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Physical Review A

Scattering theory, multiparticle detection, and time

J S Briggs, James M. Feagin

Published Nov 14, 2014• 11 citations
Publisher

Abstract

We consider the theory of multiple-particle fragmentation processes in the light of modern multihit position-sensitive detection. First, we give a formulation of time-independent many-body scattering theory as a direct generalization of standard textbook two-body potential scattering but in such a way as to emphasize position rather than momentum detection. Noteworthy is that classical asymptotic motion of fragments is shown to emerge from this quantum-mechanical time-independent theory and enables the definition of a classical time parameter. This in turn allows a transition to be made to a time-dependent scattering theory, even in the case where all Hamiltonians are time independent. Such a time-dependent description is the basis of the imaging theorem, which connects position detection to momentum detection.

DOI
10.1103/physreva.90.052712

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We consider the theory of multiple-particle fragmentation processes in the light of modern multihit position-sensitive detection. First, we give a formulation of time-independent many-body scattering theory as a direct generalization of standard textbook two-body potential scattering but in such a way as to emphasize position rather than momentum detection. Noteworthy is that classical asymptotic motion of fragments is shown to emerge from this quantum-mechanical time-independent theory and enables the definition of a classical time parameter. This in turn allows a transition to be made to a time-dependent scattering theory, even in the case where all Hamiltonians are time independent. Such a time-dependent description is the basis of the imaging theorem, which connects position detection to momentum detection.
Conventional scattering theory is incomplete in that it does not adequately describe the behaviour of the wave function at macroscopic distances from the scattering reaction volume. In scattering experiments particles are incident from sources at macroscopic distance and measured at macroscopic distance from the microscopic reaction volume. Standard experimental procedure is to describe asymptotic particle motion by classical mechanics. However, this transition to classical mechanics is not accounted for in standard quantum scattering theory. Hence, we revisit conventional scattering theory with the introduction of semiclassical propagators to describe the wave function in both incident and outgoing asymptotic scattering channels. This description leads to the manifestation of classical trajectories in the quantum wave function both before and after the collision. It includes the case of extraction with external fields as in modern multi-particle detectors. We derive the asymptotic Kemble imaging theorem (IT) limit to demonstrate how formal quantum scattering theory contains implicit classical mechanics threaded throughout the asymptotic wave functions.
We consider the measurement of multi-particle fragmentation of atomic and molecular systems in momentum space and to what extent such measurements allow the direct imaging of the spatial probability distribution of the fragments emerging from the interaction region. We present a simple derivation of the imaging theorem (IT) relating the asymptotic momentum distribution of the fragments to their spatial continuum wavefunction. We emphasize application to a real detector with finite momentum and spatial resolution. We also consider the role of particle-detector entanglement and decoherence in establishing the IT.
Stationary states are unavoidable from the practical viewpoint, since all the computations of observables, such as cross sections and other experimentally measurable physical quantities, are carried out within timeindependent scattering theory. In a scattering experiment, one attempts to determine the momentum of the impact projectile as precisely as possible. However, in the standard formulation of scattering theory, one tries to reach the same ideal goal by identifying the initial state in the remote past with the plane wave. The problem of the asymptotic convergence of scattering states cannot be formulated in the same way for both proper and improper state vectors. The optical theorem is a convenient way of computing the total cross section Q, when one knows the value of the imaginary part of the scattering amplitude in the forward direction. The scattered particles are moved away from the incident beam for a certain amount which is proportional to the value of the total cross section Q.
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.