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arXiv:chao-dyn/9509015 · Physical Review Letters

Periodic Orbit Quantization beyond the Semiclassical Theory

Gábor Vattay, Per E. Rosenqvist

Published Jan 15, 1996• 27 citations
arXivHTMLPDF

Abstract

A quantum generalization of the semiclassical theory of Gutzwiller is given. The new formulation leads to systematic orbit-by-orbit inclusion of higher $\ensuremath{\Elzxh}$ contributions to the spectral determinant. We apply the theory to billiard systems, and compare the periodic orbit quantization including the first $\ensuremath{\Elzxh}$ contribution to the exact quantum mechanical results.

DOI
10.1103/physrevlett.76.335

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A quantum generalization of the semiclassical theory of Gutzwiller is given. The new formulation leads to systematic orbit-by-orbit inclusion of higher \ensuremath\Elzxh\ensuremath{\Elzxh}\ensuremath\Elzxh contributions to the spectral determinant. We apply the theory to billiard systems, and compare the periodic orbit quantization including the first \ensuremath\Elzxh\ensuremath{\Elzxh}\ensuremath\Elzxh contribution to the exact quantum mechanical results.
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The semi-classical method has become a necessary instrument to study the classical movement of the particle.Periodic orbit theory is rapidly becoming one of most useful semi-classical tools which can be used to make direct connections between the quantized energy eigenvalues of a bound state and the classical motions for the corresponding point particle.We use a quantum spectral function which contain rich information of classical orbits in well.We study the correspondence between quantum spectra and classical orbits in the circular.Two-dimensional billiard systems have provided easily visualization examples relevant for both types of analyses.As a simple example of the application to a billiard or infinite well system of Periodic orbit theory,we compute the Fourier transform(ρL) of the quantum mechanical energy level density of two-dimensional circular billiard systems.The resulting peaks in plots of ρL2 versus L are compared to the lengths of the classical trajectories in these geometries.The locations of peaks in ρL agree with the lengths of classical orbits perfectly,which testifies the correspondence of quantum mechanics and classical mechanics.This examples show evidently that semi-classical methods provides a bridge between quantum and classical mechanics.
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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.