I've asked this question in the Physics Exchange Forums, but I think this is a more appropaite site to ask this question. How would you solve the following differential equation, and find its eigenfucntions: $$ (i\gamma^\mu \partial_\mu - m) \psi = e\gamma^\mu A_\mu \psi $$ where $\gamma^\mu$ are the gamma matrices, and $A_\mu = (\frac{e}{4\pi\varepsilon_0r}, 0, 0, 0)$. Note that we are using the Einsten Sumation Convention. For more clarification: $$ (i\gamma^\mu \partial_\mu - m\mathbb I_{4}) \psi = e\gamma^\mu A_\mu \psi $$
$$ \psi(\mathbf r, t) = \begin{pmatrix} \psi_1 \\ \psi_2 \\ \psi_3 \\ \psi_4 \end{pmatrix} $$ How would you solve this last differential equation in spherical coordinates. I do not know from where to start, because the partial derivatives get mixed up, by the $\gamma^\mu$, and you get a system of 4 differential equations. Any hint?