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grand_canonical_partition_function [2016/03/04 16:49] nikolaj |
grand_canonical_partition_function [2016/03/04 16:52] 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 $ | ||
