Supersymmetry and the Hartmann Potential of Theoretical Chemistry

Abstract

The ring-shaped Hartmann potential $V=\eta {\sigma }^2{\epsilon }_0\left(\frac{2a_0}{r}-\frac{\eta a_0^2}{r^{2}si{n}^2\theta }\right)$ was introduced in quantum chemistry to describe ring-shaped molecules like benzene. In this article, the supersymmetric features of the Hartmann potential are discussed. We first review the results of a previous paper in which we rederived the eigenvalues and radial eigenfunctions of the Hartmann potential using a formulation of one-dimensional supersymmetric quantum mechanics (SUSYQM) on the half-line $\left[0,\infty \right)$. A reformulation of SUSYQM in the full line $\left(-\infty ,\infty \right)$ is subsequently developed. It is found that the second formulation makes a connection between states having the same quantum number $L$ but different values of $\eta {\sigma }^2$ and quantum number $N$. This is in contrast to the first formulation, which relates states with identical values of the quantum number $N$ and $\eta {\sigma }^2$ but different values of the quantum number $L$.

arXiv comment24 pages; uses LaTex; to be published at the Theoretica Chimica Acta; hard copy available from the author upon request (use address: [email protected])