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arXiv:0708.2933

SUSY Shape-Invariant Hamiltonians for the Generalized Dirac-Coulomb Problem

R. de Lima Rodrigues, Arvind Narayan Vaidya

Published Aug 21, 2007• 0 citations
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Abstract

A spin $\frac 12$ relativistic particle described by a general potential in terms of the sum of the Coulomb potential with a Lorentz scalar potential is investigated via supersymmetry in quantum mechanics.

DOI
10.48550/arxiv.0708.2933

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A spin 12\frac 1221​ relativistic particle described by a general potential in terms of the sum of the Coulomb potential with a Lorentz scalar potential is investigated via supersymmetry in quantum mechanics.
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.
In the present paper, we will concern ourselves with the extended phase-space quantum mechanics of particles which have both bosonic and fermionic degrees of freedom, i.e., the quantum field theory in (0 + 1) dimensions in q -(position) and p -(momentum) spaces, exhibiting supersymmetry. We present ( N = 2) realization of extended supersymmetry algebra and discuss the vacuum energy and topology of super-potentials. Shape invariance of exactly solvable extended SUSY potentials allows us to obtain analytic expressions for the entire energy spectrum of an extended Hamiltonian with, for example, Scarf potential without ever referring to an underlying differential equation.