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Chinese Physics B

On the tunneling time of arbitrary continuous potentials and the Hartman effect

Cheng Yin, Zhi-Jing Wu, Xianping Wang, Jing-Jing Sun, Zhuangqi Cao

Published Nov 1, 2010• 6 citations
Publisher

Abstract

This paper obtains a generalized tunneling time of one-dimensional potentials via time reversal invariance. It also proposes a simple explanation for the Hartman effect using the useful concept of the scattered subwaves.
DOI
10.1088/1674-1056/19/11/117305

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Abstract In this letter we prove explicitly that if the Hartman effect exists for an arbitrary “unit cell” potential, then it also exists for a periodic system constructed using the same “unit cell” potential repeatedly. We further show that if the Hartman effect exists, the tunneling time in the limiting case of a sufficiently thick “unit cell” potential is the same as that of its periodic system. This is true for any arbitrary value of the intervening gap between the consecutive “unit cell” of the periodic system. Thus, the generalized Hartman effect always occurs for a general potential constructed using multiple copies of single potential which shows Hartman effect.
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.
I show that the MacColl-Hartman effect, namely, the saturation of the group delay time of sub-barrier quantum tunneling as a function of the barrier width, comes from the saturating behavior of a more fundamental concept - the phase of the stationary wave function. The explanation of saturation is given based on the decomposition of the stationary wave function into the spectrum of wave numbers and formulation of the initial condition for the direction of propagation of the incident matter wave. It is also shown that the saturation plateau of MacColl and Hartman actually doesn't continue indefinitely, but has a finite length. After the plateau, the sub-barrier tunneling time monotonically increases with increasing width of the potential, and this applies both to the maximum's of the wave packet and to the average tunneling time.