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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Classical tunneling as a consequence of radiation reaction forces

Frederik Denef B, Joris Raeymaekers, Urban M. Studer, Walter Troost

Published Sep 1, 1997• 11 citations
Publisher

Abstract

We show that the classical equation of motion of a radiating charged point particle (the Lorentz-Dirac equation) has ``tunneling'' solutions. For a given initial position and velocity we find that, contrary to common belief, several different physically acceptable solutions exist for a range of initial data. Both features are demonstrated for a rectangular barrier. To check that these phenomena are not dependent on the discontinuities of the potential, we also study in detail the solutions for a smoothened (single) potential step.

DOI
10.1103/physreve.56.3624

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Role of the slope of realistic potential barriers in preventing relativistic tunneling in the Klein zone

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arXiv:0706.2266•Physical Review A•Oct 4, 2007•10 citations
We show that the classical equation of motion of a radiating charged point particle (the Lorentz-Dirac equation) has ``tunneling'' solutions. For a given initial position and velocity we find that, contrary to common belief, several different physically acceptable solutions exist for a range of initial data. Both features are demonstrated for a rectangular barrier. To check that these phenomena are not dependent on the discontinuities of the potential, we also study in detail the solutions for a smoothened (single) potential step.
The classical equation of motion of a charged point particle, including its radiation reaction, is described by the Lorentz-Dirac equation. We found a new class of solutions that describe tunneling (in a completely classical context!). For nonrelativistic electrons and a square barrier, the solution is elementary and explicit. We show the persistance of the solution for smoother potentials. For a large range of initial velocities, initial conditions may leave a (discrete) ambiguity on the resulting motion.
The Lorentz-Dirac radiation reaction formula predicts that the position shift of a charged particle due to the radiation reaction is of first order in acceleration if it undergoes a small acceleration. A semi-classical calculation shows that this is impossible at least if the acceleration is due to a time-independent potential. Thus, the Lorentz-Dirac formula gives an incorrect classical limit in this situation. The correct classical limit of the position shift at the lowest order in acceleration is obtained by assuming that the energy loss at each time is given by the Larmor formula.
We make a brief review of the Kramers escape rate theory for the probabilistic motion of a particle in a potential well U(x), and under the influence of classical fluctuation forces. The Kramers theory is extended in order to take into account the action of the thermal and zero-point random electromagnetic fields on a charged particle. The result is physically relevant because we get a non null escape rate over the potential barrier at low temperatures (T -> 0). It is found that, even if the mean energy is much smaller than the barrier height, the classical particle can escape from the potential well due to the action of the zero-point fluctuating fields. These stochastic effects can be used to give a classical interpretation to some quantum tunneling phenomena. Relevant experimental data are used to illustrate the theoretical results.
It is argued that, contrary to conventional wisdom, no trustworthy universal self-force/radiative corrections to the Lorentz force equation, can be derived from the basic tenets of classical electrodynamics. This concords with the apparent randomness observed in quantum mechanical scattering experiments and with the absence of any experimental support for such universality. In a recent paper [11], the statistical effect of radiative corrections to the motion of charged bodies has been derived from the basic tenets and does take a universal form, described by quantum mechanical wave equations -- again conforming with experiment. As that derivation assumes nothing about the size, mass or composition of the body, it is conjectured that quantum mechanics is the appropriate framework for dealing also with radiative corrections to the motion of macroscopic bodies.
The transmission of fermions of mass mmm and energy EEE through an electrostatic potential barrier of rectangular shape (i.e., supporting an infinite electric field), of height U>E+mc2U>E+m{c}^{2}U>E+mc2---due to the many-body nature of the Dirac equation evidentiated by the Klein paradox---has been widely studied. Here we exploit the analytical solution, given by Sauter for the linearly rising potential step, to show that the tunneling rate through a more realistic trapezoidal barrier is exponentially depressed, as soon as the length of the regions supporting a finite electric field exceeds the Compton wavelength of the particle---the latter circumstance being hardly escapable in most realistic cases.