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Chinese Physics Letters

Quantum Spectra and Classical Orbits in Two-Dimensional Equilateral Triangular Billiards

Lin Sheng-Lu, Gao Feng, Hong Zheng-Pin, Du Meng-Li

Published Dec 23, 2004• 10 citations
Publisher

Abstract

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.

DOI
10.1088/0256-307x/22/1/003

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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.
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 a recently defined quantum spectral function and the method of closed-orbit theory we studied the spectra of the half-circular and the quarter circular billiard systems for arbitrary two points.The quantum spectra contain rich information of all classical orbits connecting two arbitrary points in well.Our calculation demonstrates that the peak positions in the Fourier-transformed quantum spectra match accurately with the lengths of the classical orbits.The two examples show evidently that semi-classical method provides a bridge between quantum and classical mechanics.
The correspondence between the quantum spectra and the classical orbits of the two-dimensional elliptic billiards is investigated. In order to study systems without analytic solutions in the future, an expansion method for stationary states (EMSS) to get highly accurate numerical solutions is applied and tested. The closed-orbit theory was extended to the open orbit case. By comparing the Fourier-transformed spectra with the classical orbits, we find that the peak positions in the quantum spectra match with the lengths of the classical orbits with small numerical differences. This is another example showing that semiclassical method provides a bridge between quantum and classical mechanics.
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.