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arXiv:chao-dyn/9505002 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Signatures of classical periodic orbits on a smooth quantum system

Daniel Provost

Published Jun 1, 1995• 3 citations
arXivHTMLPDF

Abstract

Gutzwiller's trace formula and Bogomolny's formula are applied to a nospecific, nonscalable Hamiltonian system, a two-dimensional anharmonic oscillator. These semiclassical theories reproduce well the exact quantal results over a large spatial and energy range.
DOI
10.1103/physreve.51.5396

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The M. C. Gutzwiller trace formula has played a prominent role in the study of quantum systems whose classical analogues are chaotic. Although of measure zero in a sea of chaos in the classical phase space, the periodic orbits still open a classical window to view the quantum behavior of such systems. One of the earliest applications of the trace formula in this connection was made by Strutinsky and collaborators in their studies of deformed atomic nuclei. The same physics of the dominance of short periodic orbits in finite-sized systems makes the trace formula a powerful tool in the study of mesoscopic devices. Fascinating quantal behavior in the propagation of wave packets, like the phenomenon of recurrence, is explained by a semiclassical analysis. Diffractive orbits are specially important for grazing angles of encounter and, of course, play a leading role in scattering in the forward direction.
We derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate-space formulation. Using a coherent-state basis makes it immediately obvious that classical periodic orbits make separate contributions to the trace of the quantum-mechanical time evolution operator. In addition, our approach is manifestly canonically invariant at all stages, and leads to the simplest possible derivation of Gutzwiller's formula.
Abstract We derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate‐space formulation. Using a coherent‐state basis makes it immediately obvious that classical periodic orbits make separate contributions to the trace of the quantum‐mechanical time evolution operator. In addition, our approach is manifestly canonically invariant at all stages, and leads to the simplest possible derivation of Gutzwiller's formula.
Using Feynman path integrals and the stationary-phase method, we develop a semiclassical theory for quantum trace formulas in classically hyperbolic systems. In this way, we obtain corrections to the Gutzwiller-Selberg trace formula as an asymptotic series in powers of the Planck constant. The first coefficient of this series is given explicitly. We illustrate the theory with the calculation of complex-wave-number resonances for the two-disk scatterer and show that effects beyond the Gutzwiller leading approximation are at the origin of a lengthening of the resonance lifetimes at low energy.