Classical and quantum confocal parabolic billiards
作者:
Bárbara K. Villarreal-Zepeda,
Héctor M. Iga-Buitrón,
Julio C. Gutiérrez-Vega,
期刊:
American Journal of Physics
(AIP Available online 2021)
卷期:
Volume 89,
issue 12
页码: 1113-1122
ISSN:0002-9505
年代: 2021
DOI:10.1119/10.0006018
出版商: American Association of Physics Teachers
数据来源: AIP
摘要:
The classical and quantum description of a free particle moving inside a confocal parabolic billiard is studied. From a classical approach, we derive the characteristic equations for periodic trajectories, discuss the Poincaré maps, and explore interesting geometrical properties of the orbits inside the billiard. From a quantum description, we determine the eigenstates and energy spectrum of the confined particle. The confocal parabolic billiard is an integrable system that exhibits two degrees of freedom and two constants of motion. The way to establish a correspondence between the classical and quantum solutions is by equating the constants of motion for both descriptions. The parabolic billiard provides a well-motivated and relatively straightforward example of the Hamilton–Jacobi theory in a way that is seldom discussed in the undergraduate curriculum.
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