The structure of the eigenstates of a hydrogen atom in parallel uniform electric and magnetic fields is investigated using high-order classical perturbation theory. The Kustaanheimo-Stiefel transformation is first used to convert the problem into an anharmonically perturbed four-dimensional isotropic oscillator. A canonical transformation to a set of extended ‘‘Lissajous’’ action-angle variables is then introduced that considerably simplifies the perturbation expansion, leading to a simple and compelling classification scheme for the eigenstates. Extended Lissajous action-angle variables allow the construction of rotational energy surfaces, which provide a compact geometrical picture that captures important details of the energy-level structure of the system.

• Received 17 October 1991

DOI:https://doi.org/10.1103/PhysRevA.45.4738