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Europhysics Letters (EPL)

Time Delay in the Eikonal Approximation

Takako Suzuki

Published Jul 15, 1989• 17 citations
Publisher

Abstract

The time delay or advance in the eikonal approximation is discussed in terms of the wave packet motion in the interaction region as well as the scattering amplitude. The result of the time delay or advance obtained from the wave packet motion does not strictly coincide with that given by the derivative of the phase of the amplitude.
DOI
10.1209/0295-5075/9/6/003

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•Jan 1, 1997•3 citations
The time delay of an almost monochromatic wave packet can be defined unambiguously via the energy derivative of the phase of the amplitude defining the superposition of stationary states making up the wave packet. We illustrate how this concept, originally formulated by Eisenbud and Wigner in the context of scattering theory, can be applied to quantum reflection by the nonclassical region of a long-ranged attractive potential tail. We pay special attention to the case of a wave packet incident from the near side of the nonclassical region and interpret the results with the help of regularized step potentials.
Standard eikonal methods, as asymptotic perturbation expansions in 1/\ensuremath{\Elzxh}, are incapable of describing fully the dispersive spreading of a wave packet. By simple models, we distinguish the two types of dispersion associated with a packet having initially uniform and nonuniform distributions of wave number. A nonperturbative extension of the eikonal approximation is applied to two initial-value problems for the free-particle Schr\"odinger equation and compared with exact solutions. The method maintains the ray picture fully. Applications of the method to more general classes of problems are discussed.
Abstract This is the first of two subsequent publications where the probability distribution of delay-times in scattering of wave packets is discussed. The probability distribution is expressed in terms of the on-shell scattering matrix, the dispersion relation of the scattered beam and the wave packet envelope. In the monochromatic limit (poor time resolution) the mean delay-time coincides with the expression derived by Eisenbud and Wigner and generalized by Smith more than half a century ago. In the opposite limit, and within the semi-classical approximation, the resulting distribution coincides with the result obtained using classical mechanics or geometrical optics. The general expression interpolates smoothly between the two extremes. An application for the scattering of electromagnetic waves in networks of RF transmission lines will be discussed in the next paper to illustrate the method in an experimentally relevant context.
The time delay or the time advance in the scattering of simple one-dimensional systems can be evaluated in a straightforward manner for certain potential models. It is found that when the interacting potential is attractive and has a strength such that it nearly supports an additional bound state, the time delay at small scattering energy is very large. On the other hand, if the potential supports a bound state with nearly zero binding energy, the time advance near threshold is anomalously large. The behavior of a wave packet scattering from the double delta-function potential is also investigated.
A one-dimensional scattering problem off a delta\\deltadelta-shaped potential is solved analytically and the time development of a wave packet is derived from the time-dependent Schrödinger equation. The exact and explicit expression of the scattered wave packet supplies us with interesting information about the "time delay" by potential scattering in the asymptotic region. It is demonstrated that a wave packet scattered by a spin-flipping potential can give us quite a different value for the delay times from that obtained without spin-degrees of freedom.