Dirac Hamiltonian in a supersymmetric framework

TL;DR

This paper explores embedding supersymmetry into the one-dimensional Dirac Hamiltonian with scalar and pseudoscalar potentials by constructing a quasi-Hamiltonian, revealing superpotential structures and applying the method to a magnetic field-influenced electron model.

Contribution

It introduces a novel approach using a quasi-Hamiltonian to embed supersymmetry in the Dirac Hamiltonian, providing new insights into its spectral properties.

Findings

Superpotential identified via quasi-Hamiltonian analysis.

Spectral solutions obtained for isochronous potentials.

Method applied to a magnetic field-influenced electron Hamiltonian.

Abstract

We investigate the most general form of the one-dimensional Dirac Hamiltonian $H_D$ in the presence of scalar and pseudoscalar potentials. To seek embedding of supersymmetry (SUSY) in it, as an alternative procedure to directly employing the intertwining relations, we construct a quasi-Hamiltonian $\mathcal{K}$, defined as the square of $H_D$, to explore the consequences. We show that the diagonal elements of $\mathcal{K}$ under a suitable approximation reflects the presence of a superpotential thus proving a useful guide in unveiling the role of SUSY. For illustrative purpose we apply our scheme to the transformed one-dimensional version of the planar electron Hamiltonian under the influence of a magnetic field. We generate spectral solutions for a class of isochronous potentials.