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Physical Review

Scattering of Energy-Time Wave Packets from Many-Body Systems

L. R. Dodd, Ian E. McCarthy

Published Jun 1, 1964• 5 citations
Publisher

Abstract

The wave-packet nature of an experimental beam is discussed. The theory of wave-packet scattering is applied to wave packets with exponential time dependence scattered from resonances in both the isolated-resonance and overlapping-resonance regions. It is shown how a sequential description of scattering depends on some of the resonance parameters in the scattering amplitude. The meaning and usefulness of some experimentally possible wave-packet experiments is discussed.

DOI
10.1103/physrev.134.a1136

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The wave-packet nature of an experimental beam is discussed. The theory of wave-packet scattering is applied to wave packets with exponential time dependence scattered from resonances in both the isolated-resonance and overlapping-resonance regions. It is shown how a sequential description of scattering depends on some of the resonance parameters in the scattering amplitude. The meaning and usefulness of some experimentally possible wave-packet experiments is discussed.
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.
We consider the scattering of a particle by an oscillating rectangular barrier at energies matching with the above barrier resonances. A resonance associated with the barrier has an energy which oscillates in time. With appropriately defined parameters it is possible, using a calculation entirely based on the time-independent scattering amplitudes, to show that the transmission probability is enhanced when the wave packet describing the particle arrives on the device at a time when the resonance energy matches the energy of the particle.
The scattering of a particle by a target with periodic time dependence is usually formulated in Floquet extended space. This leads to time-independent transition probabilities which correspond to the scattering of wave packets that are close to being monoenergetic. Because of the wide space extension of such packets, many oscillations take place during the time particle and target interact. The purpose of this study consists in showing that the scattering amplitudes can also be used to describe the way the particle feels the instantaneous configuration of the target at the time of the collision. The application is made to an oscillating barrier in one dimension. Several tests of the formalism validate the analysis, by changing either the initial position of the wave packet, or the phase of the particle–target interaction. Two equivalent necessary conditions can be formulated to recover the results of standard scattering theory. One is that no energies differing by at least one quantum are present in the energy range effectively covered by the initial wave packet. The other is that there is appreciable self-overlapping of the wave packet after one period of oscillation.
Aims and problems of quantum scattering theory Scattering theory is the branch of physics concerned with interacting systems on a scale of time and/or distance that is large compared with the scale of the interaction itself, and it provides the most powerful tool for studying the microscopic nature of the world. In quantum mechanics, a typical scattering process involves a beam of particles prepared with a given energy and with a more or less defined linear momentum. One then studies either the collision with a similar wave packet, or the collision of the given wave packet against a fixed target. In particular, for a two-body elastic scattering problem with conservative forces, we work in the centre of mass frame, hence reducing the original problem to the analysis of a particle in an external field of forces. Indeed, in Section 7.2 we already used such a technique in the investigation of the hydrogen atom. The crucial difference with respect to Section 7.2 is that, in scattering problems, the continuous spectrum and its perturbations are studied, whereas the Balmer formula has to do with bound states. In the following section we outline the basic aspects of what is called time-dependent scattering. Time-dependent scattering The physical situation we would like to study is as follows: a beam of particles is prepared in a state that is approximately free and allowed to evolve towards the scatterer. After the collision we detect the scattered beam far away from the interaction region.
We derive and explain the key ideas behind a time-dependent formulation of quantum scattering theory, applicable generally to systems with a finite-range scattering potential. The scattering is initiated and probed by plane wave packets, which are localized just outside the range of the potential. The asymptotic limits of conventional scattering theory (initiation in the remote past; detection in the remote future) are not taken. Instead, the differential cross section (DCS) is obtained by projecting the scattered wave packet onto the probe plane wave packets. The projection also yields a time-dependent version of the DCS. Cuts through the wave packet, just as it exits the scattering potential, yield time-dependent and time-independent angular distributions that give a close-up picture of the scattering which complements the DCS. We have previously applied the theory to interpret experimental cross sections of chemical reactions [e.g., S. C. Althorpe, F. Fern\'andez-Alonso, B. D. Bean, J. D. Ayers, A. E. Pomerantz, R. N. Zare, and E. Wrede, Nature (London) 416, 67 (2002)]. This paper gives the derivation of the theory, and explains its relation to conventional scattering theory. For clarity, the derivation is restricted to spherical-particle scattering, though it may readily be extended to general multichannel systems. We illustrate the theory using a simple application to hard-sphere scattering.