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Related Experiment Videos

Semiclassical accuracy in phase space for regular and chaotic dynamics.

L Kaplan1

  • 1Department of Physics, Tulane University, New Orleans, Louisiana 70118, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 28, 2004
PubMed
Summary

This study compares semiclassical accuracy in chaotic, stable, and mixed systems. Semiclassical errors grow differently over time, with chaotic systems showing linear error growth versus quadratic in stable systems.

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Area of Science:

  • Quantum mechanics
  • Classical mechanics
  • Chaos theory

Background:

  • Semiclassical approximations are crucial for bridging quantum and classical mechanics.
  • Understanding the accuracy of these approximations in different dynamical regimes is essential.

Purpose of the Study:

  • To compare the semiclassical accuracy for long-time and stationary observables in chaotic, stable, and mixed systems.
  • To analyze the time-dependent error growth of semiclassical approximations.
  • To investigate the semiclassical determination of eigenvalues and wave functions.

Main Methods:

  • Utilizing a phase-space semiclassical approximation valid to O(h) at short times.
  • Analyzing error scaling with time and Planck's constant (h_eff).
  • Comparing systems with distinct classical dynamics: chaotic, stable, and mixed.

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Main Results:

  • Squared semiclassical error grows linearly with time in chaotic systems, versus quadratically in stable systems.
  • In chaotic systems, eigenvalues and wave functions are unambiguously determined semiclassically at high energies.
  • Stable systems exhibit eigenvalue errors of the order of mean level spacing.
  • Eigenvalues in the chaotic sea of mixed systems are computed more accurately than those in stable islands.

Conclusions:

  • Semiclassical approximation accuracy is highly dependent on the underlying classical system dynamics.
  • Chaotic systems offer better prospects for accurate semiclassical eigenvalue and wave function determination at high energies.
  • The study provides insights into the limitations and capabilities of semiclassical methods across different dynamical regimes.