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Phys. Rev. A 59, 1758 - 1761 (1999)

Low-energy relativistic effects and nonlocality in time-dependent tunneling
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Gastón García-Calderón
Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 México, Distrito Federal, Mexico

Alberto Rubio *
Facultad de Ciencias, Universidad Autónoma de Baja California, Apartado Postal 1880, 22800 Ensenada, Baja California, Mexico

Jorge Villavicencio †
Centro de Investigación Científica y de Educación Superior de Ensenada, Apartado Postal 2732, 22800 Ensenada, Baja California, Mexico

Received 19 August 1998

We consider exact time-dependent analytic solutions to the Schrödinger equation for tunneling in one dimension with cutoff wave initial conditions at t=0. We obtain that as soon as t≠0 the transmitted probability density at any arbitrary distance rises instantaneously with time in a linear manner. Using a simple model we find that the above nonlocal effect of the time-dependent solution is suppressed by consideration of low-energy relativistic effects. Hence at a distance x0 from the potential the probability density rises after a time t0=x0/c restoring Einstein causality. This implies that the tunneling time of a particle can never be zero.


©1999 The American Physical Society

URL: http://link.aps.org/abstract/PRA/v59/p1758
DOI: 10.1103/PhysRevA.59.1758
PACS: 03.65.-w, 73.40.Gk

* Deceased.
† Also at Facultad de Ciencias, Universidad Autónoma de Baja California Apartado Postal 1880, Ensenada, Baja California, Mexico.

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