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Decoupling the One Dimensional Dirac Equations With a Scalar Potential

Francesco R. Ruggeri

Published Aug 10, 2019• 0 citations

Abstract

In a previous note (1), it was noted one could write a solution of the bound state Schrodinger equation as W(x)=Wo(x)Dn(x) where W(x) is the wavefunction and Wo(x), the ground state wavefunction. Inserting into the Schrodinger equation one may obtain the Schrodinger equation for Wo(x) and an equation coupling Wo(x) and Dn(x). In this note, we suggest such an approach may be applied to the one dimensional bound Dirac equations. This allows one to obtain decoupled equations for u1(x) and v1(x) where u(x)=uo(x)u1(x) and v(x)=v1(x)vo(x) are the full solutions. Thus, if one is able to obtain a solution for uo(x) and vo(x) by inspection, the decoupled equations allow one to find the general solution. We apply this approach to the case of a scalar potential g|x|. The Dirac problem for this potential has been solved in the literature (2) with decoupling occurring through the taking of spatial derivatives. Here we wish to suggest a general method.

DOI: 10.5281/zenodo.3365446

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In a previous note, the nonrelativistic bound state wavefunction W(x) was written as Wo(x)W1(x) with Wo(x) being the ground state wavefunction. W1(x) satisfies an eigenvalue equation, similar to the Schrodinger equation, but with the potential term replaced with a coupling term: -1/2m d/dx d/dx W1(x) - 1/m d/dx W1(x) [ d/dx Wo(x)/Wo(x) ] = (E-Eo) W1 (x) ((1)) Mathematically, ((1)) may be compared to the hypergeometric differential equation and solutions found for the Schrodinger if coefficients meet certain criteria. A similar approach applies to the Klein Gordon and Dirac equations In this note, we wish to examine possible physical significance of Wo(x)W1(x). In addition, we wish to obtain an equation, analogous to ((1)) for both the Klein Gordon and Dirac equations to compare the coupling terms.

The Schrödinger, Klein–Gordon, and Dirac equations for the hydrogen atom are solved in D dimensions, where D may be noninteger. For D=1 the nondegenerate ground state with infinite binding energy is obtained as a limit as D tends to 1 for the Schrödinger equation. The corresponding solution of the Klein–Gordon equation is shown to be unacceptable, contradicting previous work. Unexpectedly, no bound state solutions of the Dirac equation with D=1 are found at all. A few comments are made on the D=3 and 2 cases.