Relation between quantum tunneling times for relativistic particles

Abstract

A general relation between the phase time (group delay) and the dwell time is derived for relativistic tunneling particles described by the Dirac equation. It is shown that the phase time equals the dwell time plus a self-interference delay which is a relativistic generalization of previous results.Then the Hermitian conjugate of Eq.(1) is multiplied from the right by ‫ץ‬ / ‫ץ‬E, yieldingEquation (3) is then left-multiplied by † and added to Eq.(4), resulting in the expression *

Relativistic tunnelling time for electronic wave packets

“…The relativistic interaction of the Dirac particle incident on a barrier with height V 0 and length L can be divided into three cases. In the case that the potential barrier is low enough to satisfy V 0 < E − m 0 c 2 , the particle has enough energy to propagate over the potential barrier, as was discussed in [25]. No tunnelling phenomenon corresponds to this situation where non-evanescent wave propagation exists.…”

“…For particles with rest mass m 0 and total energy E in the presence of a potential V (z) restricted to a region 0 < z < L, the Dirac equation is given by [25]…”

Analogous of Hartman Effect for Relativistic Particles Through “Transparent” Barrier

“…For a relativistic particle of mass m and energy E that satisfies Dirac equation, impinging on a potential rectangular barrier of height V 0 and length L, has been demonstrated in 2004 by Winful et al [17] and [18] that the relativistic phase-time τ R ϕ has the expression 2 with κ being the decay constant for the evanescent wave within the barrier and E the total energy. Equation (2) is obtained from (4) as a nonrelativistic limit when E − mc 2 mc 2 .…”

Influence of nonparabolicity on tunneling times in semiconductor structures

“…Therefore, such a seemingly simple fundamental question of what time does the tunneling take, is of crucial importance. A variety of answers has been offered in the last six decades [3][4][5][6][7][8][9][10][11][12][13][14][15]. In the comprehensive review [16] a number of definitions of the tunneling time have been given, among them the so-called phase time or group delay time and dwell time.…”