# 状态密度

${\displaystyle \rho (E)={\frac {g(E)\exp(-\beta E)}{\int _{0}^{\infty }g(E)\exp(-\beta E)\mathrm {d} E}}}$

## 与配分函数的关系

${\displaystyle Z(\beta )=\int _{0}^{\infty }g(E)\exp(-\beta E)\mathrm {d} E}$

${\displaystyle g(E)={\frac {1}{2\pi i}}\int _{s-i\infty }^{s+i\infty }e^{\beta E}Z(\beta )\mathrm {d} \beta \quad (\Re s>0)}$

## 例子

### 经典理想气体的态密度

${\displaystyle g(E)\approx {\frac {1}{N!}}\left({\frac {V}{h^{3}}}\right)^{N}{\frac {(2\pi m)^{3N/2}}{(3N/2-1)!}}E^{3N/2-1}}$

### 理想玻色气体的态密度

${\displaystyle g(E)={\frac {1}{\hbar ^{2}\pi ^{2}c^{3}}}{\frac {E^{3}}{e^{\beta E}-1}}}$

### 零温理想费米气体的态密度

${\displaystyle g(E)={\frac {gV}{h^{3}}}4\pi p^{2}\left.{\frac {\partial {p}}{\partial {E}}}\right|_{E}}$

${\displaystyle g(E)={\frac {g(2m)^{3/2}V}{4\pi ^{2}\hbar ^{3}}}{\sqrt {E}}}$

${\displaystyle g(E)={\frac {gV}{2\pi ^{2}\hbar ^{3}c^{3}}}E^{2}}$

### 声子气体的德拜模型

${\displaystyle g(\omega )=\left\{{\begin{array}{ll}{\frac {9N}{\omega _{D}^{3}}}\omega ^{2}&\quad {\text{ for }}\omega \leq \omega _{D};\\0&\quad {\text{ for }}\omega >\omega _{D}\\\end{array}}\right.}$

## 参考文献

1. Pathria, R. K. Statistical Mechanics 2nd. Butterworth Heinemann: Elsevier. 1997. ISBN 978-0-7506-2469-5.