Semiclassical quantization of the hydrogen atom in a generalized van der Waals potential

A semiclassical quantization of the hydrogen atom in a generalized van der Waals potential is carried out using the Kustaanheimo-Stiefel transformation and Birkhoff-Gustavson normal-form procedure, employed by Kuwata, Harada, and Hasegawa [J. Phys. A 23, 3227 (1990)] for the diamagnetic Kepler problem. We derive here the generalized approximate Solov’ev constant of motion. By using appropriate action-angle variables in the normal Hamiltonian, we derive four canonically equivalent action integrals that take an especially simple form for the three classically integrable cases and provide exact quantum numbers. For near-integrable cases the semiclassical spectrum can be generated by integrating the appropriate action integrals numerically.