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arXiv:1105.2159

A mathematically rigorous approach raises questions concerning the generalized Hartman effect

Shoju Kudaka, Shuichi Matsumoto

Published May 11, 2011• 0 citations
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Abstract

With reference to a particle tunneling through two successive barriers, it seems to have been generally accepted that the tunneling time does not depend on the separation distance between the barriers. This phenomenon has been called the {\it generalized Hartman effect}. In this letter, we point out a lack of mathematical rigour in the reasoning by which this effect was deduced about ten years ago. A mathematically rigorous treatment shows us that the tunneling time does indeed depend on the length of the free space between the barriers.

DOI
10.48550/arxiv.1105.2159

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With reference to a particle tunneling through two successive barriers, it seems to have been generally accepted that the tunneling time does not depend on the separation distance between the barriers. This phenomenon has been called the {\it generalized Hartman effect}. In this letter, we point out a lack of mathematical rigour in the reasoning by which this effect was deduced about ten years ago. A mathematically rigorous treatment shows us that the tunneling time does indeed depend on the length of the free space between the barriers.
Recent papers suggest that tunneling wave packets traverse the region of allowed propagation between two potential barriers with superluminal group velocity and in a time independent of barrier separation. This phenomenon has been termed the "generalized Hartman effect" and extended to multiple barriers. Here we show that this delay time is not a transit time but a cavity lifetime. It does not imply superluminal velocity. Reported experimental verifications of this effect are reinterpreted.
We study the phenomenon of one-dimensional non-resonant tunneling through two successive potential barriers, separated by an intermediate free region R, by analyzing the relevant solutions to the Schroedinger equation. We find that the total traversal time does not depend not only on the barrier widths (the so-called 'Hartman effect'), but also on the R width: so that the effective velocity in the region R, between the two barriers, can be regarded as infinite. This agrees with the results known from the corresponding waveguide experiments, which simulated the tunnelling experiment herein considered due to the formal identity between the Schroedinger and the Helmholtz equation.
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.
We study the phenomenon of one-dimensional non-resonant tunneling through two successive (opaque) potential barriers, separated by an intermediate free region , by analyzing the relevant solutions to the Schroedinger equation. We find that the total traversal time does not depend not only on the barrier widths (the so-called "Hartman effect"), but also on the width: so that the effective velocity in the region , between the two barriers, can be regarded as practically infinite. This agrees with the results known from the corresponding waveguide experiments, which simulated the tunneling experiment herein considered due to the known formal identity between the Schroedinger and the Helmholtz equation.