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Arbitrary trajectory quantization method.

D Biswas1

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

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 17, 2001
PubMed
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The arbitrary trajectory quantization method (ATQM) was refined to include closed almost-periodic (CAP) orbits for polygonal billiards. This modification improves quasiclassical quantization at finite energies, with periodic orbits dominating at high energies.

Area of Science:

  • Quantum mechanics
  • Classical mechanics
  • Mathematical physics

Background:

  • The arbitrary trajectory quantization method (ATQM) links quantum energy spectra to classical periodic orbits.
  • Previous work suggested relaxing periodicity criteria for polygonal billiards to include closed almost-periodic (CAP) orbits.

Purpose of the Study:

  • To reinvestigate the arbitrary trajectory quantization method (ATQM).
  • To incorporate closed almost-periodic (CAP) orbit families into the ATQM framework for polygonal billiards.
  • To clarify the behavior of the quasiclassical kernel at finite and infinite energies.

Main Methods:

  • Revisiting the arbitrary trajectory quantization method (ATQM).
  • Analyzing the relationship between quantum energy spectra and classical orbits, including CAP families.

Related Experiment Videos

  • Examining the quasiclassical kernel's behavior at different energy regimes.

    • Main Results:

    • At finite energies, a modified ATQM formula including CAP families yields a smoothened quasiclassical kernel.
    • As energy approaches infinity (E-->infinity), the delta function kernel corresponding to the original periodic orbit formula is recovered.
    • The study provides clarifications on the ATQM and its application to polygonal billiards.

    Conclusions:

    • The arbitrary trajectory quantization method (ATQM) is effectively extended to include closed almost-periodic (CAP) orbits for polygonal billiards.
    • The modified ATQM accurately describes quasiclassical quantization at finite energies.
    • The original ATQM formulation remains valid in the high-energy limit.