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Duality between quantum and classical dynamics for integrable billiards.

W T Lu1, Weiqiao Zeng, S Sridhar

  • 1Department of Physics and Electronic Materials Research Institute, Northeastern University, Boston, Massachusetts 02115, USA. w.lu@neu.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|May 23, 2006
PubMed
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We found a connection between quantum wave spectra and classical Liouvillian dynamics in integrable billiards. Classical eigenmodes appear as peaks in quantum spectra, observable via wave transmission experiments.

Area of Science:

  • Quantum mechanics
  • Classical mechanics
  • Mathematical physics

Background:

  • Integrable billiards provide a simplified model system for studying quantum and classical dynamics.
  • Understanding the relationship between quantum spectra and classical dynamics is crucial for semiclassical physics.

Purpose of the Study:

  • To establish a duality between quantum wave vector spectra and classical Liouvillian dynamics.
  • To identify observable signatures of classical eigenmodes in quantum systems.
  • To derive exact classical trace formulas for specific integrable billiards.

Main Methods:

  • Semiclassical derivation.
  • Numerical calculations.
  • Analysis of wave vector spectrum correlation functions.

Related Experiment Videos

  • Derivation of classical trace formulas for resolvents.

    • Main Results:

    • A duality is established between quantum wave vector spectrum and classical Liouvillian eigenmodes for integrable billiards.
    • Peaks in the quantum wave vector spectrum's correlation function correspond to classical eigenmodes.
    • Classical eigenmodes are observable through autocorrelation of transmission coefficients in quantum billiards.
    • Exact classical trace formulas for the resolvent are derived for rectangle, equilateral triangle, and circle billiards.
    • A correspondence is found between classical periodic orbit length spectra and quantum spectra for integrable polygonal billiards.

    Conclusions:

    • The study reveals a profound connection between quantum and classical descriptions in integrable billiards.
    • The findings offer a pathway for experimental verification of theoretical predictions.
    • The derived trace formulas provide valuable tools for analyzing integrable systems.