Integrals of motion and quantum operators for hydrogenic atoms in external fields

Abstract

We report five cases of integrability for a hydrogenic atom under three static external fields: a magnetic field, an electric field, and a van der Waals interaction. Exact integrals of motion and corresponding quantum operators are obtained explicitly for each case. Integrals of motion (quantum operators) can be expressed as components of a suitably generalized Runge-Lenz vector (operator). Quadratic quantum operators are found to have the amazing property of requiring a nonclassical extra term proportional to h². The structuring of the classical phase space is investigated numerically via Poincare surfaces of section and corroborates the analytical results.