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Geometric phase, Hannay's angle, and an exact action variable

Song1

  • 1Department of Physics, Sunchon National University, Sunchon 540-742, Korea.

Physical Review Letters
|September 16, 2000
PubMed
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Researchers explored the time-periodic harmonic oscillator, defining Hannay's angle and its connection to periodic wave functions. This geometric phase relation is shown to be exact for these systems.

Area of Science:

  • * Theoretical Physics
  • * Quantum Mechanics
  • * Classical Mechanics

Background:

  • * The study of generalized time-periodic harmonic oscillators is crucial for understanding complex dynamical systems.
  • * Canonical structure and invariants are fundamental concepts in classical and quantum mechanics.
  • * Hannay's angle and geometric phase are advanced concepts in semiclassical physics.

Purpose of the Study:

  • * To determine the canonical structure of a generalized time-periodic harmonic oscillator.
  • * To define and investigate Hannay's angle in this context.
  • * To explore the relationship between Hannay's angle, periodic wave functions, and geometric phase.

Main Methods:

  • * Exact calculation of the action variable (invariant) for the system.

Related Experiment Videos

  • * Analysis of phase space trajectories under time evolution.
  • * Comparison with conditions for the existence of (quasi)periodic wave functions.

    • Main Results:

    • * The exact action variable (invariant) was found for the generalized time-periodic harmonic oscillator.
    • * The condition for the existence of Hannay's angle was identified as identical to that for complete sets of (quasi)periodic wave functions.
    • * Berry's exact relation between geometric phase and Hannay's angle was demonstrated for the considered cases.

    Conclusions:

    • * Hannay's angle provides a precise geometric interpretation for the behavior of time-periodic systems.
    • * The existence of Hannay's angle is directly linked to the system's quantum mechanical properties (periodicity of wave functions).
    • * The study confirms the exactness of semiclassical relations involving geometric phase and Hannay's angle in specific physical systems.