Quasiprobability density diffusion equations for the quantum Brownian motion in a potential

Abstract

Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] representation of the density operator as a c-number quasiprobablity distribution in phase space allowing quantum mechanical averages involving the density matrix to be calculated as phase space averages just as classical averages originally used by him to obtain quantum mechanical corrections to classical thermodynamic equilibrium i.e. to the Maxwell-Boltzmann distribution so applying to closed quantum systems is extended to open quantum systems comprising a canonical ensemble of Brownian particles in a potential. This is accomplished via an idea of Gross and Lebowitz [E. P. Gross and J. L. Lebowitz, Phys. Rev. 1956, 104, 1528]. They suggested that using Wigner's representation the connection between classical and quantum collision kernels, (i.e. in classical mechanics the Stosszahlanzatz describing the bath-particle interaction in the open system in the Boltzmann equation for the single particle distribution function) is much more transparent than in the density operator formalism. Moreover the quantum kernel should closely correspond to the classical one. Hence the idea developed in this Thesis that in the quantum Brownian motion the collision term in a quantum master equation in Wigner's representation should be described by a Kramers-Moyal like expansion truncated at the second term (leading of course in the classical limit to the Fokker-Planck equation) as in the classical Brownian motion. Imposition of the Wigner equilibrium distribution as the stationary solution of this equation (which is akin to the Fokker-Planck equation) in the manner used by Einstein [A. Einstein, in R. H. Fürth, Ed., Investigations on the Theory of the Brownian Movement, Methuen, London, 1926; reprinted Dover, New York, 1954] used to calculate diffusion coefficients in the Fokker-Planck equation by imposing the Maxwell-Boltzmann distribution as the stationary distribution then leads to the most important results of this Thesis. Namely the diffusion coefficients in the master equation become functions of the quantum parameter and the derivatives of the potential. Moreover all the solution techniques (matrix continued fractions etc) developed for the Fokker-Planck equation carry over to the quantum case as is illustrated by calculating the reaction rate and dynamical structure factor of a particle in a periodic cosine potential where the results are in agreement with those predicted by quantum reaction rate theory.