On the hydrogen‐oscillator connection: Passage formulas between wave functions

TLDR: In this article, the authors extended the hydrogen-oscillator connection to cover in a systematic (and easily computarizable) way the problem of the expansion of an R3 hydrogen wave function in terms of R4 oscillator wave functions.

Abstract: Recent works on the hydrogen‐oscillator connection are extended to cover in a systematic (and easily computarizable) way the problem of the expansion of an R3 hydrogen wave function in terms of R4 oscillator wave functions. Passage formulas from oscillator to hydrogen wave functions are obtained in six cases resulting from the combination of the following coordinate systems: spherical and parabolic coordinate systems for the hydrogen atom in three dimensions, and Cartesian, double polar, and hyperspherical coordinate systems for the isotropic harmonic oscillator in four dimensions. These coordinate systems are particularly useful in physical applications (e.g., Zeeman and Stark effects for hydrogenlike ions and coherent state approaches to the Coulomb problem).