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Physical Review

Space-Time Properties of a Scattering Interaction

J. A. McClure

Published Apr 29, 1966• 1 citation
Publisher

Abstract

The scattering of a particle wave packet by a static potential of finite range is considered. In such an interaction, the incident packet enters the potential region, is scattered, and departs, changed in a way which depends on the nature and strength of the interaction. Intuitively there is a region of space and an interval of time in which the interaction occurs. With the aid of the Schr\"odinger equation, this situation is discussed, and a function is derived which can be interpreted as describing the distribution of the interaction in space and time. The properties of the function are investigated, and arguments are given which support this interpretation. With the distribution, the space-time properties of the interaction are explored. Finally, it it is suggested that the distribution function, appropriately extended, may be a useful tool in studying other aspects of particle interactions.

DOI
10.1103/physrev.144.1316

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The scattering of a particle wave packet by a static potential of finite range is considered. In such an interaction, the incident packet enters the potential region, is scattered, and departs, changed in a way which depends on the nature and strength of the interaction. Intuitively there is a region of space and an interval of time in which the interaction occurs. With the aid of the Schr"odinger equation, this situation is discussed, and a function is derived which can be interpreted as describing the distribution of the interaction in space and time. The properties of the function are investigated, and arguments are given which support this interpretation. With the distribution, the space-time properties of the interaction are explored. Finally, it it is suggested that the distribution function, appropriately extended, may be a useful tool in studying other aspects of particle interactions.
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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.
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