A perturbative treatment for the bound states of the Hellmann potential

TL;DR

This paper introduces a perturbative approximation method to analyze the bound states of the Hellmann potential, combining Coulomb and Yukawa potentials, with accuracy depending on the potential parameters.

Contribution

A novel perturbative formalism is developed to approximate energy levels of the Hellmann potential, extending analysis to arbitrary potential strengths and screening parameters.

Findings

Accurate energy eigenvalues for weak screening and small potential strength.

Decreased accuracy as potential strength and screening increase.

The method's convergence depends on the smallness of the expansion parameter.

Abstract

A new approximation formalism is applied to study the bound states of the Hellmann potential, which represents the superposition of the attractive Coulomb potential $-a/r$ and the Yukawa potential $b\exp (-\delta r)/r$ of arbitrary strength $b$ and screening parameter $\delta $. Although the analytic expressions for the energy eigenvalues $E_{n,l\text{}}$ yield quite accurate results for a wide range of $n,\ell $ in the limit of very weak screening, the results become gradually worse as the strength $b$ and the screening coefficient $\delta $ increase. This is because that the expansion parameter is not sufficiently small enough to guarantee the convergence of the expansion series for the energy levels.