We find an exact solution to the Dirac equation in 1–1 dimension space-time in the presence of a time-dependent potential that consists of a combination of electric, scalar, and pseudoscalar terms.

arXiv:0906.0122Canadian Journal of PhysicsFeb 1, 20109 citations

Annals of Physics

On the Dirac Equation in 3 + 1 Dimensions

Garnet N. Ord, D. G. C. McKeon

Published Mar 1, 1993• 30 citations

Publisher

DOI: 10.1006/aphy.1993.1022

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We find an exact solution to the Dirac equation in 1–1 dimension space-time in the presence of a time-dependent potential that consists of a combination of electric, scalar, and pseudoscalar terms.

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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.