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arXiv:hep-th/0107170 · Journal of Physics A Mathematical and General

The Woods–Saxon potential in the Dirac equation

Piers Kennedy

Published Jan 14, 2002• 86 citations
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Abstract

The two-component approach to the one-dimensional Dirac equation is applied to the Woods–Saxon potential. The scattering and bound state solutions are derived and the conditions for a transmission resonance (when the transmission coefficient is unity) and supercriticality (when the particle bound state is at E = − m ) are then derived. The square potential limit is discussed. The recent result that a finite-range symmetric potential barrier will have a transmission resonance of zero momentum when the corresponding well supports a half-bound state at E = − m is demonstrated.

DOI
10.1088/0305-4470/35/3/314

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The two-component approach to the one-dimensional Dirac equation is applied to the Woods–Saxon potential. The scattering and bound state solutions are derived and the conditions for a transmission resonance (when the transmission coefficient is unity) and supercriticality (when the particle bound state is at E = − m ) are then derived. The square potential limit is discussed. The recent result that a finite-range symmetric potential barrier will have a transmission resonance of zero momentum when the corresponding well supports a half-bound state at E = − m is demonstrated.
The one-dimensional Dirac equation with position dependent mass in the generalized asymmetric Woods-Saxon potential is solved in terms of the hypergeometric functions. The transmission and reflection coefficients are obtained by considering the one-dimensional electric current density for the Dirac particle and the equation describing the bound states is found by utilizing the continuity conditions of the obtained wave function. Also, by using the generalized asymmetric Woods-Saxon potential solutions, the scattering states are found out without making calculation for the Woods-Saxon, Hulthen, cusp potentials, and so forth, which are derived from the generalized asymmetric Woods-Saxon potential and the conditions describing transmission resonances and supercriticality are achieved. At the same time, the data obtained in this work are compared with the results achieved in earlier studies and are observed to be consistent.
The exact analytical solutions of the Schrödinger equation for the generalized symmetrical Woods-Saxon potential are examined for the scattering, bound, and quasi-bound states in one dimension. The reflection and transmission coefficients are analytically obtained. Then, the correlations between the potential parameters and the reflection-transmission coefficients are investigated, and a transmission resonance condition is derived. Occurrence of the transmission resonance has been shown when incident energy of the particle is equal to one of the resonance energies of the quasi-bound states.
Abstract We have solved exactly the two-component Dirac equation in the presence of a spatially one-dimensional Hulthén potential, and presented the Dirac spinors of scattering states in terms of hypergeometric functions. We have derived the reflection and transmission coefficients using the matching condition on the wavefunctions, and investigated the condition for the existence of transmission resonance. Furthermore, we have demonstrated how the transmission resonance depends on the shape of the potential.
It is shown that the amplitude for reflection of a Dirac particle with arbitrarily low momentum incident on a potential of finite range is −1 and hence the transmission coefficient T = 0 in general. If, however, the potential supports a half-bound state at momentum k = 0 this result does not hold. In the case of an asymmetric potential the transmission coefficient T will be nonzero whilst for a symmetric potential T = 1. Therefore in some circumstances a Dirac particle of arbitrarily small momentum can tunnel without reflection through a potential barrier.