On a special picture of dynamical evolution of nonlinear quantum systems in the phase-space representation
In classical nonlinear dynamics, the stability of trajectories in phase-space is one of the most important problems. In quantum mechanics, the quantum flows are functionals, so the classical concept of trajectories does not make sense. Yet the problem of the stability of motion still exists, and indeed becomes more compliated. In this paper the author presents a ``special picture'' of quantum evolution in a phase-space representation. In his representation it is once again possible to make sense of trajectories. According to this picture both the symbols of operators and the distribution density carry time dependence. The symbols of operators evolve classically and the distribution density carries the remaining time dependence needed to recover the full time dependence of quantum averages. The author first develops the equations of motion in this mixed picture of quantum mechanics. He then shows how these equations can be used in the analysis of spin-boson Hamiltonian systems. The phase space structure of the spin-boson system is established by obtain asymptotic solutions for the stable regions. Then the equation of evolution of the distribution density in the mixed picture is solved analytically. The final section computes asymptotic quantum averages of observables.
scientific article; zbMATH DE number 5612810
scientific article