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Вестник Санкт-Петербургского университета. Серия 4. Физика. Химия

Релятивистское уравнение Шрёдингера со ступенчатым потенциалом

Александр Викторович Головин, Владимир Меерович Лагодинский

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

In this work a version of relativistic quantum mechanics based on a local definition of a square root operator is used to solve the problem of a particle reflection on a step potential. It is shown that such an approach does not lead to Klein paradox and other problems of mainstream formalism.

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•Jan 1, 2013•0 citations
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.
We present a new resolution of the Klein paradox by considering the problem of electron incidence on a step potential barrier using modified Dirac equations (MDE). Contrary to the generally accepted interpretation, which is based on the Dirac equation (DE) and predicts electron tunneling through an infinitely high barrier, our calculations show that tunneling is impossible if the electron kinetic energy is less than the barrier height. We propose to investigate MDE as an alternative DE relativistic quantum mechanical equation for spin ½ particles․
This paper explores the topic of relativistic particles with zero spins from a unique perspective. Our approach is derived from the Dunkl derivative, which we used to investigate this issue. By examining the ([Formula: see text])-dimensional DKP equation, we obtain eigenfunctions. Additionally, we replace the standard partial derivative with the Dunkl derivative and solve the relativistic particle problem in a box using the new formalism. We then determine the energy spectrum for this scenario. Following this, we investigate the scattering of the potential step problem and the Ramsauer–Townsend effect separately. Finally, we calculate the coefficients of transmission and reflection.
Abstract The Dirac equation requires a treatment of the step potential that differs fundamentally from the traditional treatment, because the Dirac plane waves, besides momentum and spin, are characterized by a quantum number with the physical meaning of sign of charge. Since the Hermitean operator corresponding to this quantum number does not commute with the step potential, the time displacement parameter used in the ansatz of the stationary state does not have the physical meaning of energy. Therefore there are no paradoxal values of the ‘energy’. The new solution of the Dirac equation with a step potential is obtained. This solution, again, allows for phenomena of the Klein paradox type, but in addition it contains a positron amplitude localized at the threshold point of the step potential.
We derive the equations of relativistic quantum mechanics from a modified version of classical electrodynamics, where probability is replaced by potential. As a result, a particle is not a localized entity, in the classical sense, but has a localized energy extremum. The particle/wave aspect of matter is inherent in the particle/wave equation describing elementary particles. Furthermore, the Heisenberg uncertainty and Planck-Einstein-de Broglie relations, and the Klein-Gordon, Dirac and Proca equations follow naturally from the particle/wave equation. In addition, we incorporate a new and more physical interpretation of spin angular momentum.