Studies on some singular potentials in quantum mechanics

TL;DR

This paper introduces an efficient generalized pseudospectral method for accurately calculating energies and properties of singular potentials in quantum mechanics, demonstrating its effectiveness on two complex potentials.

Contribution

The paper presents a novel, efficient computational scheme using the generalized pseudospectral method for singular potentials, achieving high accuracy and reporting new quantum states.

Findings

Accurate energies for two singular potentials are computed.

Good agreement with existing literature data is achieved.

New quantum states are reported for the first time.

Abstract

A simple methodology is suggested for the efficient calculation of certain central potentials having singularities. The generalized pseudospectral method used in this work facilitates {\em nonuniform} and optimal spatial discretization. Applications have been made to calculate the energies, densities and expectation values for two singular potentials of physical interest, {\em viz.,} (i) the harmonic potential plus inverse quartic and sextic perturbation and (ii) the Coulomb potential with a linear and quadratic term for a broad range of parameters. The first 10 states belonging to a maximum of $\ell=8$ and 5 for (i) and (ii) have been computed with good accuracy and compared with the most accurate available literature data. The calculated results are in excellent agreement, especially in the light of the difficulties encountered in these potentials. Some new states are reported here for the first time. This offers a general and efficient scheme for calculating these and other similar potentials of physical and mathematical interest in quantum mechanics accurately.