Highly excited states of a hydrogen atom in a strong magnetic field

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Abstract
Classical trajectories and semiclassical energy eigenvalues are calculated for an atomic electron in a high Rydberg state in an external magnetic field. With the use of perturbation theory, a classical trajectory is described as a Kepler ellipse with orbital parameters evolving slowly with time. As they evolve, the ellipse rocks, tilts, and flips in space, but the length of its major axis remains approximately constant. Exact numerical calculations verify that perturbation theory is quite accurate for the cases considered (principal quantum number ≃ 30, magnetic field ≲ 6 T). Action variables are calculated from perturbation theory and from exact trajectories, and semiclassical eigenvalues are obtained by quantization of action. Excellent agreement is found with observations.