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arXiv:1704.04108 · European Journal of Physics

The Klein paradox: a new treatment

Egon Truebenbacher

Published Jul 15, 2015• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1088/0143-0807/36/5/055015

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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.
The Klein paradox is one of the cornerstones in the development of quantum mechanics, and its consequences were used in various branches of physics, ranging from elementary particles to solid state. Yet its mathematical derivation is questionable in a number of steps, resulting in the wrong solution of Dirac equation. In this paper, the paradox is analysed in more detail, and its mathematical content is emphasized in order to show that in the study of extreme conditions of matter simple arguments may result in erroneous predictions.
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.
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․
The early papers by Klein, Sauter and Hund which investigate scattering off a high step potential in the context of the Dirac equation are discussed to derive the 'paradox' first obtained by Klein. The explanation of this effect in terms of electron-positron production is re-assessed. It is shown that a potential well or barrier in the Dirac equation can become supercritical and emit positrons or electrons spontaneously if the potential is strong enough. If the well or barrier is wide enough, a seemingly constant current is emitted. This phenomenon is transient whereas the tunnelling first calculated by Klein is time-independent. It is shown that tunnelling without exponential suppression occurs when an electron is incident on a high barrier, even when the barrier is not high enough to radiate. Klein tunnelling is therefore a property of relativistic wave equations and is not necessarily connected to particle emission. The Coulomb potential is investigated and it is shown that a heavy nucleus of sufficiently large Z will bind positrons. Correspondingly, as Z increases the Coulomb barrier should become increasingly transparent to positrons. This is an example of Klein tunnelling. Phenomena akin to supercritical positron emission may be studied experimentally in superfluid 3 He.
The well-known Klein paradox for the relativistic Dirac wave equation consists in the computation of possible "negative probabilities" induced by certain potentials in some regimes of energy. The paradox may be resolved by employing the notion of electron–positron pair production in which the number of electrons present in a process can increase. The Klein paradox also exists in the Maxwell's equations viewed as the wave equation for photons. In a medium containing "inverted energy populations" of excited atoms, e.g. in a LASER medium, one may again compute possible "negative probabilities." The resolution of the electromagnetic Klein paradox is that when the atoms decay, the final state may contain more photons then were contained the initial state. The optical theorem total cross-section for scattering photons from excited state atoms may then be computed as negative within a frequency band with matter induced amplification.
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.