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Acta Physica Sinica

Two-dimensional quantum spectra and classical orbits of isosceles- right triangular billiards

Liu Xiang-Long, Zhu Man-Zuo, Lu Lu

Published Jan 1, 2012• 1 citation
Publisher

Abstract

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.

DOI
10.7498/aps.61.220301

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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.
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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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.