# 状态密度

$\rho(E) = \frac{g(E)\exp(-\beta E)}{\int_0^\infty g(E)\exp(-\beta E) \mathrm{d}E }$

## 与配分函数的关系

$Z(\beta) = \int_0^\infty g(E)\exp(-\beta E) \mathrm{d}E$

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

## 例子

### 经典理想气体的态密度

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

### 理想玻色气体的态密度

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

### 零温理想费米气体的态密度

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

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

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

### 声子气体的德拜模型

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

## 参考文献

1. Pathria, R. K.. Statistical Mechanics 2nd. Butterworth Heinemann: Elsevier. 1997. ISBN ISBN:978-0-7506-2469-5 请检查|isbn=值 (帮助).