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Parametric dependent Hamiltonians, wave functions, random matrix theory, and quantal-classical correspondence.

D Cohen1, T Kottos

  • 1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 20, 2001
PubMed
Summary

We analyzed how quantum wave functions change in a chaotic system as its shape is deformed. The study reveals insights into quantum-classical correspondence and the limits of random matrix theory.

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Area of Science:

  • Quantum mechanics
  • Classical chaos
  • Statistical physics

Background:

  • Chaotic systems exhibit complex dynamics.
  • Quantum-classical correspondence is a fundamental problem.
  • Hamiltonian systems are used to model physical phenomena.

Purpose of the Study:

  • To analyze the evolution of quantum eigenstates in a classically chaotic system.
  • To investigate the quantum-classical correspondence by examining a parametric kernel.
  • To explore both perturbative and nonperturbative regimes and assess random matrix theory.

Main Methods:

  • Numerical analysis of a two-dimensional Hamiltonian system.
  • Calculation of the parametric kernel P(n/m) = ||^2.
  • Studying the kernel's dependence on system parameter changes (Δx).

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Main Results:

  • The parametric kernel characterizes wave function shape and local density of states.
  • A well-defined classical limit for the kernel was established.
  • Limitations of the random matrix theory approach were identified.

Conclusions:

  • The study provides insights into quantum-classical correspondence in chaotic systems.
  • The parametric kernel serves as a crucial tool for analyzing quantum state evolution.
  • Findings highlight the boundaries of applicability for random matrix theory in such systems.