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Integrability and action operators in quantum Hamiltonian systems.

V V Stepanov1, G Müller

  • 1Department of Physics, University of Rhode Island, Kingston, Rhode Island 02881-0817, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 21, 2001
PubMed
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This study analyzes quantum Hamiltonians, finding their classical limit convergence is often nonuniform. This explains degeneracy in quantum systems and provides insights into integrable models.

Area of Science:

  • Quantum mechanics
  • Classical mechanics
  • Mathematical physics

Background:

  • Investigating the relationship between quantum and classical mechanics is fundamental.
  • Integrable systems offer a unique perspective for studying this quantum-classical correspondence.
  • Understanding Hamiltonian operator functional dependence is key to analyzing system dynamics.

Purpose of the Study:

  • To analyze the functional dependence of quantum Hamiltonians on action operators.
  • To compare this dependence with the classical limit's functional relationship.
  • To explain empirical results regarding level degeneracy in parametric quantum systems.

Main Methods:

  • Analysis of the functional dependence H=HQ(J1,J2) for integrable quantum systems.

Related Experiment Videos

  • Comparison with the classical limit functional relationship H(p1,q1;p2,q2)=HC(J1,J2).
  • Case studies on an integrable one-parameter two-spin model and the circular billiard model.

    • Main Results:

    • The quantum Hamiltonian's functional dependence converges to the classical limit's, but often nonuniformly.
    • The existence of H=HQ(J1,J2) in integrable regimes explains observed dimensionalities of degenerate level manifolds.
    • Analysis of specific models (two-spin, circular billiard) confirms these findings.

    Conclusions:

    • Nonuniform convergence is a characteristic of the quantum-classical transition in these systems.
    • The functional form of the quantum Hamiltonian provides a theoretical basis for degeneracy phenomena.
    • Integrable models serve as crucial testbeds for understanding fundamental quantum-classical relationships.