Confined quantum systems: The parabolically confined hydrogen atom

We investigate a hydrogen-like atom (or any other system with a Coulomb potential) confined to a space which is bounded by a paraboloid. The nucleus of the atom resides at the focus of the paraboloid and we require the electronic wavefunction to vanish on the paraboloid. We derive an exact implicit analytic solution to the problem and also explicit analytic expressions for the wavefunctions and eigenenergies in the so-called strong-shift regime. We also discuss the influence of the boundary on the permanent dipole moments of the eigenstates. Finally, we investigate this system in WKB-approximation and give the Bohr-Sommerfeld quantization rule which is different from the usual rule due to the new boundary condition.