O paradoxo de Klein em diferentes acoplamentos na equação de Klein-Gordon

Abstract

In this work, the Klein-Gordon equation was developed and used in the one-dimensional step potential through an arbitrary mixing of distinct couplings. For contextualization, the Schrödinger equation is studied and applied in the same potential and is later used as a basis for comparison with relativistic theory. Current issues like the continuity equation, free particle, addition of external potentials, and stationary states are explored in both theories. In the addition of external potentials, two couplings are distinguished for the wave equation, the vector coupling and the scalar one. Then, an arbitrary mixing of both couplings is applied to the Klein-Gordon equation. Thereafter, the one-dimensional Klein-Gordon equation is applied to the step potential, where the theory leads to three classes of solutions in its positive interval. In one of these classes, it is possible to learn about the Klein paradox and analyze the influence of scalar coupling on the mechanism of particle-antiparticle pair prodution. It is showed, for example, that the mechanism occurs only when the vector coupling exceeds in proportion the scalar one. Moreover, the presence of a scalar coupling, even a small one, contributes for increasing the threshold energy for the pair production. Lasty, it is observed that the use of a scalar potential also influences the accuracy of the localization of a particle in relativistic quantum mechanics, leading to the introduction of the effective Compton wavelength concept to avoid apparent paradoxes.