Solving for three-dimensional central potentials using matrix mechanics

TL;DR

This paper demonstrates how matrix mechanics can be used to solve three-dimensional central potential problems in quantum mechanics, including both solvable and non-solvable cases, with less mathematical complexity.

Contribution

It introduces a practical approach using matrix mechanics for solving complex 3D central potential problems, expanding beyond traditional analytical methods.

Findings

Excellent agreement with analytical solutions for Coulomb potential

Ability to solve non-solvable potentials like Yukawa using matrix methods

Reduced mathematical complexity for solving quantum problems

Abstract

Matrix mechanics is an important component of an undergraduate education in quantum mechanics. In this paper we present several examples of the use of matrix mechanics to solve for a number of three dimensional problems involving central forces. These include examples with which the student is familiar, such as the Coulomb interaction. In this case we obtain excellent agreement with exact analytical methods. More importantly, other interesting `non-solvable' examples, such as the Yukawa potential, can be solved as well. Much less mathematical expertise is required for these methods, while some minimal familiarity with the usage of numerical diagonalization software is necessary.