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Physical Review A

Semiclassical periodic-orbit theory for identical particles

Hans A. Weidenmüller

Published Sep 1, 1993• 24 citations
Publisher

Abstract

Gutzwiller's semiclassical approximation is applied to systems of identical particles. The special case of antisymmetry under particle exchange is treated as an example. The Pauli principle is shown to cause a modification of Gutzwiller's periodic-orbit sum: The amplitude with which a periodic orbit contributes to the sum is changed, but new orbits do not arise in the sum. The case of noninteracting particles is used for purposes of illustration of the procedure.

DOI
10.1103/physreva.48.1819

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Gutzwiller's semiclassical approximation is applied to systems of identical particles. The special case of antisymmetry under particle exchange is treated as an example. The Pauli principle is shown to cause a modification of Gutzwiller's periodic-orbit sum: The amplitude with which a periodic orbit contributes to the sum is changed, but new orbits do not arise in the sum. The case of noninteracting particles is used for purposes of illustration of the procedure.
Semiclassical periodic orbit theory is used in many branches of physics. However, most applications of the theory have been to systems that involve only single-particle dynamics. In this paper, we develop a semiclassical formalism to describe the density-of-states for two noninteracting particles. This includes accounting properly for particle exchange symmetry. As specific examples, we study two identical particles in a disk and in a cardioid. In each case, we demonstrate that the semiclassical formalism correctly reproduces the quantum densities.
Gutzwiller's trace formula has a central place in quantum chaos because it provides semiclassical approximations for quantum energy levels in classically chaotic systems by linking them to classical periodic orbits. In this didactic article, we discuss a derivation of the trace formula starting from the Feynman path integral. We then describe how the trace formula is used to explain universal features in the distribution of the quantum energy levels that are described by random matrix theory, and we give an overview of related work.
The semiclassical limit of the quantum few-body problem has not been studied in general terms from the point of view of periodic orbit theory. In a previous paper, we studied noninteracting two-body systems [Phys. Rev. A 62, 042109 (2000)] and discussed the fact that the periodic orbits occur in continuous families. Interactions destroy the periodic orbit families leaving a discrete set of isolated periodic orbits. In this paper, we consider the effect of weak two-body interactions, which can be thought of as symmetry-breaking perturbations and can thus be analyzed using a theory developed by Creagh [Ann. Phys. (N.Y.) 248, 1 (1996)]. The P\"oschl-Teller two-body system confined in a square well is analyzed to illustrate the use of the formalism. It is shown that the effect of the interaction can be evaluated for all two-particle periodic orbits, and that the coarse-grained quantum density of states can be fully reproduced from simply summing the perturbed contributions of each periodic orbit family. Good numerical estimates of the quantum singlet energies can actually be obtained, but it is found that that perturbed trace formulas cannot reproduce the multiplet splittings predicted from quantum mechanics. Several interesting properties are observed depending on the range of the interaction and on whether the interaction is attractive or repulsive.
Harmonic inversion has already been proven to be a powerful tool for the analysis of quantum spectra and the periodic orbit orbit quantization of chaotic systems. The harmonic inversion technique circumvents the convergence problems of the periodic orbit sum and the uncertainty principle of the usual Fourier analysis, thus yielding results of high resolution and high precision. Based on the close analogy between periodic orbit trace formulae for regular and chaotic systems the technique is generalized in this paper for the semiclassical quantization of integrable systems. Thus, harmonic inversion is shown to be a universal tool which can be applied to a wide range of physical systems. The method is further generalized in two directions: firstly, the periodic orbit quantization will be extended to include higher order ~ corrections to the periodic orbit sum. Secondly, the use of cross-correlated periodic orbit sums allows us to significantly reduce the required number of orbi...
Harmonic inversion is introduced as a powerful tool for both the analysis of quantum spectra and semiclassical periodic orbit quantization. The method allows to circumvent the uncertainty principle of the conventional Fourier transform and to extract dynamical information from quantum spectra which has been unattainable before, such as bifurcations of orbits, the uncovering of hidden ghost orbits in complex phase space, and the direct observation of symmetry breaking effects. The method also solves the fundamental convergence problems in semiclassical periodic orbit theories -- for both the Berry-Tabor formula and Gutzwiller's trace formula -- and can therefore be applied as a novel technique for periodic orbit quantization, i.e., to calculate semiclassical eigenenergies from a finite set of classical periodic orbits. The advantage of periodic orbit quantization by harmonic inversion is the universality and wide applicability of the method, which will be demonstrated in this work for v...