Representation of exact and semiclassical eigenfunctions via coherent states. Hydrogen atom in a magnetic field

Abstract

Global formulas for eigenfunctions and solutions to the Cauchy problem, including the path integral representation, are obtained using the coherent states technique. The reduction of coherent states via symmetry groups is studied for a transformation from “Bessel” to “hypergeometric” states. The eigenfunctions of the Hamiltonian for the hydrogen atom in a homogeneous magnetic field are expressed in terms of Bessel coherent states. For a small field, after quantum averaging, the Hamiltonian is represented in terms of generators with quadratic commutation relations. The irreducible representations of this quadratic algebra are realized on hypergeometric states. The notion of deformed hypergeometric states is also introduced for this quadratic algebra as an analog of squeezed Gaussian packets of the Heisenberg algebra. The asymptotic equations of eigenfunctions with respect to a small field and a large leading quantum number are derived using these states and their “deaveraging.” Some explicit formulas for the Zeeman splitting of the spectrum are obtained up to the fourth order with respect to the field, as well as for lower and upper levels in the cluster, including the case of “incidence on the center.”