The hydrogen atom in phase space

Abstract

The Hamiltonian of the three-dimensional hydrogen atom is reduced, in parabolic coordinates, to the Hamiltonians of two bidimensional harmonic oscillators, by doing several space-time transformations,separating the movement along the three parabolic directions (ξ,η,φ), and introducing two auxiliary angular variables ψ and ψ′, 0≤ψ, ψ′≤2π. The Green’s function is developed into partial Green’s functions, and expressed in terms of two Green’s functions that describe the movements along both the ξ and η axes. Introducing auxiliary Hamiltonians allows one to calculate the Green’s function in the configurational space, via the phase-space evolution function of the two-dimensional harmonic oscillator. The auxiliary variables ψ and ψ′ are eliminated by projection. The thus-obtained Green’s function, save for a multiplicating factor, coincides with that calculated following the path-integral formalism.