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American Journal of Physics

Periodic orbit theory analysis of the circular disk or annular billiard: Nonclassical effects and the distribution of energy eigenvalues

Richard W. Robinett

Published Jan 1, 1999• 38 citations

Abstract

Periodic orbit (PO) theory can be used to make connections between the quantum energy eigenvalue spectrum and the closed orbits of the corresponding classical system. The two-dimensional annular billiard or circular disk system (namely, a particle in the plane confined between inner and outer infinite circular walls at fR≡Rin<Rout≡R where 0<f<1) is examined in the context of periodic orbit theory to illustrate several novel aspects of the PO analysis. Important features of this problem include (i) the appearance and disappearance of various features in the classical path length spectrum as a parameter (in this case f=Rin/Rout), is continuously varied, (ii) the presence of path length features which do not correspond to classical trajectories, but are rather due to purely wavelike phenomena (namely, diffraction around the inner annulus), and (iii) the study of the contribution of different regions in the energy eigenvalue space to different classes of classical trajectories. This last feature is a general observation about periodic orbit theory which can be usefully applied to a number of billiard systems. In addition, since the annular disk system is seldom studied in detail in either classical or quantum mechanics, we provide an appendix which reviews some of the more traditional connections between the classical and quantum probability densities for this geometry, exemplifying some of the important differences between one- and two-dimensional systems which arise from purely geometrical rather than kinematical effects.

DOI: 10.1119/1.19193

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Periodic orbit (PO) theory can be used to make connections between the quantum energy eigenvalue spectrum and the closed orbits of the corresponding classical system. The two-dimensional annular billiard or circular disk system (namely, a particle in the plane confined between inner and outer infinite circular walls at fR≡Rin<Rout≡R where 0<f<1) is examined in the context of periodic orbit theory to illustrate several novel aspects of the PO analysis. Important features of this problem include (i) the appearance and disappearance of various features in the classical path length spectrum as a parameter (in this case f=Rin/Rout), is continuously varied, (ii) the presence of path length features which do not correspond to classical trajectories, but are rather due to purely wavelike phenomena (namely, diffraction around the inner annulus), and (iii) the study of the contribution of different regions in the energy eigenvalue space to different classes of classical trajectories. This last feature is a general observation about periodic orbit theory which can be usefully applied to a number of billiard systems. In addition, since the annular disk system is seldom studied in detail in either classical or quantum mechanics, we provide an appendix which reviews some of the more traditional connections between the classical and quantum probability densities for this geometry, exemplifying some of the important differences between one- and two-dimensional systems which arise from purely geometrical rather than kinematical effects.

We compute the Fourier transform [ρ(L)] of the quantum-mechanical energy level density for the problem of a particle in an annular infinite well (or circular disk) of outer (inner) radius R max =RR min =fR. For various values of f=R min /R we then explicitly compare ρ(L) to the lengths of the classical periodic orbits. We also comment on the diffraction peak seen by Reimann et al. in this system, noting that at least one recurrence of that feature is present in the spectrum, and discussing the corresponding effective path lengths for these features.

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.

We compute the Fourier transform (ρ(L)) of the quantum mechanical energy level density for the problem of a particle in a two-dimensional circular infinite well (or circular billiard) as well as for several special generalizations of that geometry, namely the half-well, quarter-well, and the circular well with a thin, infinite wall along the positive x-axis (hereafter called a circular well plus baffle). The resulting peaks in plots of |ρ(L)|2 versus L are compared to the lengths of the classical closed trajectories in these geometries as a simple example of the application of periodic orbit (PO) theory to a billiard or infinite well system. We then solve the Schrödinger equation for the general case of a circular well with infinite walls both along the positive x-axis and at an arbitrary angle Θ (a circular “slice”) for which the half-well (Θ=π), quarter-well (Θ=π/2), and circular well plus baffle (Θ=2π) are then all special cases. We perform a PO theory analysis of this general system and calculate |ρ(L)|2 for many intermediate values of Θ to examine how the peaks in ρ(L) attributed to periodic orbits change as the quasi-circular wells are continuously transformed into each other. We explicitly examine the transitions from the half-circular well to the circle plus baffle case (half-well to quarter-circle case) as Θ changes continuously from π to 2π (from π to π/2) in detail. We then discuss the general Θ→0 limit, paying special attention to the cases where Θ=π/2n, as well as deriving the formulae for the lengths of closed orbits for the general case. We find that such a periodic orbit theory analysis is of great benefit in understanding and visualizing the increasingly complex pattern of closed orbits as Θ→0.

We analyze the quantum spectra of the two-dimensional annular billiard system based on the open-orbit quantum spectrum function, and respectively calculate the Fourier-transformed quantum spectra for various values of a parameter （in the case f=Rin/Rout）. The results show that the peak positions of quantum spectrum match with the lengths of classical orbits of particle motion very well ，and the semiclassical closed-orbit theory affords a good explanation. While the inner circle radius becomes comparaber with the de Brogile wavelength of the particle, the character of quantum spectrum changes essentially, turning out to be similar with the pattern of optical diffraction, which is a purely wavelike phenomenon and accords with Fresnel-Kirchhoff diffraction theorem. Our treatment provides a basis theory for researching the property of billiard system and microjunction transport, and opens a new way to investigate the diffraction of crystal and spectral analysis.

Based on the extended closed-orbit theory together with spectral analysis, this paper studies the correspondence between quantum mechanics and the classical counterpart in a two-dimensional annular billiard. The results demonstrate that the Fourier-transformed quantum spectra are in very good accordance with the lengths of the classical ballistic trajectories, whereas spectral strength is intimately associated with the shapes of possible open orbits connecting arbitrary two points in the annular cavity. This approach facilitates an intuitive understanding of basic quantum features such as quantum interference, locations of the wavefunctions, and allows quantitative calculations in the range of high energies, where full quantum calculations may become impractical in general. This treatment provides a thread to explore the properties of microjunction transport and even quantum chaos under the much more general system.