Critique of the Wigner tunneling speed and a proposed alternative

Abstract

In the context of superluminal propagation of wave packets through potential barriers, the tunneling speed is usually characterized by the Wigner velocity. We propose an alternative speed that takes into account the interference between the incoming and the reflected waves and leads to a better estimation of arrival time for a wave packet entering the tunneling region. This arrival time is derived by an extrapolation from inside the barrier. The analytical theory is based on the stationary phase approximation whose validity is justified by a comparison with the numerical solution of the time-dependent Dirac equation.

Tunnelling through two barriers

“…One can in particular question, in this context, the use of the terminology "super-luminal velocities". However, since similar results have more recently been obtained with the Dirac equation [3,4], there is good reason to be perplexed and to invoke further analysis.We address in this paper a variant of this subject which connects closely to a recent paper upon above-barrier diffusion [5]. In recent years, the Hartman analysis has been extended to a potential model with two successive barriers separated by a free propagation region.…”

Dirac equation studies in the tunneling energy zone

“…With the Schrödinger equation the conclusion that the transition time is independent of barrier width l, when l → ∞, is known as the Hartman effect [11]. Such a result is hard to avoid and, if the same occurs for the Dirac equation (subject matter of this paper and previously discussed by other authors [12][13][14]), we would have to face the unpalatable feature of superluminal velocities within the barrier. We warn that more than one type of transition time has been defined in the literature [10,[15][16][17] and for details we refer to reference [18,19].…”

Delay time computation for relativistic tunneling particles

“…The tunneling zone, V 0 − m < E < V 0 + m, for which only evanescent waves exist [34,35] in the barrier region, is that of interest in this work. In fact, the usual definition of Klein tunneling, which involves the Klein paradox [30], has been studied in the literature just for the Dirac equation [27,28,35,36,37].…”