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Physical Review A

Klein paradox with spin-resolved electrons and positrons

Piotr Krekora, Qichang C. Su, R. Grobe

Published Dec 30, 2005• 55 citations
Publisher

Abstract

Using numerical solutions to relativistic quantum field theory with space-time resolution, we illustrate how an incoming electron wave packet with a definite spin scatters off a supercritical potential step. We show that the production rate is reduced of only those electrons that have the same spin as the incoming electron is reduced. This spin-resolved result further clarifies the importance of the Pauli-exclusion principle for the Klein paradox.

DOI
10.1103/physreva.72.064103

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Using numerical solutions to relativistic quantum field theory with space-time resolution, we illustrate how an incoming electron wave packet with a definite spin scatters off a supercritical potential step. We show that the production rate is reduced of only those electrons that have the same spin as the incoming electron is reduced. This spin-resolved result further clarifies the importance of the Pauli-exclusion principle for the Klein paradox.
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.
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.
We review the steps needed to reduce quantum field theory to quantum mechanics as its limiting case. While processes involving a changing number of particles require a quantum field theoretical description, some works used quantum mechanical data based on the single particle Dirac equation to predict the outcome. These non-quantum field theoretical approaches have led to conceptual controversies that are often related to the negative energy states, which are characteristic of the Dirac equation. As a result, the issues about the physical reality of the relativistic localisation phenomenon, the Zitterbewegung and the Klein paradox are widely discussed in many textbooks. In order to contribute to this discussion, we review recent computer simulations of oversimplified model systems based on quantum field theory. The direct visualisation of intrinsically quantum field theoretical process with full spatial and temporal resolutions can provide us with some first insight into the spontaneous electron–positron pair creation from the vacuum triggered by a strong external (supercritical) force field and with the resolution of the Klein paradox.
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.
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.