Semiclassical Theory of the Structure of the Hydrogen Spectrum in Near-Perpendicular Electric and Magnetic fields: Derivations and Formulas for Einstein-Brillouin-Keller-Maslov Quantization and Description of Monodromy

In a previous paper [Schleif and Delos, Phys. Rev. A 76, 013404 (2007)] we described the spectrum of hydrogen atoms in near-perpendicular electric and magnetic fields. We displayed a number of previously unrecognized structures in the spectrum, most of which are connected with a classical phenomenon called “nontrivial monodromy of action and angle variables in a Hamiltonian system,” or simply “monodromy.” In that paper, we presented only the results, giving predictions of what to look for in various ranges of electric and magnetic fields. Here we present the underlying 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