Application of regularization maps to quantum mechanical systems in 2 and 3 dimensions

TL;DR

This paper extends classical regularization methods to quantum systems in 2D and 3D, deriving spectra and mappings for hydrogen-like atoms with additional potentials using perturbation theory and transformations.

Contribution

It introduces quantum analogs of classical regularization mappings, enabling analysis of hydrogen-like systems with extra potentials through perturbative and transformation techniques.

Findings

Eigen spectrum of hydrogen with harmonic potential derived

Mapping between shifted harmonic oscillator and hydrogen atom established

Solutions obtained via perturbation and Bohlin-Sundman transformation

Abstract

We extend the Levi-Civita (L-C) and Kustaanheimo-Stiefel (K-S) regularization methods that maps the classical system where a particle moves under the combined influence of $\frac{1}{r}$ and $r^2$ potentials to a harmonic oscillator with inverted sextic potential and interactions to corresponding quantum mechanical counterparts, both in 2 and 3 dimensions. Using the perturbative solutions of the Schr\"odinger equation of the later systems, we derive the eigen spectrum of the Hydrogen atom in presence of an additional harmonic potential. We have also obtained the mapping of a particle moving in the shifted harmonic potential to H-atom using Bohlin-Sundman transformation, for quantum regime. Exploiting this equivalence, the solution to the Schr\"odinger equation of the former is obtained from the solutions of the later.