INTEGRABILITY AND NONINTEGRABILITY OF QUANTUM-SYSTEMS .2. DYNAMICS IN QUANTUM PHASE-SPACE

Based on the concepts of integrability and nonintegrability of a quantum system presented in a previous paper [Zhang, Feng, Yuan, and Wang, Phys. Rev. A 40, 438 (1989)], a realization of the dynamics in the quantum phase space is now presented. For a quantum system with dynamical group G and in one of its unitary irreducible-representation carrier spaces h-lambda, the quantum phase space is a 2M lambda-dimensional topological space, where M-lambda is the quantum-dynamical degrees of freedom. This quantum phase space is isomorphic to a coset space G/H via the unitary exponential mapping of the elementary excitation operator subspace of g (algebra of G), where H (subset-of G) is the maximal stability subgroup of a fixed state in H-lambda. The phase-space representation of the system is realized on G/H, and its classical analogy can be obtained naturally. It is also shown that there is consistency between quantum and classical integrability. Finally, a general algorithm for seeking the manifestation of "quantum chaos" via the classical analogy is provided. Illustrations of this formulation in several important quantum systems are presented.