Chemistry Reference
In-Depth Information
one uses the set of states used in the definition of
ρ
:
n
mn
Tr
Aρ
=
A
Aρ
A
=
ψ
n
|
|
ψ
n
=
ψ
n
|
|
ψ
m
ψ
m
|
ρ
|
ψ
n
=
A
nm
ρ
mn
mn
(14)
This result coincides with Eq. (2). It can be proven that the trace of an operator is
independent of the choice of the basis in Tr
{···}
and operators can be cyclically
permuted under the trace symbol, so that Tr
Aρ
=
Tr
ρ A
.
This formalism
does not require that the system is in a pure state.
If the system interacts with its environment described by the basis
|
φ
, the
total DO has the form
ρ
=
ρ
m,n
|
ψ
m
φ
ψ
n
φ
|≡
ρ
m,n
|
ψ
m
ψ
n
|⊗|
φ
φ
|
m,n
m,n
(15)
Calculating any observable
A
that corresponds to the small system can be done
in two steps: First, calculating the trace over the variables of the bath and then
calculating the trace over the basis states of the small system using Eq. (14). The
first step yields the reduced density operator for the small system
ρ
s
=
Tr
b
ρ
≡
Tr
ρ
(16)
ρ
s
mn
and
that has the form of Eq. (11) with
ρ
mn
⇒
ρ
s
mn
=
ρ
m,n
(17)
If the whole system is in a pure state, this formula coincides with Eq. (9).
Let us write the Hamiltonian of the whole system in the form
H
H
0
+
V,
H
0
≡
H
s
+
H
b
,
V
H
s
−
b
=
≡
(18)
where
H
s
is the Hamiltonian of the small system,
H
b
is the Hamiltonian of the
environment (the bath), and
V
is the interaction. At equilibrium, the DO of the
whole system assumes the form
Z
exp
H
k
B
T
1
ρ
eq
=
−
(19)
where
Z
is the partition function. Because of applications in quantum statistics,
the density operator is also called the
statistical operator
.
In obtaining Eq. (19), it is tacitly assumed that the whole system (small system
+ bath) is surrounded by the so-called superbath. A very small coupling to the
superbath (that is never considered explicitly) ensures that the whole system tends
to thermal equilibrium with the temperature
T
of the superbath. On the other hand,
