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We wish to obtain an equation of motion for the effective density matrix of the system at time
,
![$\displaystyle {\rho}(t)\equiv {\rm Tr}_B [\chi(t)].$](img57.png) |
|
|
(11) |
This object is sufficient to calculate expectation values of system operators
:
Now use
Note that the interaction picture
involves
only the free System Hamiltonian
and not
,
 |
|
|
(14) |
Using
 |
|
|
(15) |
for system operators, one has
![$\displaystyle \langle A_S \rangle_t = {\rm Tr}_S\left[ \tilde{\rho}(t)\tilde{A}_S(t)\right]
= {\rm Tr}_S\left[ {\rho}(t){A}_S(t)\right].$](img70.png) |
|
|
(16) |
Next: Equation of Motion for
Up: Perturbation Theory in the
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Tobias Brandes
2004-02-18