Connection between closeness of classical orbits and the factorization of radial Schr\"{o}dinger equation

TL;DR

This paper explores the connection between classical orbit closeness and the factorization of the radial Schrödinger equation, highlighting the role of symmetry operators like the Runge-Lenz vector in quantum systems.

Contribution

It reveals a potential link between classical orbit properties and quantum factorization methods, extending the understanding of symmetries in quantum mechanics.

Findings

Runge-Lenz vector relates to raising and lowering operators

Factorization of radial Schrödinger equation connects to classical orbit closeness

Discussion includes factorization of 1D Schrödinger equation

Abstract

It was shown that the Runge-Lenz vector for a hydrogen atom is equivalent to the raising and lowering operators derived from the factorization of radial Schr\"{o}dinger equation. Similar situation exists for an isotropic harmonic oscillator. It seems that there may exist intimate relation between the closeness of classical orbits and the factorization of radial Schr\"{o}dinger equation. Some discussion was made about the factorization of a 1D Schr\"{o}dinger equation.