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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 repeatedsupersymmetrization of the infinite square well, here defined over the region$(0,a)$. We review the derivation of this hierarchy of potentials and then usethe methods of supersymmetric quantum mechanics, as well as more familiartextbook techniques, to derive compact closed-form expressions for thenormalized solutions, $\psi_n^{(S)}(x)$, for all $V^{(S)}(x)$ in terms ofwell-known special functions in a pedagogically accessible manner. We also notehow the solutions can be obtained as a special case of a family ofshape-invariant potentials, the trigonometric P\"oschl-Teller potentials, whichcan be used to confirm our results. We then suggest additional avenues forresearch questions related to, and pedagogical applications of, thesesolutions, including the behavior of the corresponding momentum-space wavefunctions $\phi_n^{(S)}(p)$ for large $|p|$ and general questions about thesupersymmetric hierarchies of potentials which include an infinite barrier.

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