Quantum Transport and the Wigner Function

Abstract

Classical transport is based on the Boltzmann transport equation:

$$\frac{{\partial {\text{f}}}}{{\partial {\text{t}}}} + {\text{v}}\cdot{\vec \nabla _{\text{r}}}{\text{f + F}}\cdot{\vec \nabla _{\text{p}}}{\text{f = }}{\left( {\frac{{\partial {\text{f}}}}{{\partial {\text{t}}}}} \right)_{{\text{collision}}}}$$

((1))

where

$${\text{v}} = {\vec \nabla _{\text{p}}}{\text{E}}\left( {\text{p}} \right){\text{; p = }}\hbar {\text{k; F = qE}}$$

((2))

and f=f(r, p, t) is the probability of finding a typical particle within a unit volume around the point r, in momentum state p, at time t, and

$${\left( {\frac{{\partial {\text{f}}}}{{\partial {\text{t}}}}} \right)_{{\text{collision}}}} = \int {dp'{\mkern 1mu} \left[ {{\text{f}}\left( {{\text{p'}}} \right){\text{W}}\left( {{\text{p'}},{\mkern 1mu} {\text{p}}} \right){\mkern 1mu} {\mkern 1mu} - {\mkern 1mu} f\left( p \right){\text{W}}\left( {{\text{p,}}{\mkern 1mu} {\text{p'}}} \right)} \right]} $$

((3))

Here W(p′, p) is the transition rate from p to p′ and depends on the source of scasttering.