Semiclassical perturbation theory for the hydrogen atom in a uniform magnetic field

We study a perturbative semiclassical expansion for the energy levels of the hydrogen atom in a magnetic field. A recursion relation is derived for the calculation of the ground-state energy in each azimuthal subspace. Four terms of the series have been obtained analytically and higher-order corrections have been computed by numerical iteration of the recursion relation. The semiclassical series yields accurate results for field strengths up to the intermediate region, and a simple Shanks extrapolation extends its applicability into the region of very strong fields. A similar calculation for the Stark Hamiltonian is also outlined.