Introducing supersymmetric quantum mechanics via the Dirac equation

Yukihisa Nogami, F. Masafumi Toyama

The Dirac equation in one dimension with a Lorentz scalar potential is associated with a supersymmetric pair of Schr\"odinger Hamiltonians ${\mathit{H}}_{1}$ and ${\mathit{H}}_{2}$. The ${\mathit{H}}_{1}$ and ${\mathit{H}}_{2}$ 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 ${\mathit{H}}_{1}$ and ${\mathit{H}}_{2}$ is exploited to construct transparent potentials for the Dirac equation in one dimension.

Barry A. Freedman

M. Kibler, Mohammed Daoud A

Ju Guo-Xing, Zhongzhou Ren

The Dirac equations with vector and scalar potentials of the Coulomb types in two and three dimensions are solved using the supersymmetric quantum mechanics method. For the system of such potentials, the analytical expressions of the matrix elements for both position and momentum operators are obtained.

A. Frank, P. Van Isacker, Jan Jolie

Sahar Arbabi Moghadam, M. Eshghi, Hossein Mehraban

In this work, exact solutions of the Dirac equation are studied for the pseudo-harmonic oscillatory ring-shaped potential by using the supersymmetry method. The un-normalized eigenfunctions and the eigenvalue equation of the Dirac equation are presented. Some numerical results are given. We also discuss the special cases of this potential. In addition, valence energy states can be produced from our solution for the whole state by taking the appropriate transformation of parameters.