Method of approximation for potentials in impenetrable boxes: Harmonic Oscillator and Coulomb potentials

TL;DR

This paper develops an analytical approximation method for potentials within impenetrable boxes, specifically applied to harmonic oscillator and Coulomb potentials, ensuring correct quantum energy limits.

Contribution

It introduces a novel approach to derive analytical expressions for potentials in confined systems, aligning with quantum limits and exploring connections to quasi-exactly solvable potentials.

Findings

Analytical expressions for harmonic oscillator and Coulomb potentials in impenetrable boxes.

Energy levels respect quantum mechanical limits.

Method demonstrates potential for broader applications in quantum confinement problems.

Abstract

In this work we develop an approach to obtain analytical expressions for potentials in an impenetrable box. It is illustrated through the particular cases of the harmonic oscillator and the Coulomb potential. In this kind of system the energy expression respect the correct quantum limits, which is a very important quality. The similarity of this kind of problem with the quasi exactly solvable potentials is explored in order to accomplish our goals.