Stationary phase method and delay times for relativistic and non-relativistic tunneling particles

TL;DR

This paper explores the concepts of transit times in quantum tunneling, comparing stationary phase and dwell times for relativistic and non-relativistic particles, clarifying their relation and implications for superluminal interpretations.

Contribution

It provides a detailed analysis of the relation between phase and dwell times, extends non-relativistic tunneling formalism to relativistic equations, and clarifies misconceptions about superluminal tunneling.

Findings

Explicit connection between phase and dwell times for symmetric collisions.

Conditions for accelerated tunneling transmission probabilities.

Reevaluation of superluminal tunneling interpretations.

Abstract

This report deals with the basic concepts on deducing transit times for quantum scattering: the stationary phase method and its relation with delay times for relativistic and non-relativistic tunneling particles. We notice that the applicability of this method is constrained by several subtleties in deriving the phase time that describes the localization of scattered wave packets. We investigate the general relation between phase times and dwell times for quantum tunneling/scattering. Considering a symmetrical collision of two identical wave packets with an one-dimensional barrier, we demonstrate that these two distinct transit time definitions are explicitly connected. The traversal times are obtained for a symmetrized (two identical bosons) and an antisymmetrized (two identical fermions) quantum colliding configuration. Multiple wave packet decomposition shows us that the phase time (group delay) describes the exact position of the scattered particles and, in addition to the exact relation with the dwell time, leads to correct conceptual understanding of both transit time definitions. At last, we extend the non-relativistic formalism to the solutions for the tunneling zone of a one-dimensional electrostatic potential in the relativistic (Dirac to Klein-Gordon) wave equation where the incoming wave packet exhibits the possibility of being almost totally transmitted through the potential barrier. The conditions for the occurrence of accelerated tunneling transmission probabilities are all quantified and the problematic superluminal interpretation based on the non-relativistic tunneling dynamics is revisited.