CONFINED QUANTUM-SYSTEMS - THE PARABOLICALLY CONFINED HYDROGEN-ATOM

We investigate a hydrogen-like atom (or any other system with a Coulom b 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 req uire the electronic wavefunction to vanish on the paraboloid. We deriv e an exact implicit analytic solution to the problem and also explicit analytic expressions for the wavefunctions and eigenenergies in the s o-called strong-shift regime. We also discuss the influence of the bou ndary on the permanent dipole moments of the eigenstates. Finally, we investigate this system in WKB-approximation and give the Bohr-Sommerf eld quantization rule which is different from the usual rule due to th e new boundary condition.