Accurate calculation of the bound states of Hellmann potential

TL;DR

This paper presents a highly accurate and efficient numerical method to calculate bound states of the Hellmann potential, improving precision over previous results and reporting new states for the first time.

Contribution

The authors develop a generalized pseudospectral method that achieves thirteen to fourteen significant figures in energy calculations for the Hellmann potential, including new states and a wide parameter range.

Findings

Energy eigenvalues with 13-14 significant figures

First-time reporting of some bound states

Significant accuracy improvement over existing methods

Abstract

Bound states of the Hellmann potential, which is a superposition of the attractive Coulomb ($-A/r$) and the Yukawa ($Be^{-Cr}/r$) potential, are calculated by using a generalized pseudospectral method. Energy eigenvalues accurate up to thirteen to fourteen significant figures, and densities are obtained through a nonuniform, optimal spatial discretization of the radial Schr\"odinger equation. Both ground and excited states are reported for arbitrary values of the potential parameters covering a wide range of interaction. Calculations have been made for higher states as well as for stronger couplings. Some new states are reported here for the first time, which could be useful for future works. The present results are significantly improved in accuracy over all other existing literature values and offers a simple, accurate and efficient scheme for these and other singular potentials in quantum mechanics.