Integrability and Action Operators in Quantum Hamiltonian Systems

Authors

Vyacheslav V. Stepanov
Gerhard Müller, University of Rhode IslandFollow

Document Type

Article

Date of Original Version

2001

Abstract

For a (classically) integrable quantum-mechanical system with two degrees of freedom, the functional dependence Ĥ=H_Q(J₁,J₂) of the Hamiltonian operator on the action operators is analyzed and compared with the corresponding functional relationship H(p₁,q₁;p₂,q₂)=H_C(J₁,J₂) in the classical limit of that system. The former converges toward the latter in some asymptotic regime associated with the classical limit, but the convergence is, in general, nonuniform. The existence of the function Ĥ=H_Q(J₁,J₂) in the integrable regime of a parametric quantum system explains empirical results for the dimensionality of manifolds in parameter space on which at least two levels are degenerate. The analysis is carried out for an integrable one-parameter two-spin model. Additional results presented for the (integrable) circular billiard model illuminate the same conclusions from a different angle.

Citation/Publisher Attribution

Stepanov, V. V., & Müller, G. (2001). Integrability and action operators in quantum Hamiltonian systems. Physical Review E, 63(5), 056202-1/9.

Available at: http://dx.doi.org/10.1103/PhysRevE.63.056202