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American Journal of Physics

Supersymmetric quantum mechanics: Examples with dirac δ functions

J. Goldstein, C. Lebiedzik, Richard W. Robinett

Published Jul 1, 1994• 16 citations
Publisher

Abstract

We examine the bound-state spectra of several one-dimensional supersymmetric quantum mechanical systems containing Dirac δ-function potentials. Specifically, we investigate the cases of two attractive δ functions, as well as both the infinite well and simple harmonic potentials with an added attractive or repulsive δ function.
DOI
10.1119/1.17536

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We study quantum potentials containing Dirac delta function (DDF) within its supersymmetric (isospectral) construction via a discontinuous superpotential, instead of being simply an added part. This construction comprises of two independently solvable sectors joined at the singular point by matching conditions imposed by the corresponding self-adjoint extension. Needing careful regularization, these boundary conditions impose independent algebraic constraints on each of the eigenstates, that restrict and truncate the spectrum. The number of surviving bound states, which are required to be parity-even, is thus fixed by the available system parameters rather than by the usual quantization rule. The specific cases of the Harmonic oscillator and the Rosen-Morse potential isospectrally infused with the DDF demonstrate such highly restrictive spectra, while the similar combination of the DDF with a Calogero-type singularity turns out to be largely incompatible as the regularization breaks down. Therefore, localized singularities are capable of controlling and engineering the discrete spectrum of quantum systems, with possible use in myriads of physical systems with very localized interactions.
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We review SUSY in non-relativistic quantum mechanics as a powerful tool for studying one dimensional potentials and discuss some applications. The Dirac operatorial factorization method for the harmonic oscillator in QM is generalized to a broader scope. A Schr¨odinger second order operator can be factorized into two first-order ones by solving a Riccati equation and determining the corresponding superpotential. We apply the technique to square well, P¨oschl-Teller and finite barrier potentials, analyzing the resulting energy spectra and the transmission and reflection coefficients. Both discrete and continuous energy spectrum cases are discussed.
The Dirac equation in one dimension with a Lorentz scalar potential is associated with a supersymmetric pair of Schr\"odinger Hamiltonians H1{\mathit{H}}_{1}H1​ and H2{\mathit{H}}_{2}H2​. The H1{\mathit{H}}_{1}H1​ and H2{\mathit{H}}_{2}H2​ share the same energy spectrum and scattering phases. The shared spectrum includes the lowest states unless the Dirac equation allows a zero mode (a zero-energy bound state). This situation is unlike the common examples of supersymmetric quantum mechanics. The Dirac equation admits a zero mode only if the scalar potential has certain ``topology.'' Various such features are illustrated through explicit examples. In particular, the phase equivalence between H1{\mathit{H}}_{1}H1​ and H2{\mathit{H}}_{2}H2​ is exploited to construct transparent potentials for the Dirac equation in one dimension.