Integrals of motion and quantum operators for hydrogenic atoms in externalfields - art. no. 043410

We report five cases of integrability for a hydrogenic atom under three sta tic external fields: a magnetic field, an electric field, and a van der Waa ls interaction. Exact integrals of motion and corresponding quantum operato rs are obtained explicitly for each case. Integrals of motion (quantum oper ators) can be expressed as components of a suitably generalized Runge-Lenz vector (operator). Quadratic quantum operators are found to have the amazin g property of requiring a nonclassical extra term proportional to (h) over bar(2). The structuring of the classical phase space is investigated numeri cally via Poincare surfaces of section and corroborates the analytical resu lts.