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

Abstract

One-dimensional potentials defined by V (S) (x) = S(S + 1)h 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 (S) n (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 (S) n (p) for large |p| and general questions about the supersymmetric hierarchies of potentials which include an infinite barrier.

Quantum control and quantum speed limits in supersymmetric potentials

“…and since the eigenstates of the full hierarchy of the supersymmetric Hamiltonians related to the infinite box can be written in terms of Chebyshev polynomials of the second kind [31], it can be exactly evaluated. However, these general expressions can become unwieldly for excited states in higher order Hamiltonians very quickly, and we therefore restrict ourselves here to an illustrative example for the groundstate and first excited state of H (α) , which allows us to contrast both groundstate and isospectral STAs.…”

“…This will allow us to gain insight into the effects on the QSL stemming from the energy for different states in the spectrum vs the ones coming from the distance between the initial and the final state in Hilbert space. In particular we consider the example of an expanding infinite box and the related higher order supersymmetric partner Hamiltonians [30,31] and show analytically that the knowledge of the shortcut for the initial Hamiltonian translates through the superpotential to all higher order ones. This then allows to efficiently construct the control parameters for any Hamiltonian in this connected supersymmetric hierarchy.…”

Nonequilibrium many-body dynamics in supersymmetric quenching

“…(1) n = n 2 π 2 2 /(2mL 2 ). The infinite box potential is a common system for which supersymmetric partners have been studied extensively [35], and, indeed, the superpotential for V (1) (x) and its partner potential V (2) (x) can be calculated as W (2)…”

“…where the superscript (2) references the target potential. The exact eigenfunctions of the supersymmetric Hamiltonian H (2) are known and the expression for the general superpotential for higher partners is given by [35,48]…”

“…To calculate the WPD for finite temperatures we include the thermal ensemble of the initial state p (1) n −N µ) , which in analogy with Eq. (35) gives…”