The Zeeman effect for the 'anistropic hydrogen atom' in the complex WKB approximation: I. Quantization of closed orbits for the Pauli operator with spin-orbit interaction

Abstract

Using the complex WKB method, new semiclassical spectral series for perturbed states of an 'anisotropic' hydrogen atom, taking into account the electron's spin polarization in a homogeneous magnetic held (the anomalous Zeeman effect) are constructed. They are valid uniformly for all values of the field's magnitude. These series correspond to families (stable in the linear approximation) of periodic and conditionally periodic motions of a classical electron in the plane orthogonal to the direction of the field. The corresponding semiclassical wavefunctions have the property of quantum 'superscaring' in the coordinate space near these classical trajectories of motion. We obtain the limit cases of our semiclassical energy levels for strong and weak magnetic fields. We compare them with the only known approximations, i.e. for the hydrogen atom.