The semiclassical description of the energy spectrum of hydrogen in near-perpendicular fields

We examine the energy spectrum of hydrogen in weak near-perpendicular electric and magnetic fields using quantum computations and semiclassical analysis. The structure of the quantum spectrum is displayed in a lattice constructed by plotting the difference between total energy and first order energy versus first order energy, for all states of a given principal quantum number n. For some field parameters, the lattice structure is not regular, but has a lattice defect structure which may be characterized by the transport of lattice vectors. We find that in near-perpendicular fields the structure of the spectrum is divided into six distinct parameter regions, which we characterize by the presence and type of lattice defect. to explain this structure we examine a corresponding classical system which we have derived by classical perturbation theory. Starting from Kepler action and angle variables, we give a derivation of a classical Hamiltonian to second order in perturbation theory; the derivation is different from, but the final result agrees with previous work. We focus especially on the topological structure of the reduced phase space and on the resulting topological structure of the trajectories. We show that construction of action variables by the obvious methods leads to variables that have discontinuous derivatives. Smooth continuation of these primitive action variables leads to action variables that are multivalued. We show how these multivalued actions lead to lattice defects in the quantum spectrum. Finally we present a few correlation diagrams which show how quantum eigenvalues evolve from one region of near-perpendicular parameter space to another and show how the structure of the quantum correlations is related to structures in the classical phase space