Use of the Wigner Representation in Scattering Problems

Remler, E.A.

doi:10.1016/0003-4916(75)90065-2

Use of the Wigner Representation in Scattering Problems

E.A. Remler
(
- William-Mary Coll.
)

1975

41 pages

Published in:

Annals Phys. 95 (1975) 455-495

DOI:

10.1016/0003-4916(75)90065-2

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Abstract: (Elsevier)

The basic equations of quantum scattering are translated into the Wigner representation. This puts quantum mechanics in the form of a stochastic process in phase space. Instead of complex valued wavefunctions and transition matrices, one now works with real-valued probability distributions and source functions, objects more responsive to physical intuition. Aside from writing out certain necessary basic expressions, the main purpose of this paper is to develop and stress the interpretive picture associated with this representation and to derive results used in applications published elsewhere. The quasiclassical guise assumed by the formalism lends itself particularly to approximations of complex multiparticle scattering problems. We hope to be laying the foundation for a systematic application of statistical approximations to such problems. The form of the integral equation for scattering as well as its multiple scattering expansion in this representation are derived. Since this formalism remains unchanged upon taking the classical limit, these results also constitute a general treatment of classical multiparticle collision theory. Quantum corrections to classical propogators are discussed briefly. The basic approximation used in the Monte Carlo method is derived in a fashion that allows for future refinement and includes bound state production. The close connection that must exist between inclusive production of a bound state and of its constituents is brought out in an especially graphic way by this formalism. In particular one can see how comparisons between such cross sections yield direct physical insight into relevant production mechanisms. Finally, as a simple illustration of some of the formalism, we treat scattering by a bound two-body system. Simple expressions for single- and double-scattering contributions to total and differential cross sections, as well as for all necessary shadow corrections thereto, are obtained. These are compared to previous results of Glauber and Goldberger.