Exact results for the infinite supersymmetric extensions of the infinite square well
One-dimensional potentials defined by $V^{(S)}(x) =S(S+1) \hbar^2 π^2 /[2ma^2\sin^2(πx/a)]$ (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, $ψ_n^{(S)}(x)$, 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 Pöschl-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 $ϕ_n^{(S)}(p)$ for large $|p|$ and general questions about the supersymmetric hierarchies of potentials which include an infinite barrier.