A general method for central potentials in quantum mechanics

TL;DR

This paper presents a generalized pseudospectral method for accurately and efficiently solving the Schrödinger equation with various central potentials, including singular and non-singular cases, demonstrating high precision across multiple quantum systems.

Contribution

The paper introduces a versatile pseudospectral approach that effectively handles a wide range of central potentials, surpassing existing methods in accuracy and applicability in quantum mechanics.

Findings

Accurate solutions for diverse potentials like Hulthén, Yukawa, and Coulomb.

High-quality results for ground and excited states across different systems.

Method outperforms or matches the best existing results in literature.

Abstract

We focus on a recently developed generalized pseudospectral method for accurate, efficient treatment of certain central potentials of interest in various branches in quantum mechanics, usually having singularity. Essentially this allows optimal, nonuniform spatial discretization of the pertinent single-particle Schrodinger equation satisfying Dirichlet boundary condition leading to standard diagonalization of symmetric matrices. Its validity and feasibility have been demonstrated for a wide range of important potentials such as Hulth\'en, Yukawa, generalized spiked harmonic oscillators, Hellmann, Coulomb potentials without/with various perturbations (for instance, linear and quadratic) etc. Although initially designed for singular potentials, this has also been remarkably successful for various other cases such as power-law, logarithmic, harmonic potentials containing higher order perturbations, 3D rational potentials as well as confinement studies. Furthermore, a large number of low-, moderately high-, high-lying multiply excited Rydberg states such as singly, doubly excited He as well as triply excited hollow $2l2l'2l'' (n \ge 2)$ and $3l3l'3l''$ doubly-hollow resonances in many-electron atoms have been treated by this approach within a KS DFT with great success. This offers very high-quality results for both ground and higher lying states for arbitrary values of potential parameters (covering both weak and strong coupling) with equal ease and efficacy. In all cases, excellent agreement with literature results are observed; in many cases this surpasses the accuracy of all other existing results while in other occasions our results are comparable to the best ones available in literature.