Quantum dynamics in canonical and micro-canonical ensembles. Part II.   Tunneling in double well potential
  In the second part of this paper in micro canonical ensemble the new numerical approach for consideration of quantum dynamics and calculations of the average values of quantum operators and time correlation functions in the Wigner representation of quantum statistical mechanics has been developed. The numerical results have been obtained for series of the average values of quantum operators as well as for the time correlation function characterizing the energy level structure, the momentum flow of tunneling particles at barrier crossing and the absorption spectra of electron in potential well. The developed quantum dynamics method was tested by comparison of numerical results with analytical estimations. Tunneling transitions and the effect of the quasi stationary state has been considered as the reason of the peculiarities in behaviour of the time correlation functions and position and momentum dispersions. Possibility of applying the developed approach to the theory of classical wave propagation in random media have been also considered. For classical waves some results have been obtained for Gaussian beam propagation in 2D and 3D waveguides. 
V. Filinov, Yu. Lozovik, A. Filinov, I. Zacharov, A. Oparin