Low momentum scattering in the Dirac equation

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 non-zero 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.

Full transmission within a wide energy range and supercriticality in relativistic barrier scattering

“…Recently, electron transport through electrostatic barriers in single and bi-layer graphene has been studied using the Dirac equation [8]. The study of transmission resonances in relativistic wave equations in external potentials has been discussed extensively in the literature [9][10]. In this case, for given values of the energy and shape of the barrier, the probability of transmission reaches unity even if the potential strength is larger than the energy of the particle, a phenomenon that is not present in the nonrelativistic case.…”

Relativistic Treatment of Massive Klein-Gordon Particle by Modified Generalized Hulthen Potential

“…In this paper, we assume that the mass of the Klein-Gordon particle depends on the spatial coordinate as [29] 0 1…”

Reflectionless {\cal P}{\cal T} -symmetric potentials in the one-dimensional Dirac equation

“…of the first equation, k 2 > 0 for scattering states, while for bound states, k 2 < 0 implies an imaginary value of k, corresponding to poles of the transmission coefficient. In the limiting case k = 0, both normalizable bound states and non-normalizable half-bound states [30], corresponding to transmission resonances, are possible, depending on the potentials under consideration. When c V = −c S = c ′′ , ψ 1 and ψ 2 exchange their role, since ψ 2 now satisfies a Schrödinger-like equation with the original f (x) and the energy-dependent strength s ′′ (E) = 2c ′′ (E − m), while ψ 1 is proportional to the space derivative of ψ 2 .…”