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