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arXiv:2303.01873 · Physical Review A

Influence of spin on tunneling times in the super-relativistic regime

Said Lantigua, Jonas Maziero

Published Aug 23, 2023• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.108.022218

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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.
In this work, we present a study of the intriguing Hartman Effect, through two approaches. The first one is a relativistic approach of quantum theory using the Dirac equation, and the second one is a geometrodynamic approach using Bohmian mechanics. Therefore, we first developed a mathematical model from the Dirac equation in an alternative representation, which allowed us to deduce a group (or tunneling) time as the sum of the residence times inside the potential barrier for particles with spin up and spin down, and the self-interaction time between the incident and reflected wave functions of the spin-up particles. In other words, we obtained a model that allows us to observe the influence of spin on the tunneling time, in the absence of an external magnetic field, which offers a less satisfactory explanation of the Hartman Effect in terms of group time. Subsequently, through a geometrodynamic approach of Bohmian mechanics, we constructed a general solution for the problem of scattering of incident particles on a constant potential barrier under the hypothesis that such particles undergo quantum tunneling along geodesic trajectories in an Alcubierre-type spacetime. Consequently, from the solution, we determined the mathematical expressions for the quantum potential, momentum, position, and tunneling time in terms of the spacetime geometry. That is, we obtained a mathematical model that allows us to explain the Hartman Effect as a consequence of the spacetime distortion generated by the quantum potential inside the barrier.
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.
We study the relativistic quantum mechanical problem of a Dirac particle tunneling through two successive electrostatic barriers. Our aim is to study the emergence of the so-called generalized Hartman effect, an effect observed in the context of nonrelativistic tunneling as well as in its electromagnetic counterparts and which is often associated with the possibility of superluminal velocities in the tunneling process. We discuss the behavior of both the phase (or group) tunneling time and the dwell time, and show that in the limit of opaque barriers the relativistic theory also allows the emergence of the generalized Hartman effect. We compare our results with the nonrelativistic ones and discuss their interpretation.