Integrability and nonintegrability of quantum systems. II. Dynamics in quantum phase space

Abstract

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 scrG and in one of its unitary irreducible-representation carrier spaces ${\mathit{gerh}}_{\mathrm{\Lambda }}$, the quantum phase space is a 2${\mathit{M}}_{\mathrm{\Lambda }}$-dimensional topological space, where ${\mathit{M}}_{\mathrm{\Lambda }}$ is the quantum-dynamical degrees of freedom. This quantum phase space is isomorphic to a coset space scrG/scrH via the unitary exponential mapping of the elementary excitation operator subspace of scrg (algebra of scrG), where scrH (⊂scrG) is the maximal stability subgroup of a fixed state in ${\mathit{gerh}}_{\mathrm{\Lambda }}$. The phase-space representation of the system is realized on scrG/scrH, 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.