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Communications in Theoretical Physics

Supersymmetry and Solution of Dirac Equation with Vector and Scalar Potentials

Ju Guo-Xing, Zhongzhou Ren

Published Feb 1, 2008• 16 citations
Publisher

Abstract

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.
DOI
10.1088/0253-6102/49/2/14

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In this work, a spin 12\frac 1221​ 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.
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.
Abstract By using the fundamental principles of supersymmetric quantum mechanics methodology and parametric Nikiforov-Uvarov method respectively, we obtained both the positive and negative (spin and pseudospin symmetries) energy equations of the Dirac equation with scalar and vector potentials. The nonrelativistic limit of the spin symmetry was obtained, and this was used to calculate Fisher information for both position space and momentum space respectively. The behaviour of eigenvalue energy with both the angular frequency and equilibrium bond length respectively were studied. Five special cases of the potential under consideration were also studied. Finally, we examined the feature of Fisher information with the angular frequency which was found to obey Heisenberg uncertainty principle.