Exact results for the infinite supersymmetric extensions of the infinite square well

TL;DR

This paper derives exact, normalized solutions for a hierarchy of supersymmetric quantum potentials based on the infinite square well, using supersymmetric quantum mechanics and special functions, with implications for pedagogical and research applications.

Contribution

It provides explicit closed-form solutions for all potentials in the supersymmetric hierarchy of the infinite square well, connecting them to shape-invariant P"oschl-Teller potentials.

Findings

Explicit normalized solutions for all S in the hierarchy.

Connections established with shape-invariant P"oschl-Teller potentials.

Discussion of momentum-space wave functions and further research directions.

Abstract

One-dimensional potentials defined by $V^{(S)}(x) =S(S+1) \hbar^2 \pi^2 /[2ma^2\sin^2(\pi x/a)]$ (for integer $S$) arise in the repeated supersymmetrization of the infinite square well, here defined over the region $(0,a)$. We review the derivation of this hierarchy of potentials and then use the methods of supersymmetric quantum mechanics, as well as more familiar textbook techniques, to derive compact closed-form expressions for the normalized solutions, $\psi_n^{(S)}(x)$, for all $V^{(S)}(x)$ in terms of well-known special functions in a pedagogically accessible manner. We also note how the solutions can be obtained as a special case of a family of shape-invariant potentials, the trigonometric P\"oschl-Teller potentials, which can be used to confirm our results. We then suggest additional avenues for research questions related to, and pedagogical applications of, these solutions, including the behavior of the corresponding momentum-space wave functions $\phi_n^{(S)}(p)$ for large $|p|$ and general questions about the supersymmetric hierarchies of potentials which include an infinite barrier.