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Acta Physica Sinica

From quantum spectra to classical orbits: the rectangular billiards

Jun Lu, Du Meng-Li, 中国科学院理论物理研究所，北京 100080

Published Jan 1, 2004• 8 citations
Publisher

Abstract

A new quantum spectrum function is defined using the eigenvalues and eigenfunctions of the system considered. We study the two-dimensional rectangular billiard,and find that the Fourier transform of the spectrum function reveals the classical orbits from one point to another.
DOI
10.7498/aps.53.2450

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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.
The quantum spectra are derived from the wave-functions and the energy-functions of the isosceles-right triangular model. Although the eigenstates of the billiard system are not separable, the problem of functions with two variables is exactly solvable. The numerical results of the Fourier transform of quantum spectral functions are compared with the results from the classical orbits whose lengths match well with the positions of the spectra peaks. This result gives a new evident for the correspondence of quantum and classical mechanics.
Using the periodic orbit theory,the quantum spectra in the two dimensional rectangular billiard are calculated and analyzed.The results show that each peak in the Fourier transformed spectrum of this system corresponds to the contribution of one classical periodic orbit;and with the increase of the length of the orbit,the number of the classical periodic orbit increases,and the peaks in the corresponding quantum spectra have also increased.This study suggests the good correspondence between the quantum spectra and the classical orbits in the rectangular billiards.Therefore,the correctness of the periodic orbit theory has been verified,which further deepen people's awareness of the relations between the classical and quantum mechanics.