Quantum dynamics in Wigner and tomography representations

In the standard formulation of quantum mechanics states of a system are described by wave functions or density operators . However quantum description of the system can be given in many other ways , for example, in Wigner-Moyal or Feynman formulations . All these representation are equivalent in the the sense of physical results, but the form of presentation of quantum mechanics is different as wave functions, Wigner-Liouville functions or other representations of density operators differ in essential way from each other. In this work we are dealing with Wigner and probability representation of quantum mechanics. The probability representations or tomography representation was recently proposed in terms of marginal distribution functions (MDF). MDF describing the quantum states are positive distribution functions connected with wave functions or density matrices by known integral transformation. Sign conservation of the of the MDF can be valuable in computer simulations to overcome the "sign problem". Important advantage of the probability representation is that the quantum transitions between quantum states are descried by non-negative probabilities. This work is devoted to the development of a new stochastic approaches to numerical solution of the evolution equations for Wigner-Liouviile (WL) and marginal distribution functions. To solve the WL equation we combine both molecular dynamics and Monte Carlo methods and compute traces of the dynamical operators. The results obtained for a system of electrons and random scatterers clearly demonstrate that the many-particle interaction between the electrons leads to an enhancement of the conductivity compared to the case of noninteracting electrons. To obtain evolution of MDF we developed new stochastic approach to solution of the generalized Langevin equation (GLE). GLE is derived from Kolmogorov equations for Green function of evolution equation for MDF. We discuss the basic relations and main ideas of this approach, c- - ompare obtained numerical results with results of independent finite-difference calculations for quantum oscillator and quantum particles crossing the finite well and tunneling through the Gaussian barrier.