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Physical Review

Rearrangement Collisions

Bernard A. Lippmann

Published Apr 1, 1956• 192 citations
Publisher

Abstract

We discuss the quantum-mechanical theory of rearrangement collisions. In particular, we consider: (a) the transformation of the state vector from the original basis to the basis of the rearranged system; (b) the conditions for the equivalence of the "post" and "prior" interactions when computing matrix elements; and, (c) the extension of Schwinger's variational principle to rearrangement collisions.

DOI
10.1103/physrev.102.264

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We discuss the quantum-mechanical theory of rearrangement collisions. In particular, we consider: (a) the transformation of the state vector from the original basis to the basis of the rearranged system; (b) the conditions for the equivalence of the "post" and "prior" interactions when computing matrix elements; and, (c) the extension of Schwinger's variational principle to rearrangement collisions.
We derive the conditions which give a well-defined meaning to the mathematical expressions appearing in the current formal scattering theory. These conditions remove some restrictions that are usually imposed on the spectra of H and H0. By these conditions we can systematically discuss many problems, such as the equivalence between two kinds of S-matrices given by Heisenberg and Dyson respectively, or as the adiabatic theorem in the quantum field theory. The rearrangement collisions can be investigated from this view-point.
This work demonstrates that the quantum mechanical moments of a state described by the density matrix correspond to discrete spherical harmonic moments of the classical multipole expansion of the spatial distribution of the angular momentum vectors. For the diagonal density matrix elements, this work exploits the fact that the quantum mechanical vector coupling (Clebsch-Gordan) coefficients become increasingly accurate discrete representations of spherical harmonics as j increases. A Schwinger-type basis accounts for nonaxially symmetric angular distributions, which result in nonzero off-diagonal elements of the density matrix. The resulting discrete minimum uncertainty picture of the classical moments has a stringent equivalence with the quantum mechanical one for all j and provides an unambiguous connection for the classical and quantum moments in the large j limit. The equivalence is numerically tested for simple models, and there is a satisfying equivalence even for small j. Applications, implications, and extensions are indicated, and the relevance of this work for the interpretation of classical mechanical simulations of inelastic and reactive molecular collisions will be documented elsewhere.