贡献者： addis
1. 微分关系
\begin{equation}
H = E + PV~.
\end{equation}
\begin{equation}
G = E + PV - ST~.
\end{equation}
\begin{equation}
\mu \,\mathrm{d}{N} + N \,\mathrm{d}{\mu} = \,\mathrm{d}{G} = V \,\mathrm{d}{P} - S \,\mathrm{d}{T} + \mu \,\mathrm{d}{N} ~.
\end{equation}
2. 微正则系综
\begin{equation}
\,\mathrm{d}{S} = \frac1T \,\mathrm{d}{E} + \frac{P}{T} \,\mathrm{d}{V} - \frac{\mu }{T} \,\mathrm{d}{N} ~.
\end{equation}
3. 正则系综
\begin{equation}
- kT\ln Q = F = E - ST~.
\end{equation}
\begin{equation}
\,\mathrm{d}{F} = -S \,\mathrm{d}{T} - P \,\mathrm{d}{V} + \mu \,\mathrm{d}{N} ~.
\end{equation}
\begin{equation}
E = - \frac{\partial}{\partial{\beta}} \ln Q~/~.
\end{equation}
4. 巨正则系综
\begin{equation}
- PV = \Phi = E - ST - \mu N~.
\end{equation}
\begin{equation}
\Phi = - kT\ln \Xi ~.
\end{equation}
\begin{equation}
\,\mathrm{d}{\Phi} = - P \,\mathrm{d}{V} - S \,\mathrm{d}{T} - N \,\mathrm{d}{\mu} ~.
\end{equation}
\begin{equation}
\left\langle n_i \right\rangle = \frac{\partial \Phi}{\partial \varepsilon_i} ~.
\end{equation}
5. 理想气体
\begin{equation}
V_n = \frac{\pi^{n/2} R^n}{\Gamma(1 + n/2)}
\qquad
\text{（$N$ 维球体）}~.
\end{equation}
\begin{equation}
\Omega_0 = \frac{V^N}{N! h^3} \frac{(2\pi mE)^{3N/2}}{(3N/2)!}
\qquad \text{（$N$ 粒子能级密度）}~.
\end{equation}
\begin{equation}
a(\varepsilon) = \frac{2\pi V{(2m)^{3/2}}}{h^3} \varepsilon^{1/2}
\qquad \text{（单粒子能及密度）}~.
\end{equation}
\begin{equation}
S = Nk \left(\ln \frac{V}{N\lambda^3} + \frac52 \right)
\qquad
\text{（熵）}~.
\end{equation}
\begin{equation}
N = zQ_1 \Rightarrow \mu = kT\ln \frac{N\lambda^3}{V}
\qquad
\text{（化学势）}~.
\end{equation}
\begin{equation}
\Xi = \ln N
\qquad
\text{（巨势）}~.
\end{equation}
6. 量子气体
\begin{equation}
N = Q_1 g_{3/2} (z) = \frac{V}{\lambda^3} g_{3/2} (z)
\qquad\text{（$BE$）}~.
\end{equation}
\begin{equation}
N = Q_1 f_{3/2} (z)
\qquad\text{（$FD$）}~.
\end{equation}
\begin{equation}
\frac{PV}{kT} = Q_1 g_{5/2} (z) = \frac{V}{\lambda^3} g_{5/2} (z)
\qquad\text{（$BE$）}~.
\end{equation}
\begin{equation}
\frac{PV}{kT} = Q_1 f_{5/2} (z)
\qquad\text{（$FD$）}~.
\end{equation}
\begin{equation}
PV = NkT\frac{g_{5/2}(z)}{g_{3/2}(z)}
\qquad\text{（$BE$）}~.
\end{equation}
\begin{equation}
PV = NkT\frac{f_{5/2}(z)}{f_{3/2}(z)}
\qquad\text{（$FD$）}~.
\end{equation}
\begin{equation}
E = \frac32 PV
\qquad\text{（$BE$ 和 $FD$）}~.
\end{equation}
理论上可以通过三式中的任意两式消去 $z$， 但是不能写成解析形式。
\begin{equation} \begin{aligned}
g_n(z) = z + \frac{z^2}{2^n} + \frac{z^3}{3^n}\dots\\
f_n(z) = z - \frac{z^2}{2^n} + \frac{z^3}{3^n}\dots
\end{aligned} ~\end{equation}
7. $BE$ 凝聚态
\begin{equation}
N = \frac{V}{\lambda_c^3} g_{3/2} (1) \Rightarrow T_c = \frac{h^2}{2\pi mk} \left(\frac{N}{2.612V} \right) ^{2/3}~.
\end{equation}
\begin{equation}
\frac{N_e}{N} = \frac{\lambda^3}{\lambda_c^3} \Rightarrow N_e = N \left(\frac{T}{T_c} \right) ^{3/2} \Rightarrow {N_0} = N \left[1 - \left(\frac{T}{T_c} \right) ^{3/2} \right] ~.
\end{equation}
\begin{equation}
N_0 = \frac{1}{ \mathrm{e} ^{(\varepsilon_0 - \mu )/kT} - 1} = \frac{kT}{\varepsilon_0 - \mu}~.
\end{equation}
\begin{equation}
\varepsilon_0 - \mu \ll \varepsilon_1 - \varepsilon_0 \Rightarrow \varepsilon_0 - \mu \ll \varepsilon_1 - \mu ~.
\end{equation}
\begin{equation}
N_1 = \frac{1}{ \mathrm{e} ^{(\varepsilon_1 - \mu )/kT} - 1} < \frac{kT}{\varepsilon_1 - \mu} \ll \frac{kT}{\varepsilon_0 - \mu } = N_0~.
\end{equation}
8. 范德瓦尔斯方程
\begin{equation}
\left(P + \frac{aN^2}{V^2} \right) (V - bN) = NkT~.
\end{equation}
9. 量子转子能级
角量子数 $l$ 决定能级
\begin{equation}
E_l = l (l + 1)\frac{\hbar^2}{2IkT}~.
\end{equation}
$2l+1$ 重简并，其中 $I = m_1 m_2 r_{12}^2/(m_1 + m_2)$ 为质心转动惯量。当 $l$ 为偶数时，两粒子的波函数具有交换对称，奇数时反对称。两原子核的自旋共有 $s^2 = (2I + 1)^2$ 种状态，其中对称态占 $s(s + 1)/2$ 种，反对称太占 $s(s - 1)/2$ 种。若两粒子都是费米子（$I$ 为半整数），则总波函数反对称，即 $l$ 为单数核自旋对称，或 $l$ 为偶数核自旋反对称。
10. 弹簧振子能级
\begin{equation}
E_n = \hbar \omega \left(n + \frac12 \right) ~
\end{equation}
非简并。
为什么书上说 $m = 0 $（能级密度与 $\varepsilon^m$ 成正比）不能产生凝聚态，然而我在模拟中做到了？