The Semiclassical Few-Body Problem

Abstract

<p>The few-body problem has not been studied in a general context from the point of view of semiclassical periodic orbit theory. The purpose of this thesis is to give a periodic orbit analysis of the quantum mechanical few-body problem. In particular, the goal is to study two important special cases: noninteracting and weakly-interacting few-body systems. The standard Gutzwiller theory does not apply to the case of noninteracting many-body systems since the single- article energies are conserved causing the periodic orbits to occur in continuous families in phase space. The unsymmetrized few-body problem is analyzed using the formalism of Creagh and Littlejohn [33], who have studied semiclassical dynamics in the presence of continuous symmetries. The symmetrized few-body problem further requires using the formalism of Robbins [30], and it is shown how the various exchange terms of symmetrized densities of states can be understood in terms of pseudoperiodic orbits. Numerical studies of two- and three-particle systems in a fully chaotic cardioid billiard are used throughout to illustrate and test the results of the theory. For weak interactions, the Gutzwiller theory also fails. Although the continuous families are destroyed, the periodic orbits are not sufficiently isolated for the standard theory to apply. The interaction can be thought of as a symmetry-breaking perturbation, and then it is possible to apply the results of semiclassical perturbation theory as developed by Creagh [38]. The periodic orbit structure is affected by the symmetry breaking, and this can be understood through asymptotic analysis. An approximate quantization of a two-body nonscaling nonintegrable system is achieved, although explicit quantization of nonintegrable systems is generally not possible. Issues related to the convergence of periodic orbit expansions arise in several places. The convergence of a two-particle system in a disk billiard is explored using an amplitude ordering of the sum. Problems related to truncation and ordering are further explored using two other expansions from number theory.</p>