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Related Experiment Videos

Quantization of chaotic systems.

Gregor Tanner1, Dieter Wintgen

  • 1Fakulatat fur Physik der Universitat, Hermann-Herder-Str. 3, 7800 Freiburg, Germany.

Chaos (Woodbury, N.Y.)
|January 1, 1992
PubMed
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This study introduces a novel method combining cycle expansion and functional equations to calculate semiclassical eigenvalues for chaotic Hamiltonian systems. The approach accurately determines highly excited eigenvalues, even for complex systems like the anisotropic Kepler problem.

Area of Science:

  • Quantum mechanics and chaos theory
  • Mathematical physics
  • Dynamical systems

Background:

  • Calculating semiclassical eigenvalues for chaotic Hamiltonian systems is a significant challenge in quantum mechanics.
  • Existing methods often require extensive classical data or lack accuracy for highly excited states.

Purpose of the Study:

  • To develop and demonstrate a robust method for obtaining highly excited semiclassical eigenvalues.
  • To apply this method to the anisotropic Kepler problem, a well-known chaotic system.

Main Methods:

  • Utilized the semiclassical dynamical zeta function as a starting point.
  • Employed a combination of the cycle expansion method and a functional equation.
  • Applied Bogomolny's transfer matrix approach for computational efficiency.

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Main Results:

  • Successfully obtained highly excited semiclassical eigenvalues with high accuracy.
  • Demonstrated the method's effectiveness on the anisotropic Kepler problem, showcasing its power for strongly chaotic systems.
  • The transfer matrix approach significantly reduced the required classical input data.

Conclusions:

  • The combined cycle expansion and functional equation method provides an accurate and efficient way to compute semiclassical eigenvalues.
  • This approach offers a powerful tool for studying chaotic Hamiltonian systems and their quantum properties.