Spatio-temporal evolution of initial coherent state under the action of quantum integrable and nonintegrable systems

According to the exactly formulated concepts of quantum integrability and nonintegrability, an initial coherent state \Phi (0)(gamma)] evolving under the action of an integrable Hamiltonian having the same dynamical symmetry is expressed as an analytical function of gamma and t. The expectation values of various dynamical variables should be expressed as correlated analytical functions of gamma and t. Thus, the general characters of initial coherent state would be maintained during the spatio-temporal evolution under the action of such a system even if the wavepacket is slightly distorted. In contrast, when the integrability condition is destroyed, these features will be lost during the evolution. This theoretical prediction is illustrated with numerical results computed for the two-level Lipkin model with a double-well potential and dynamical explanations are given. And the Eherenfest time is verified with this model.