Exercise #5

a) consider the following interaction between a one dimensional bath (B) and spin $1/2$ which we will consider our system (S)

$H_{SB}=(|\uparrow><\downarrow|+|\downarrow><\uparrow|) \otimes |x><x| V(x)$

and the following density matrix (S+B)

$\rho = |\uparrow><\uparrow| P_\uparrow(x)+|\downarrow><\downarrow|P_\downarrow(x)$

with

$\int dx ( P_\uparrow(x)+P_\downarrow(x))=1$

Determine the operator

$A = tr_B [\rho, H_{SB}$]

Show that is not zero but its trace on the system variables is zero.

b) Consider the Langevin equation for the momentum and determine the mean square diffusion $<|x(t)-x(0)|^2>$ for large times

c) Given the Langevin equation

$\dot{x}(t) = F(x(t))+\sqrt{\epsilon}\eta(t)$

with $<\eta(t)\eta(t')>=\delta(t-t')$

carry on the explicit calculations that leads to the $U(z)$ and $W(z)$ terms in the proof of Fokker-Planck equation.