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Physics-Uspekhi

On the multiple internal reflections of particles and photons tunneling in one, two, or three dimensions

V. S. Olkhovsky

Published Nov 2, 2014• 5 citations
Publisher

Abstract

We analyze multiple internal reflections of particles and photons undergoing one-, two-, and three-dimensional tunneling. Results obtained by using the time-dependent Schrdinger equation for nonrelativistic particles and those obtained with the time-dependent Helmholtz equation for electromagnetic waves are presented. The paper closes with conclusions and considerations for future research.

DOI
10.3367/ufne.0184.201411h.1255

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We analyze multiple internal reflections of particles and photons undergoing one-, two-, and three-dimensional tunneling. Results obtained by using the time-dependent Schrdinger equation for nonrelativistic particles and those obtained with the time-dependent Helmholtz equation for electromagnetic waves are presented. The paper closes with conclusions and considerations for future research.
The formal mathematical analogy between time-dependent quantum equation for the nonrelativistic particles and time-dependent equation for the propagation of electromagnetic waves had been studied in [A. I. Akhiezer and V. B. Berestezki, Quantum Electrodynamics (FM, Moscow, 1959) [in Russian] and S. Schweber, An Introduction to Relativistic Quantum Field Theory, Chap. 5.3 (Row, Peterson & Co, Ill, 1961)]. Here, we deal with the time-dependent Schrödinger equation for nonrelativistic particles and with time-dependent Helmholtz equation for electromagnetic waves. Then, using this similarity, the tunneling and multiple internal reflections in one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) particle and photon tunneling are studied. Finally, some conclusions and future perspectives for further investigations are presented.
A non-stationary method for tunneling description of non-relativistic particles and photons through a barrier on the basis of consideration of the multiple internal reflections of vawe packets in relation of barrier boundaries is presented. The method is described in details and proved in the case of the one-dimentional tunneling of the particle through the rectangular barrier. For problems of the tunneling of the particle through the spherically symmetric barrier and of the photon through the one-dimensional barrier the amplitudes of transmitted and reflected wave packets in relation to the barrier, times of the tunneling and the reflection are found using of the method. Hartman's and Fletcher's effect is analysed.
The non-stationary solution method for a problem of tunneling of nonrelativistic particles and photons through a barrier on the basis of multiple internal re ections of wave packets in relation of barrier boundaries is submitted.The method is described and proved in solving an one-dimentional problem of tunneling of the particle through a rectangular barrier.For problems of tunneling of the particle through a spherically symmetric barrier and the photon through an one-dimensional barrier the amplitudes of transmitted and re ected wave packets in relation to the barrier, times of the tunneling and the re ection are found using of the method.Hartman's and Fletcher's eect is analysed.
A method of a non-stationary description of tunneling of a particle through the one-dimensional and spherically symmetric rectangular barriers on the basis of analisis of multiple internal reflections of wave packets in relation on the barrier boundaries, named as the Method of multiple internal reflections, is presented at the first time. For the one-dimensional problem the applicability of this method is proved, its specific features are analyzed. For the spherically symmetric problem the amplitudes of the transmitted and reflected wave packets, times of tunneling and reflection in relation to the barrier are calculated using this method. The effect of Hartman-Fletcher is analyzed.
We study the two-dimensional above-barrier penetration and the sub-barrier tunneling of non-relativistic particles and photons, described in the quasi-monochromatic approximation by simple plane waves. Our scheme represents the motion from the left free-motion zero-potential region to the right zero-potential region through the intermediate region with a one-dimensional rectangular potential barrier along the axis, normal to the both parallel interfaces between all three regions, and with the zero potential along the axis, parallel to the those interfaces. We have firstly obtained the analytical expressions for the infinite series of multiple internal and external reflections and also of multiple transmitted waves of particles and photons, with equal shifts between them along the interfaces for the above-barrier penetration and with various shifts between them in the case of the sub-barrier tunneling. Finally the Hartman and Fletcher effect for any transmitted wave is established.