Chemistry Reference
In-Depth Information
this coupling is so small that it does not influence the dynamics of the whole system.
This dictates the dual approach: Dynamically the whole system is considered as a
pure (isolated) system, but then it is assumed that the bath is at thermal equilibrium.
We will concentrate on the common case in which the coupling
V
in Eq. (18)
is a small perturbation. Then
V
can be neglected in Eq. (19) and the equilibrium
DO factorizes:
ρ
eq
=
ρ
s
,
eq
ρ
b
,
eq
(20)
where, for example,
exp
H
s
k
B
T
1
Z
s
ρ
s
,
eq
=
−
(21)
where
Z
is the partition function of the small system. Defining the density matrix
ρ
s
mn
with respect to the basis of eigenstates of
H
s
in Eq. (12), one arrives at Eq.
(10). The eigenstate basis is the most convenient for calculating thermal averages
of physical quantities. However, as pointed out above, this could be done with the
help of any other basis.
The role of the small interaction
V
neglected in Eq. (19) is in providing the
source of relaxation of the small system toward the equilibrium from any ini-
tial state. To describe the dynamics of this process, we will need the interaction
representation considered in Section I.C.
C.
Temporal Evolution and Interaction Representation
Temporal evolution of the DM or DO of an isolated system, such as the small
system + bath, obeys the equation that follows from the Schr odinger equation. If,
for example, the whole system is in a pure state
|
,
its density operator is given
by
ρ
=|
|
(22)
cf. Eq. (13). Then with the help of the Schr odinger equation and its conjugate
∂
∂t
|
∂
∂t
H
H
i
=
|
,
−
i
|=
|
(23)
one obtains the quantum Liouville equation
∂t
=
H, ρ
∂ ρ
i
(24)
Note that this equation is
not
an equation of motion for an operator in the Heisen-
berg representation that has another sign. The DO consists of states that have their
