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Physical Review A

Supersymmetry aspects of the Dirac equation in one dimension with a Lorentz scalar potential

Yukihisa Nogami, F. Masafumi Toyama

Published Mar 1, 1993• 69 citations

Abstract

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.

DOI: 10.1103/physreva.47.1708

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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.

Supersymmetric Dirac Hamiltonians in (

1+1

) dimensions revisited

Abstract The most general Dirac Hamiltonians in

(1+1)

( 1 + 1 ) dimensions are revisited under the requirement to exhibit a supersymmetric structure. It is found that supersymmetry allows either for a scalar or a pseudo-scalar potential. Their spectral properties are shown to be represented by those of the associated non-relativistic Witten model. The general discussion is extended to include the corresponding relativistic and non-relativistic resolvents. As example, the well-studied relativistic Dirac oscillator is considered and the associated resolved kernel is found in a closed-form expression by utilising the energy-dependent Green’s function of the non-relativistic harmonic oscillator. The supersymmetric quasi-classical approximation for the Witten model is extended to the associated relativistic model.

In this work, a spin

\frac 12

relativistic particle described by a generalized potential containing both the Coulomb potential and a Lorentz scalar potential in Dirac equation is investigated in terms of the generalized ladder operators from supersymmetry in quantum mechanics. This formalism is applied for the generalized Dirac-Coulomb problem, which is an exactly solvable potential in relativistic quantum mechanics. We obtain the energy eigenvalues and calculate explicitly the energy eigenfunctions for the ground state and the first excited state.

We develop a unified treatment of scattering and discrete spectra for the one-dimensional Dirac equation with scalar and vector interactions. Under the spin-symmetry condition, the coupled first-order Dirac system maps exactly onto an effective Sturm--Liouville (Schrö\-din\-ger-like) problem for a single spinor component. This mapping provides a convenient framework for analyzing transmission, reflection, and analytic continuation. As an explicit application, we consider effective interactions of hyperbolic Pöschl--Teller type and exploit supersymmetric quantum mechanics and shape invariance to obtain a closed-form expression for the transmission probability. The bound-state spectrum is then recovered from the poles of the analytically continued transmission amplitude, reproducing known results and offering a unified description of scattering and bound states. For the barrier configuration, we briefly comment on the resulting pole pattern in the complex momentum plane and its connection with resonance and quasi-normal-mode behavior. Moreover, we use the chiral transformation to relate the spin- and pseudospin-symmetry sectors and translate results between them without repeating the full derivation.

We extend the notion of Dirac oscillator in two dimensions, to construct a set of potentials. These potentials become exactly and quasi-exactly solvable potentials of nonrelativistic quantum mechanics when they are transformed into a Schrödinger-like equation. For the exactly solvable potentials, eigenvalues are calculated and eigenfunctions are given by confluent hypergeometric functions. It is shown that, our formulation also leads to the study of those potentials in the framework of the supersymmetric quantum mechanics.