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The Journal of Chemical Physics

Classical Limit of Fredholm Theory for Elastic and Inelastic Scattering; Inability of Phase Space Integrals to Describe Inelastic Transitions

William Hughes Miller

Published Jan 15, 1972• 8 citations
Publisher

Abstract

Using a classical phase space integral to approximate certain quantum mechanical operations, the classical limit of the Fredholm determinant for a general multichannel scattering system is derived. This statistical form of the classical limit of scattering is found to give exactly the same ``classical S matrix'' for the case of potential scattering as does a dynamical treatment of the classical limit. For inelastic scattering, however, the classical S matrix which results from this Fredholm determinant is only an approximation to that obtained by exact classical dynamics. Reasons for this failure of the statistical version of the classical limit for inelastic scattering are discussed.

DOI
10.1063/1.1677225

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Using a classical phase space integral to approximate certain quantum mechanical operations, the classical limit of the Fredholm determinant for a general multichannel scattering system is derived. This statistical form of the classical limit of scattering is found to give exactly the same ``classical S matrix'' for the case of potential scattering as does a dynamical treatment of the classical limit. For inelastic scattering, however, the classical S matrix which results from this Fredholm determinant is only an approximation to that obtained by exact classical dynamics. Reasons for this failure of the statistical version of the classical limit for inelastic scattering are discussed.
We develop a determinantal method in a quantum scattering system for direct evaluation of various quantities including the local field amplitude and transition amplitude (SSS-matrix element). This method is applicable to multichannel elastic and inelastic scattering systems without rearranging the particles. The underlying principle of our formulation is that requested information can be extracted from a wave operator without solving the wave equation. The wave operator is the master operator maintaining all of the information about a system; the determinant of the inverse wave operator is just the Fredholm determinant. In the 1960s, a similar determinantal method was developed for the SSS-matrix element or equivalently the far-field amplitude of the wave function. Until now, however, no determinantal method for near-field quantities existed, although de Broglie and electromagnetic near fields are observable using scanning probe microscopes. Additionally, we prove that our formula for SSS-matrix element covers the known formula (the Le Couteur-Newton formula).
The space-time propagator for multichannel scattering is cast in the form of a Feynman path integral where the action is a matrix and the interactions are nonlocal in time. The SSS matrix is evaluated in the semiclassical limit, leading to trajectories which naturally follow individual channel potentials in noninteracting regions and adiabatic-like ones in strongly interacting ones. A numerical example is provided for the curve-crossing problem, and the agreement with exact quantum results is good even when the interaction region is large and/or encompasses classical turning points.
The classical correspondence of the S matrix for scattering processes is examined with a view to obtain the classical approximation and also to clarify the nature of the approximations required for the classical formula. In this discussion the Schrödinger equation in the momentum representation is employed instead of the usual coordinate representation in order to provide a more direct route to the classical limit of the S matrix. The S matrix thus obtained essentially corresponds to the WKB approximation, and, with a further approximation, leads to Miller's expression except for a multiplicative factor and an additional phase which are in general not equal to 1 and 0, respectively. The classical limit is also discussed in terms of Wigner-Moyal phase space distribution function, and when the classical limit of this function is taken, an additional factor and phase appear in the S matrix.
A new class of sum rules in nonrelativistic scattering theory is obtained. By extending the concept of the Fredholm determinant to a local form defined at each space point xxx, a function with analytic properties analogous to the Fredholm determinant, but containing more information, is found. These rules assume the form of the power-moment problem and are valid in both quantum and classical mechanics.