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Measuring billiard eigenfunctions with arbitrary trajectories.

Debabrata Biswas1

  • 1Theoretical Physics Division, Bhabha Atomic Research Centre, Mumbai 400 085, India.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 15, 2003
PubMed
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We present a novel method for measuring quantum eigenfunctions in polygonal billiards. This quasiclassical approach accurately predicts quantum behavior, validated using the stadium billiard model.

Area of Science:

  • Quantum mechanics
  • Mathematical physics
  • Classical mechanics

Background:

  • Billiard systems are fundamental in studying classical and quantum chaos.
  • Previous methods struggled to accurately approximate quantum eigenfunctions in complex geometries.
  • Polygonal billiards offer a tractable yet rich framework for theoretical investigation.

Purpose of the Study:

  • To develop a quasiclassical method for measuring approximate quantum eigenfunctions.
  • To establish a connection between quasiclassical and Schrödinger equations for polygonal billiards.
  • To validate the proposed method using the stadium billiard geometry.

Main Methods:

  • A quasiclassical evolution operator with a smoothed Perron-Frobenius kernel was employed.
  • A phase factor derived from quantum considerations was incorporated.

Related Experiment Videos

  • A plane wave ansatz was used to derive the eigenfunction condition.
  • The method was applied to the polygonalized stadium billiard.

    • Main Results:

    • The condition for quasiclassical eigenfunctions matches the Schrödinger equation for polygonalized billiards.
    • Approximate quantum eigenfunctions were successfully determined for the stadium billiard.
    • Comparison with exact quantum eigenfunctions demonstrated the method's accuracy.

    Conclusions:

    • The proposed quasiclassical method provides an effective way to measure approximate quantum eigenfunctions.
    • This technique bridges classical and quantum descriptions in complex billiard systems.
    • The findings have implications for understanding quantum chaos in geometrically confined systems.