### 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≡R_{in}<R_{out}≡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=R_{in}/R_{out}), 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.

Original language | English (US) |
---|---|

Pages (from-to) | 67-77 |

Number of pages | 11 |

Journal | American Journal of Physics |

Volume | 67 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1999 |

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### All Science Journal Classification (ASJC) codes

- Physics and Astronomy(all)

### Cite this

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**Periodic orbit theory analysis of the circular disk or annular billiard : Nonclassical effects and the distribution of energy eigenvalues.** / Robinett, Richard Wallace.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Periodic orbit theory analysis of the circular disk or annular billiard

T2 - Nonclassical effects and the distribution of energy eigenvalues

AU - Robinett, Richard Wallace

PY - 1999/1/1

Y1 - 1999/1/1

N2 - 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 0in/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.

AB - 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 0in/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.

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U2 - 10.1119/1.19193

DO - 10.1119/1.19193

M3 - Article

AN - SCOPUS:0033484719

VL - 67

SP - 67

EP - 77

JO - American Journal of Physics

JF - American Journal of Physics

SN - 0002-9505

IS - 1

ER -