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Journal of Mathematical Physics

Transition from time-dependent to time-independent multichannel quantum scattering theory

Colston Chandler, Archie G. Gibson

Published Oct 1, 1973• 45 citations
Publisher

Abstract

Rigorous proofs are given of several theorems establishing the connection between time-dependent and time-independent multichannel scattering theory. The method of proof involves a two-Hilbert-space formulation of time-dependent multichannel theory and the theory of spectral integrals. In particular, the time-independent theory in the form proposed by Alt, Grassberger, and Sandhas is derived.

DOI
10.1063/1.1666185

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Rigorous proofs are given of several theorems establishing the connection between time-dependent and time-independent multichannel scattering theory. The method of proof involves a two-Hilbert-space formulation of time-dependent multichannel theory and the theory of spectral integrals. In particular, the time-independent theory in the form proposed by Alt, Grassberger, and Sandhas is derived.
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.
A time-dependent multichannel nonrelativistic quantum scattering theory is established for systems which contain identical particles. The symmetry properties of the theory are rooted in the abstract algebraic relationships between permutations. The formalism of second quantization is thus avoided. Appropriate wave and scattering operators are defined, including those for systems with long range (e.g., Coulomb) interactions. A correctly symmetrized dynamical equation for the symmetric form of the transition operator is derived and used to develop an optical potential formalism.NUCLEAR REACTIONS Nonrelativistic multichannel scattering theory for identical particles, wave and scattering operators, dynamical equations, optical potentials.
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.
Derivations are given for some transition and resolvent operator equations for multichannel quantum scattering with short-range potentials. The basic difference between these and previous equations is that the unknown operators act only on the channel subspaces. This is made possible by utilizing, and extending, the two-Hilbert-space formulation previously given by the authors [in J. Math. Phys. 14, 1328 (1973)]. The equations in abstract form are of the Lippmann–Schwinger type, differing only in the appearance of certain injection operators from one Hilbert space to the other. When applied to multichannel quantum scattering, the abstract theory yields a new system of equations for the transition and resolvent operators. Uniqueness of the solution to the equations is proved.