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Related Experiment Videos

Periodic orbit quantization of the anisotropic Kepler problem.

Freddy Christiansen1, Predrag Cvitanovic

  • 1Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark.

Chaos (Woodbury, N.Y.)
|January 1, 1992
PubMed
Summary

This study tested methods for calculating quantum spectra in chaotic systems using periodic orbit quantization. Researchers found no evidence supporting four common claims about the accuracy of these semiclassical estimation techniques.

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Area of Science:

  • Quantum mechanics
  • Classical chaos
  • Mathematical physics

Background:

  • Periodic orbit quantization is a method to study quantum spectra of classically chaotic systems.
  • Several claims exist regarding the efficacy of different semiclassical estimation techniques, particularly those involving zeta functions and cycle expansions.

Purpose of the Study:

  • To test four specific claims about calculating quantum spectra for the anisotropic Kepler problem.
  • To evaluate the accuracy of curvature expansions and cycle expansions of quantum mechanical zeta functions.
  • To assess the necessity of cycle expansions and the sufficiency of irreducible cycles for eigenvalue estimation.

Main Methods:

  • Computed the stability and action of approximately 2000 shortest periodic orbits.
  • Calculated the eigenvalue spectrum of the anisotropic Kepler problem using these orbits.

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  • Assessed the validity of four proposed semiclassical estimation claims.

    • Main Results:

    • No evidence was found to support the claim that curvature expansions provide the best semiclassical estimates.
    • The study found no support for the claim that real parts of cycle expansions offer the best estimates.
    • Evidence did not support the idea that cycle expansions are superfluous or that irreducible cycles alone suffice for accurate eigenvalue estimates.

    Conclusions:

    • The tested claims regarding semiclassical estimation techniques for quantum spectra in chaotic systems were not supported by this study.
    • The results suggest that current methods, including cycle expansions and curvature expansions, may not be as effective as previously suggested for the anisotropic Kepler problem.