This problem is taken from Landau (1996). A spinless electron is bound by the Coulomb potential $V(r)=-Z e^{2} / r$ in a stationary state of total energy $E \leq m$. You can incorporate this interaction into the Klein-Gordon equation by using the covariant derivative with $V=-e \Phi$ and $\mathbf{A}=0$.
a. Assume that the radial and angular parts of the equation separate, and that the wave function can be written as $e^{-i E t}\left[u_{l}(r) / r\right] Y_{l m}(\theta, \phi)$ and show that the radial equation becomes
$$
\frac{d^{2} u}{d \rho^{2}}+\left[\frac{2 E Z \alpha}{\gamma \rho}-\frac{1}{4}-\frac{l(l+1)-(Z \alpha)^{2}}{\rho^{2}}\right] u_{l}(\rho)=0
$$
where $\alpha=e^{2}, \gamma^{2}=4\left(m^{2}-E^{2}\right)$, and $\rho=\gamma r$.
b. Assume that this equation has a solution of the usual form of a power series times the $\rho \rightarrow \infty$ and $\rho \rightarrow 0$ solutions, that is
$$
u_{l}(\rho)=\rho^{k}\left(1+c_{1} \rho+c_{2} \rho^{2}+\cdots\right) e^{-\rho / 2}
$$
and show that
$$
k=k_{\pm}=\frac{1}{2} \pm \sqrt{\left(l+\frac{1}{2}\right)^{2}-(Z \alpha)^{2}}
$$
and that only for $k_{+}$is the expectation value of the kinetic energy finite and that this solution has a nonrelativistic limit which agrees with the solution found for the Schrödinger equation.
c. Determine the recurrence relation among the $c_{i}$ for this to be a solution of the Klein-Gordon equation, and show that unless the power series terminates, the wave function will have an incorrect asymptotic form.
d. In the case where the series terminates, show that the energy eigenvalue for the $k_{+}$solution is
$$
E=\frac{m}{\left(1+(Z \alpha)^{2}\left[n-l-\frac{1}{2}+\sqrt{\left(l+\frac{1}{2}\right)^{2}-(Z \alpha)^{2}}\right]^{-2}\right)^{1 / 2}}
$$
where $n$ is the principal quantum number.
e. Expand $E$ in powers of $(Z \alpha)^{2}$ and show that the first-order term yields the Bohr formula. Connect the higher-order terms with relativistic corrections, and discuss the degree to which the degeneracy in $l$ is removed.
Jenkins and Kunselman, Phys. Rev. Lett., 17 (1966) 1148, report measurements of a large number of transition energies for $\pi^{-}$atoms in large $Z$ nuclei. Compare some of these to the calculated energies and discuss the accuracy of the prediction. (For example, consider the $3 d \rightarrow 2 p$ transition in ${ }^{59} \mathrm{Co}$ which emits a photon with energy $384.6 \pm 1.0 \mathrm{keV}$.) You will probably either need a computer to carry out the energy differences with high enough precision, or else expand to higher powers of $(Z \alpha)^{2}$.