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Semiclassical approximations based on complex trajectories.

A D Ribeiro1, M A M de Aguiar, M Baranger

  • 1Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, Unicamp, 13083-970, Campinas, Sao Paulo, Brazil.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|July 13, 2004
PubMed
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The study reveals that semiclassical approximations for quantum wave packet return probabilities require excluding specific classical trajectories. Filtering these trajectories, particularly those near bifurcations, leads to excellent agreement with exact quantum calculations.

Area of Science:

  • Quantum mechanics
  • Classical mechanics
  • Chaos theory

Background:

  • The semiclassical limit of quantum mechanics connects quantum behavior to classical trajectories.
  • Coherent state propagators describe the evolution of quantum wave packets.

Purpose of the Study:

  • To investigate the accuracy of semiclassical approximations for wave packet return probabilities.
  • To identify conditions under which classical trajectories must be excluded from semiclassical formulas.

Main Methods:

  • Calculation of the coherent state propagator in the semiclassical limit.
  • Analysis of complex classical trajectories for a soft chaotic system.
  • Comparison of exact return probabilities with semiclassical approximations.

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Main Results:

  • Exact return probabilities reveal quantal images of classical periodic orbits.
  • Two conditions for excluding trajectories from semiclassical formulas were identified: exponential divergence and bifurcations.
  • Near bifurcations, semiclassical formulas diverge; post-bifurcation, only one of multiple valid trajectories contributes.

Conclusions:

  • Semiclassical approximations require careful exclusion of specific classical trajectories to match exact quantum results.
  • The phenomenon is related to Stokes lines and highlights limitations of semiclassical methods near bifurcations.