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arXiv:hep-th/0112156 · Journal of Physics A Mathematical and General

Low momentum scattering in the Dirac equation

Piers Kennedy, Norman Dombey

Published Jul 25, 2002• 35 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/0305-4470/35/31/309

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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.
In one-dimensional scattering, the portion of particles that is transmitted in general vanishes as the kinetic energy of the incident particles approaches zero. Based upon published results from the literature, the present work shows that for a potential consisting of two Dirac δ functions of arbitrary strength, a finite portion of the incident particles is transmitted at threshold for certain choices of the set of parameters defining the potential. The underlying reasons are shown to be related to the structure of the bound levels of the potential. It is shown that for a potential with a finite range but of general shape the reflection coefficient in general tends to unity near threshold, which is what is usually observed. However, in the presence of a bound level at the onset of the continuum, a finite portion of the incident particles is transmitted at threshold for a potential with a general shape and, furthermore, it can be shown that if the potential V is symmetric with V(x)=V(−x) then resonance transmission is observed, i.e., the reflection coefficient vanishes at threshold.
We calculate the tunneling process of a Dirac particle across two square barriers separated by a distance d, as well as the scattering by a double-cusp barrier, where the centers of the cusps are separated by a distance larger than their screening lengths. Using the scattering matrix formalism, we obtain the transmission and reflection amplitudes for the scattering processes of both configurations. We show that the presence of transmission resonances modifies the Lorentzian shape of the energy resonances and induces the appearance of additional maxima in the transmission coefficient in the range of energies where transmission resonances occur. We calculate the Wigner time delay and show how their maxima depend on the position of the transmission resonance.
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.
It is well known that, Klein paradox is one of the most exotic and counterintuitive consequences of quantum theory. Nevertheless, many discussions about the Klein paradox are based upon single-particle Dirac equation in quantum mechanics rather than quantum field method. By using the path integral formalism, we evaluate the reflection and transmission coefficients up to the lowest order for the electron scattering by the finite square barrier potential. Within the context of assuming the step potential is the limiting case of the finite square barrier potential, we explain the Klein paradox that is caused by the ill-definition of the step potential.
The reflection of wave packets from a quantum well bounded by rectangular potential barriers, and which has a tunneling transmission resonance, is discussed. Numeric solutions to Schrödinger’s time-dependent equation show that the shape of the probability function ψψ* for a reflected wave packet can be quite different from that of the incident packet. Either a symmetric double hump or an asymmetric shape are possible for the reflected ψψ*, depending on the sharpness of the resonance. These shapes are explained analytically in terms of the reflection coefficient.