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grand_canonical_partition_function [2016/03/04 16:47] nikolaj |
grand_canonical_partition_function [2016/03/04 16:49] 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 e^{-\beta\sum_r N\varepsilon_r} = $\sum_{N=0}^{N^\text{max}}\mathrm e^{-\beta\sum_r N\ (\varepsilon_r-\mu)} = \prod_r \sum_{N=0}^{N^\text{max}} (e^{-\beta\ (\varepsilon_r-\mu)})^N = \begin{cases} \prod_r \frac{1}{1-e^{-\beta\ (\varepsilon_r-\mu)}} & \mathrm{if}\ N^\text{max}=\infty \\\\ \prod_r(1+e^{-\beta\ (h_r-\mu)}) & \mathrm{if}\ N^\text{max}=1 \end{cases}$ | + | $\sum_{N=0}^{N^\text{max}}({\mathrm e}^{\beta\sum_r \mu})^N {\mathrm e}^{-\beta\sum_r N\varepsilon_r}$ |
| + | |||
| + | $= $\sum_{N=0}^{N^\text{max}}\mathrm e^{-\beta\sum_r N\ (\varepsilon_r-\mu)} $ | ||
| + | |||
| + | $= \prod_r \sum_{N=0}^{N^\text{max}} (e^{-\beta\ (\varepsilon_r-\mu)})^N $ | ||
| + | |||
| + | $= \begin{cases} \prod_r \frac{1}{1-e^{-\beta\ (\varepsilon_r-\mu)}} & \mathrm{if}\ N^\text{max}=\infty \\\\ \prod_r(1+e^{-\beta\ (h_r-\mu)}) & \mathrm{if}\ N^\text{max}=1 \end{cases}$ | ||
| where $\varepsilon_r$ are the energy eigenvalues we obtain | where $\varepsilon_r$ are the energy eigenvalues we obtain | ||
