QUANTUM DYNAMICS IN CANONICAL AND MICRO-CANONICAL ENSEMBLES - PART I - ANDERSON LOCALIZATION OF ELECTRONS

A new numerical approach for consideration of quantum dynamics and cal culations of the average values of quantum operators and time correlat ion functions in the Wigner representation of quantum statistical mech anics has been developed. The time correlation functions have been pre sented in the form of integrals of the Weyl's symbol of considered ope rators and Fourier transforms of the product of matrix elements of the dynamic propagators. For the latter functions the integral Wigner-Lio uville's type equation has been derived. The initial condition for thi s equation has been obtained in the form of the Fourier transform of t he Wiener path integral representation of the matrix elements of the p ropagators at initial time. The numerical procedure for solving this e quation combining both molecular dynamics and Monte carlo methods has been developed An application of the developed approach to the micro c anonical ensemble has also been considered in the second part of this paper. For electrons in disordered systems of scatterers numerical res ults have been obtained for series of the average values of the quantu m operators including position and momentum dispersions, average energ y, energy distribution function as well as for the frequency dependenc ies of tensors of electron conductivity and permittivity according to quantum Kubo formula Zero or very small values of the static conductiv ity have been considered as a manifestation of Anderson localization o f electrons in the 1D case. Independent evidence of Anderson localizat ion comes from the behaviour of the calculated time dependence of posi tion dispersion. Nevertheless for localized electrons the energy distr ibution function obtained has a long exponentially decaying tail, whic h is the reason for the exponentially rare appearance of large values of quantum particle virtual energy that strongly affects the behaviour of the position dispersion.