Klein's paradox and the relativistic point interaction

Abdulaziz D Alhaidari, A. Al‐Hajry, Ahmad Umar +4 more

Paweł Kurzyński

Alberto Molgado, Ociel Morales, José A. Vallejo

Whenever we consider any relativistic quantum wave equation we are confronted with the Klein paradox, which asserts that incident particles will suffer a surplus of reflection when dispersed by a discontinuous potential. Following recent results on the Dirac equation, we propose a solution to this paradox for the Klein-Gordon case by introducing virtual beams in a natural well-posed generalization of the method of images in the theory of partial differential equations. Thus, our solution considers a global reflection coefficient obtained from the two contributions, the reflected particles plus the incident virtual particles. Despite its simplicity, this method allows a reasonable understanding of the paradox within the context of the quantum relativistic theory of particles (according to the original setup for the Klein paradox) and without resorting to any quantum field theoretic issues.

T Shivalingaswamy, Nagalakshmi A. Rao, B. A. Kagali

In this article, we discuss a paradoxical result that exists for positive energy Klein-Gordon particles. It may be called “Klein paradox” of another kind. We have shown its existence in one, two and three dimensions. The paradox seems to indicate the limitations of the single particle Klein-Gordon equation in the presence of strong potentials.

Francis W. Sears

M. Alkhateeb, A. Matzkin

The “particle-in-a-box” problem is investigated for a relativistic particle obeying the Klein–Gordon equation. To find the bound states, the standard methods known from elementary non-relativistic quantum mechanics can only be employed for “shallow” wells. For deeper wells, when the confining potentials become supercritical, we show that a method based on a scattering expansion accounts for Klein tunneling (undamped propagation outside the well) and the Klein paradox (charge density increase inside the well). We will see that in the infinite well limit, the wave function outside the well vanishes, and Klein tunneling is suppressed: Quantization is, thus, recovered, similar to the non-relativistic particle in a box. In addition, we show how wave packets can be constructed semi-analytically from the scattering expansion, accounting for the dynamics of Klein tunneling in a physically intuitive way.