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Physics-Uspekhi

The Klein paradox and the zitterbewegung of an electron in a field with a constant scalar potential

S.V. Vonsovskii, Svirskii

Published May 31, 1993• 4 citations
Publisher

Abstract

The zitterbewegung of an electron in the case of a field with a constant nonzero scalar potential is examined. The relationship of the average velocity of the particle and its momentum is discussed, and also the relation between the energy and the scalar potential. It is shown that taking this relationship into account enables one to resolve the well-known Klein paradox in Dirac's relativistic quantum theory of the electron.

DOI
10.1070/pu1993v036n05abeh002182

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The zitterbewegung of an electron in the case of a field with a constant nonzero scalar potential is examined. The relationship of the average velocity of the particle and its momentum is discussed, and also the relation between the energy and the scalar potential. It is shown that taking this relationship into account enables one to resolve the well-known Klein paradox in Dirac's relativistic quantum theory of the electron.
Considers the Dirac equation with external electromagnetic fields and gives a careful formulation of the Stuckelberg-Feynman interpretation. This electron-positron theory may be applied whenever electron and positron states can be separated, which is possible for a wide class of time-independent potentials vanishing at infinity. One then may introduce a system of observables with a position operator X(t) which is free of Zitterbewegung and yields the usual relativistic connection between momentum and velocity. The Stuckelberg-Feynman interpretation is usually associated with a scattering theory (propagator formalism), which differs from the one of a one-particle interpretation (formalism imitating non-relativistic techniques). For the above-mentioned potentials there is a simple relationship between these physically different formalisms. There are, however, more general potentials for which the Stuckelberg-Feynman interpretation may still be applied asymptotically. With the help of Klein's paradox the authors show that the Dirac equation may then have unphysical solutions not conserving the total charge.
The Hamiltonian form of the Klein–Gordon equation is used to study the Zitterbewegung of spin- zero particles. It is shown that the time dependence of the velocity and position operators in the Heisenberg picture is formally the same as that of a spin- 1/2 particle. It is also shown that the Zitterbewegung is the result of the interference between positive and negative energy states. A representation-free expression is found for the mean position operator. This operator is free from Zitterbewegung. The Klein–Gordon equation is solved with an electrostatic potential originally used by Sauter in his investigation of the Klein paradox for spin- 1/2 particles. It is shown that the transmission to states with negative kinetic energy is very small when the electric field is weak.
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.
Using space-time resolved solutions in relativistic quantum field theory we analyze the electron-positron pair creation process from vacuum and resolve the Klein-paradox. We show that there are in principle no limitations to the localization length of an electron and demonstrate that its spatial probability density can be much narrower than the Compton wavelength. We also argue that quantum field theory prohibits the occurrence of Zitterbewegung for an electron in contrast to claims found in textbooks.
Dirac's Relativistic Wave Equation implies a measured electron velocity of ±c\pm c±c in any direction, in contradiction to Special Relativity and observation. It is shown in this article that this anomalous electron velocity reveals an internal structure of the electron whereby the mass and the charge of the electron cannot be described by the same position operator. The measured velocity of electron mass is always less than ccc in any direction but charge can be displaced at the speed of light. This speed is realizable only when the electron is in a state that is a superposition of positive and negative energy states, also known as a zitterbewegung state. It is shown that in zitterbewegung it is the charge and not the mass that undergoes rapid spatial oscillation, and that there are measurable consequences of this charge zitterbewegung. Zitterbewegung of charge also occurs in an entangled electron-positron pair created by a strong electric field.