Numerical and Analytical Study of the Bound States of the $-\alpha/x^2$ Potential

TL;DR

This paper investigates the bound states of a modified $-rac{ ext{alpha}}{x^2}$ potential using analytical and numerical methods, providing well-defined solutions and insights into this quantum mechanical problem.

Contribution

It introduces a cutoff in the potential near the origin and employs both analytical and numerical approaches, offering a comprehensive case study suitable for undergraduate research.

Findings

Solutions are well-defined and normal with the cutoff introduced.

Analytical and numerical methods complement each other in solving the problem.

Provides insights into the anomalous bound states of the potential.

Abstract

The quantum mechanical bound states of the $-{\alpha}/x^2$ potential are truly anomalous. We revisit this problem by adopting a slightly modified version of this potential, one that adopts a cutoff in the potential arbitrarily close to the origin. The resulting solutions are completely well-defined and ``normal.'' We present results here as a case study in undergraduate research --- two independent methodologies are used: one analytical (with very unfamiliar non-elementary functions) and one numerical (with very straightforward methodology). These play complementary roles in arriving at solutions and achieving insights in this problem.