Parabolic Coordinates and the Hydrogen Atom in Spaces H_{3} and S_{3}

TL;DR

This paper develops a method using generalized parabolic coordinates to solve the Schrödinger equation with Coulomb potential in curved spaces H_{3} and S_{3}, deriving explicit energy spectra and wave functions for bound states.

Contribution

It introduces a novel approach employing real and complex parabolic coordinates for solving the Coulomb problem in curved spaces, extending separation of variables techniques.

Findings

Explicit energy spectra for bound states in H_{3} and S_{3}

Wave functions constructed in closed form for both spaces

Connections established with Runge-Lenz operators in curved spaces

Abstract

The Coulomb problem for Schr\"{o}dinger equation is examined, in spaces of constant curvature, Lobachevsky H_{3} and Riemann S_{3} models, on the base of generalized parabolic coordinates. In contrast to the hyperbolic case, in spherical space S_{3} such parabolic coordinates turn to be complex-valued, with additional constraint on them. The technique of the use of such real and complex coordinates in two space models within the method of separation of variables in Schr\"{o}dinger equation with Kepler potential is developed in detail; the energy spectra and corresponding wave functions for bound states have been constructed in explicit form for both spaces; connections with Runge-Lenz operators in both curved space models are described.