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International Journal of Modern Physics A

Relativistic tunneling time under the generalized uncertainty principle

Lin-Fang Deng, Heyao Zhang, Chao‐Yun Long

Published May 26, 2022• 0 citations
Publisher

Abstract

The one-dimensional Dirac equation is studied for square-well potential in the presence of generalized uncertainty principle (GUP). Based on the exact solution, the general and explicit relation between the phase time and the dwell time is given and quantified. The phase time, dwell time and self-interference delay are calculated as well and it is shown that the GUP has effects on the tunneling time. Moreover, the effect of GUP on the tunneling time showed a slight change with the change of mass, and goes to zero with the increase of kinetic energy.

DOI
10.1142/s0217751x22501007

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The one-dimensional Dirac equation is studied for square-well potential in the presence of generalized uncertainty principle (GUP). Based on the exact solution, the general and explicit relation between the phase time and the dwell time is given and quantified. The phase time, dwell time and self-interference delay are calculated as well and it is shown that the GUP has effects on the tunneling time. Moreover, the effect of GUP on the tunneling time showed a slight change with the change of mass, and goes to zero with the increase of kinetic energy.
For the relativistic tunneling effect described using Dirac's equation, Winful et al. [Phys. Rev. A 70, 052112 (2004)] presented the deduction of a general result that allows for the determination of the phase time (group delay) as the sum of the particle dwell time inside a potential barrier and of the self-interference delay associated with the incident and reflected wave functions interaction. In this paper, a mathematical model is derived through a construction analogous to the proposal mentioned above, but based on an alternative representation for Dirac's equation. This representation is similar to the one introduced by Ajaib [Found. Phys. 45, 1586 (2015)]. Thus, from the application of this model in the study of the tunneling effect in the absence of an external magnetic field, the influence of spin on the tunneling times is described. More specifically, the tunneling time is obtained as the sum of the dwell times inside the potential barrier for particles with spin up and spin down and the self-interaction time associated with the incident and reflected wave functions for particles with spin up.
There remains the old question of how long a quantum particle takes to tunnel through a potential barrier higher than its incident kinetic energy. In this article a solution of the question is proposed on the basis of a realistic explanation of quantum mechanics. The explanation implies that the tunneling particle has a certain chance to borrow enough energy from self-interference to high-jump over the barrier. The root-mean-square velocity and the effective tunneling time of an electron tunneling through a rectangular barrier are numerically calculated. No superluminal effect (Hartman effect) is found for the tunneling electron. Heisenberg's energy-time uncertainty relation for the tunneling effect is verified by calculating an introduced coefficient representing uncertainty. The present author argues that phase time, dwell time and Bütticker-Landauer time are not appropriate expressions for the actual transit time in a tunneling process. A quantum high-jumping model is presented to resolve the paradox that kinetic energy of the tunneling particle is negative and its momentum is imaginary.