{max}}\mathrm e^{-\beta\sum_r N(\varepsilon_r-\mu)} $+$\sum_{N=0}^{N^\text{max}}(\prod_r{\mathrm e}^{\beta \mu} {\cdot} {\mathrm e}^{-\beta \varepsilon_r})^N$
  
 $= \prod_r \sum_{N=0}^{N^\text{max}} (e^{-\beta\ (\varepsilon_r-\mu)})^N $ $= \prod_r \sum_{N=0}^{N^\text{max}} (e^{-\beta\ (\varepsilon_r-\mu)})^N $
  
-$= \begin{cases} \prod_r \frac{1}{1-e^{-\beta\ (\varepsilon_r-\mu)}} & \mathrm{if}\ N^\text{max}=\infty \\\\ \prod_r(1+e^{-\beta\ (h_r-\mu)}) & \mathrm{if}\ N^\text{max}=1 \end{cases}$ ​+$= \begin{cases} \prod_r \frac{1}{1-e^{-\beta\ (\varepsilon_r-\mu)}} & \mathrm{if}\ N^\text{max}=\infty \\\\ \prod_r(1+e^{-\beta\ (\varepsilon_r-\mu)}) & \mathrm{if}\ N^\text{max}=1 \end{cases}$ ​
  
 where $\varepsilon_r$ are the energy eigenvalues we obtain where $\varepsilon_r$ are the energy eigenvalues we obtain
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 The $\propto$ expresses takes into account the possible multiplicity of states $r,​r',​\cdot$ with equal energy $\varepsilon_r,​\varepsilon_{r'​},​\dots$. The $\propto$ expresses takes into account the possible multiplicity of states $r,​r',​\cdot$ with equal energy $\varepsilon_r,​\varepsilon_{r'​},​\dots$.
  
-The above derivation presupposes that the different energy values are given by the discretely indexed expressions of the form $\sim N \varepsilon$. This is the historical beginning of quantum ​mechanics.  +The above derivation presupposes that the different energy values are given by the discretely indexed expressions of the form $\sim N \varepsilon$. This is the historical beginning of quantum ​mechaniscs
- +
-For spin 1 bosons, if we take a continuum limit (see discussing in [[Classical density of states|density of states]]) and write $\varepsilon_\omega\equiv \hbar \omega$, the [[Statistical internal energy|internal energy]] $U\equiv\langle H\rangle$ is +
- +
-^ $U=V\int_0^\infty \left(\frac{\hbar\omega^3}{\pi^2 c^3}\frac{1}{\mathrm e^{\beta\hbar\omega}-1}\right)\mathrm d\omega$ ^ +
- +
-The formlua for the integrand, the energy density, is known as //​Planck'​s law//.+
  
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