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Interaction Picture

We define an interaction picture by writing

$\displaystyle H \equiv H_0+V,\quad H_0\equiv H_S+H_B, \quad V\equiv H_{SB}$     (4)

with the Hamiltonian $ H_0$ describing the time evolution of the uncoupled system and bath, and the perturbation $ V$ describing the interaction $ H_{SB}$.

We define $ \chi(t)$ as the total density matrix (system + bath) which obeys the Liouville-von-Neumann equation ,

$\displaystyle \frac{d}{dt}\chi(t)=-i[H,\chi(t)] \leadsto \chi(t)= e^{-iHt} \chi(t=0) e^{iHt},$     (5)

where we start with the initial condition $ \chi(t=0)$ at time $ t=0$. In the interaction picture,
$\displaystyle \tilde{\chi}(t)$ $\displaystyle \equiv$ $\displaystyle e^{iH_0t} \chi(t) e^{-iH_0t}$ (6)
$\displaystyle \tilde{A}(t)$ $\displaystyle \equiv$ $\displaystyle e^{iH_0t} A e^{-iH_0t}.$ (7)

The equation of motion for the density operator in the interaction picture becomes
$\displaystyle \frac{d}{dt} \tilde{\chi}(t)$ $\displaystyle =$ $\displaystyle i [H_0,\tilde{\chi}(t)] +
e^{iH_0t} \frac{d}{dt}\chi(t) e^{-iH_0t}$  
  $\displaystyle =$ $\displaystyle i [H_0,\tilde{\chi}(t)] -i
e^{iH_0t} [H,\chi(t)] e^{-iH_0t}$  
  $\displaystyle =$ $\displaystyle i [H_0,\tilde{\chi}(t)] -i
e^{iH_0t} [H_0+V,\chi(t)] e^{-iH_0t}$  
  $\displaystyle =$ $\displaystyle i [H_0,\tilde{\chi}(t)] -i
[H_0+\tilde{V}(t),\tilde{\chi}(t)]$  
  $\displaystyle =$ $\displaystyle -i
[\tilde{V}(t),\tilde{\chi}(t)].$ (8)

In integral form, this can be written as
$\displaystyle \tilde{\chi}(t)$ $\displaystyle =$ $\displaystyle {\chi}(t=0) -i \int_0^t dt'[\tilde{V}(t'),\tilde{\chi}(t')]$ (9)

which we insert into Eq. (7.8) to obtain
$\displaystyle \fbox{$ \begin{array}{rcl} \displaystyle
\frac{d}{dt} \tilde{\chi...
...]
-\int_0^t dt'[\tilde{V}(t),[\tilde{V}(t'),\tilde{\chi}(t')]].
\end{array}$\ }$     (10)

Up to here, everything is still exact.


next up previous contents index
Next: Perturbation Theory in the Up: Master Equation I: Derivation Previous: Master Equation I: Derivation   Contents   Index
Tobias Brandes 2004-02-18