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

A coherent states technique is applied to obtain global formulas for eigenfunctions as well as for solutions of the Cauchy problem, including a path integral representation. The reduction of coherent states by symmetry groups is studied using a transformation from “Bessel” states to “hypergeometric” ones. The eigenfunctions of the Hamiltonian of the hydrogen atom in a homogeneous magnetic field are represented via Bessel coherent states. In the case of small field, after a quantum averaging, the Hamiltonian is expressed by means of generators with quadratic commutation relations. Irreducible representations of this quadratic algebra are realized on the hypergeometric states. The notion of deformed hypergeometric states is also introduced for this quadratic algebra; it is an analog of squeezed Gaussian packets usually related to the Heisenberg algebra. The asymptotics of eigenfunctions with respect to small field and to high leading quantum number is derived using these states and their deaveraging. Explicite formulas for the Zeeman spectrum splitting are obtained up to the fourth order with respect to the field, for lower and upper levels in the claster as well, including the case of “passing through the center”.