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arXiv:nucl-th/0406035

Method of multiple internal reflections in description of tunneling evolution through barriers

V. S. Olkhovsky, Sergei P. Maydanyuk

Published Jun 12, 2004• 11 citations
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Abstract

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.

DOI
10.48550/arxiv.nucl-th/0406035

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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.
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 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 problem of scattering of a nonrelativistic particle on a nucleus is considered, the interaction potential between which has a spherically symmetrical view. As the further development of the time analysis of processes of tunneling the non-stationary method of the solution of a problem is represented, due to which begins the possibility to describe in a time dependence the propagation (tunneling) of a particle and in details to study this process in an interesting instant or concerning a concrete point of space. At calculating the time parameters the method has shown itself simple and convenient. At finding the expressions for stationary wave functions for problems with an interaction potential, which radial part has more composite view than rectangular, this method is more effective than reference stationary approaches.
Using a recently developed procedure---multiple wave packet decomposition---here we study the phase time formulation for tunneling or reflecting particles colliding with a potential barrier. To partially overcome the analytical difficulties which frequently arise when the stationary phase method is employed for deriving phase (tunneling) time expressions, we present a theoretical exercise involving a symmetrical collision between two identical wave packets and an one-dimensional rectangular potential barrier. Summing the amplitudes of the reflected and transmitted waves---using a method we call multiple peak decomposition---is shown to allow reconstruction of the scattered wave packets in a way which allows the stationary phase principle to be recovered.
The stationary phase method is often employed for computing tunneling phase times of analytically-continuous gaussian or infinite-bandwidth step pulses which collide with a potential barrier. The indiscriminate utilization of this method without considering the barrier boundary effects leads to some misconceptions in the interpretation of the phase times. After re-examining the above barrier diffusion problem where we notice the wave packet collision necessarily leads to the possibility of multiple reflected and transmitted wave packets, we study the phase times for tunneling/reflecting particles in a framework where an idea of multiple wave packet decomposition is recovered. To partially overcome the analytical incongruities which rise up when tunneling phase time expressions are obtained, we present a theoretical exercise involving a symmetrical collision between two identical wave packets and a one-dimensional squared potential barrier, where the scattered wave packets can be recomposed by summing the amplitudes of simultaneously reflected and transmitted waves.
Abstract: After reexamining the above- barrier diffusion problem where we notice that the wave packet collision implies the existence of multiple reflected and transmitted wave packets, we analyze the way of obtaining phase times for tunneling/reflecting particles in a particular colliding configuration where the idea of multiple peak decomposition is recovered. To partially overcome the analytical incongruities which frequently arise when the stationary phase method is adopted for computing the (tunneling) phase-time expressions, we present a theoretical exercise involving a symmetrical collision between two identical wave packets and a unidimensional squared potential barrier where the scattered wave packets can be recomposed by summing the amplitudes of simultaneously reflected and transmitted wave components so that the conditions for applying the stationary phase principle are totally recovered. Lessons concerning the use of the stationary phase method are drawn