Rewriting the Hydrogen Schrodinger Equation as a system of differential equations I have only ever seen the Schrodinger equation for the hydrogen atom written out in a form like this:
$$
-\frac{\hbar^2}{2\mu}\left[\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial \psi}{\partial r}\right) + \frac{1}{r^2\sin{\theta}}\frac{\partial}{\partial \theta}\left(\sin{\theta}\frac{\partial\psi}{\partial\theta}\right)+\frac{1}{r^2\sin^2{\theta}}\frac{\partial^2\psi}{\partial \phi^2}\right]-\frac{Ze^2}{4\pi\epsilon_0 r}\psi=E\psi
$$
I'm still learning the necessary skills to solve PDEs, let alone get to the point of solving this problem, but I wanted to know if someone could show me what this differential equation would look like in a matrix notation or as a system of differential equations.
 A: If you assume separability of the wave function, i.e., $\psi(\mathbf x)=u(x)v(y)w(z)$, you can solve the individual components separately:
\begin{align}
-\frac{\hbar^2}{2\mu}\frac{d^2u(x)}{dx^2}+V_1(x)u(x)&=E_1u(x)\\
-\frac{\hbar^2}{2\mu}\frac{d^2v(y)}{dy^2}+V_2(y)v(y)&=E_2v(y)\tag{1}\\
-\frac{\hbar^2}{2\mu}\frac{d^2w(z)}{dz^2}+V_3(z)w(z)&=E_3w(z)
\end{align}
with the further constraint that
$$
E_1+E_2+E_3=E
$$
We can express (1) as the matrix differential equation,
$$
\mathbf u''=A\mathbf u,\tag{2}
$$
in which case $A$ is clearly diagonal and $\mathbf u=(u(x),\,v(y),\,w(z))^T$. In the case that the wave-function is not separable, then this method is not appropriate as you'd have a single scalar equation.
For your case of the spherical wave function, you can solve the radial component and the angular component separately, $\psi(\mathbf r)=R(r)Y(\theta,\phi)$ with $Y(\theta,\phi)$ the spherical harmonics, as
\begin{align}
\frac{1}{R}\frac{d}{dr}\left(r^2\frac{dR(r)}{dr}\right)&=\lambda \\
\frac1Y\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial Y}{\partial\theta}\right)+\frac1Y\frac1{\sin^2\theta}\frac{\partial^2Y}{\partial\phi^2}&=-\lambda
\end{align}
where $\lambda$ is a parameter to be discovered. This is the typical method of solving this particular problem in quantum mechanics textbooks.
