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American Journal of Physics

Time-dependent scattering of wave packets in one dimension

C. L. Hammer, T. A. Weber, Vaughn Stephen Zidell

Published Oct 1, 1977• 29 citations
Publisher

Abstract

This paper treats the scattering of wave packets in one space dimension at the mathematical level of the first-year graduate student. The analytic properties of the general-scattering solutions are emphasized, thereby making the analysis useful as introductory material for a more sophisticated S-matrix treatment. Resonant scattering, time delay, and line shape, as well as the relationship between bound and resonant states, all follow as a natural consequence of the analysis.

DOI
10.1119/1.10854

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Physical Review•Jun 1, 1964•5 citations
This paper treats the scattering of wave packets in one space dimension at the mathematical level of the first-year graduate student. The analytic properties of the general-scattering solutions are emphasized, thereby making the analysis useful as introductory material for a more sophisticated S-matrix treatment. Resonant scattering, time delay, and line shape, as well as the relationship between bound and resonant states, all follow as a natural consequence of the analysis.
In this paper we extend our previous analysis of the scattering of wave packets in one dimension to the case of the square-well potential. The analytic properties of the general scattering solution are emphasized thereby making the analysis useful as introductory material for a more sophisticated S-matrix treatment. The square-well model is particularly interesting because of its application to the deuteron problem. Resonance scattering, barrier penetration, time delay, and line shape are discussed at the level of the first-year graduate student.
Two-body scattering is analyzed using a time-dependent formalism and wave packets of arbitrary shape as the initial wavefunctions. In particular, the case of single-channel resonant scattering is discussed in some detail. In addition to the usual amplitudes resulting from the potential term and the resonant term, a third amplitude is found which interferes with the other two. Another interesting result is that in this formalism the final state is not necessarily represented by spherical outgoing waves, but can be interpreted instead as two separate outgoing wave packets whose shape is different from the initial packets. The Wigner time delay is obtained as a natural consequence of the analysis, and the resonant cross sections are expressed as density matrices constructed from the wave packets.
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.
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-dependent theory of scattering is reformulated. In §1, it is developed on the basis of a new limiting process which is self-consistent, and the equivalence is shown of Heisenberg's S-matrix and Dyson's one, when the total Hamiltonian permits the existence of bound states. In §2, a theory of scattering of wave packet is proposed in conformity with the physical picture. The damping factor e−ε|t| is derived from the amplitude of the wave packet. In §3, the rearrangement scattering is treated on the basis of the wave packet formalism.