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Journal of Atomic and Molecular Physics

Quantum spectra and classical periodic orbits in a rectangular billiard

Chuan‐Lu Yang

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

By using the representation of SU(2) coherent states,we construct the wave functions related to the classical periodic orbits in the 2D rectangular billiard.The relation between the classical periodic orbits and the quantum wave functions provide a more physical description of a phenomenon than the true eigenstates in mesoscopic systems.Furthermore,by using the periodic orbit theory,we calculate the Fourier transform ρ(L) of the quantum spectra of the two-dimensional rectangular billiard system.The resulting of the peaks in plots of |ρN(L)|2 versus L are compared to the lengths of the classical periodic orbits of the particle moving in the 2D rectangular cavity.The agreement of the peaks between |ρN(L)|2 and the length of the classical periodic orbit suggests the correspondence of quantum mechanics and classical mechanics.

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By using the representation of SU(2) coherent states,we construct the wave functions related to the classical periodic orbits in the 2D rectangular billiard.The relation between the classical periodic orbits and the quantum wave functions provide a more physical description of a phenomenon than the true eigenstates in mesoscopic systems.Furthermore,by using the periodic orbit theory,we calculate the Fourier transform ρ(L) of the quantum spectra of the two-dimensional rectangular billiard system.The resulting of the peaks in plots of |ρN(L)|2 versus L are compared to the lengths of the classical periodic orbits of the particle moving in the 2D rectangular cavity.The agreement of the peaks between |ρN(L)|2 and the length of the classical periodic orbit suggests the correspondence of quantum mechanics and classical mechanics.
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.
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.
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.
We study the correspondence between quantum spectra and classical orbits in the equilateral triangular billiards. The eigenstates of such systems are not separable functions of two variables even though the problem is exactly solvable. We calculate the Fourier transform of a quantum spectral function and find that the positions of the peaks match well with the lengths of the classical orbits. This is another example showing that the quantum spectral function provides a bridge between quantum and classical mechanics.
We study the correspondence between quantum spectra and classical orbits in the equilateral triangular billiards. The eigenstates of such systems are not separable functions of two variables even though the problem is exactly solvable. We calculate the Fourier transform of a quantum spectral function and find that the positions of the peaks match well with the lengths of the classical orbits. This is another example showing that the quantum spectral function provides a bridge between quantum and classical mechanics.