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

Abstract

The transmission of fermions of mass m and energy E through an electrostatic potential barrier of rectangular shape (i.e. supporting an infinite electric field), of height U > E + m c 2 -due to the many-body nature of the Dirac equation evidentiated by the Klein paradox -has been widely studied. We exploit here the analytical solution, given by Sauter for the linearly rising potential step, to show that the tunnelling 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 wavelenght of the particle -the latter circumstance being hardly escapable in most realistic cases.

Charge Conservation, Klein’s Paradox and the Concept of Paulions in the Dirac Electron Theory

“…Although the result for the transmission, obtained in [32], contains a small factor due to an exponential diminution of the electron ψ -function under the barrier, it cannot be correct in view of joining of pieces of ψ -functions with different charge indices. It is obvious then that the recent theoretical analysis [12] of anomalous high-energy electron transmission through high potential barriers, based on the above-mentioned results by Klein and Sauter, cannot be correct either.…”

“…Now we are turning to the situation with V 0 > M which demands special attention as the Klein paradox is assumed [2][3][4][5][6]12] to take place just in this case for energy values M < E < V 0 − M (the so-called Klein zone). Our approach, based on the paulion concept and charge-index conservation law, leads to radically different results which disavow the notion of the Klein paradox.…”

“…A striking common feature of all theoretical works on the Klein paradox (see [11] for a historical review and [12] for recent results on an anomalous electron transmission through high potential barriers) is the fact that they do not notice the charge non-conservation hidden in the phenomenon under study. In this communication we show that the Klein paradox is a result of fallacious theoretical analysis.…”

“…That is why, despite a considerable recent progress [21][22][23] in this field, a number of fundamental theoretical problems, which are related to the Dirac equation physics and are a subject of our investigations, remained unsolved up to the present moment. This particularly concerns the Klein paradox, the Dirac measurements, localization and motion of the Dirac wave packets [3][4][5][6], and the anomalous transmission of particles through high potential barriers [12].…”

Relativistic time-dependent quantum dynamics across supercritical barriers for Klein-Gordon and Dirac particles

“…[1][2][3][4]) give different, often conflicting accounts of the Klein paradox. This situation is reflected even in recent works [5][6][7][8][9][10][11][12][13][14][15], that reach different conclusions generally based on time independent considerations.…”