贡献者： addis
1本文使用原子单位制。使用相对坐标，令约化质量为 $\mu$，有
\begin{equation}
-\frac{1}{2\mu} \boldsymbol{\nabla}^2 \psi - \frac{Z}{r} \psi = E\psi~.
\end{equation}
在抛物线坐标系中变为
\begin{equation}
-\frac{1}{2m} \left\{\frac{4}{\xi + \eta} \left[ \frac{\partial u}{\partial \xi} \left(\xi \frac{\partial}{\partial{\xi}} \right) + \frac{\partial u}{\partial \eta} \left(\eta \frac{\partial}{\partial{\eta}} \right) \right] + \frac{1}{\xi\eta} \frac{\partial^{2}{u}}{\partial{\phi}^{2}} \right\} \psi - \frac{2Z}{\xi + \eta}\psi = E\psi~.
\end{equation}
分离变量，令
\begin{equation}
\psi(\xi, \eta, \phi) = f(\xi) g(\eta) \Phi(\phi)~.
\end{equation}
和球坐标同理，$\Phi(\phi) = \exp\left( \mathrm{i} m \phi\right) $。令另外两个分离变量常数满足
\begin{equation}
\nu_1 + \nu_2 = Z~.
\end{equation}
有
\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}{\xi}} \left(\xi \frac{\mathrm{d}{f}}{\mathrm{d}{\xi}} \right) + \left(\frac{\mu E \xi}{2} - \frac{m^2}{4\xi} + \nu_1 \right) f = 0~,
\end{equation}
\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}{\eta}} \left(\eta \frac{\mathrm{d}{g}}{\mathrm{d}{\eta}} \right) + \left(\frac{\mu E\eta}{2} - \frac{m^2}{4\eta} + \nu_2 \right) g = 0~,
\end{equation}
可以化简为 Kummer-Laplace 微分方程。解出后，$\nu_1, \nu_2$ 分别对应两个整数 $n_1, n_2$，称为抛物线量子数，和主量子数 $n$ 的关系为
\begin{equation}
n = n_1 + n_2 + \left\lvert m \right\rvert + 1~.
\end{equation}
令 $\rho_1 = Z\xi / n$，$\rho_2 = Z\eta/n$，定态波函数为
\begin{equation}
\begin{aligned}
\psi_{n_1,n_2,m}(\xi,\eta,\phi) &= \frac{(Z/a_\mu)^{3/2}}{\sqrt{\pi} n^2} \sqrt{\frac{n_1!n_2!}{[(n_1+ \left\lvert m \right\rvert )!(n_2+ \left\lvert m \right\rvert )!]^3}} \\
&\times \mathrm{e} ^{-(\rho_1+\rho_2)/2}(\rho_1\rho_2)^{ \left\lvert m \right\rvert /2} L_{n_1 + \left\lvert m \right\rvert }^{ \left\lvert m \right\rvert }(\rho_1) L_{n_2 + \left\lvert m \right\rvert }^{ \left\lvert m \right\rvert }(\rho_2) \mathrm{e} ^{ \mathrm{i} m\phi}~,
\end{aligned}
\end{equation}
其中 $a_\mu = M/(M + 1)$ 是约化玻尔半径（未完成）。
能量本征值为
\begin{equation}
E_n = -\frac{Z^2}{2n^2}~.
\end{equation}
1. ^ 参考 [1] Chap. 3.5 One-electron atoms in parabolic coordinates.
[1] ^ Bransden, Physics of Atoms and Molecules, 2ed