Environmental Engineering Reference
In-Depth Information
the external force is applied.
When the system is in thermal equilibrium, it is characterized by
the density operator
1
Z
e
−βH
0
Z
=Tr
e
−βH
0
,
ρ
0
=
;
(3
.
1
.
1)
where
H
0
is the (effective) Hamiltonian,
Z
is the (grand) partition func-
tion, and
β
=1
/k
B
T
. Since we are only interested in the linear part of
the response, we may assume that the weak external disturbance
f
(
t
)
gives rise to a linear time-dependent perturbation in the total Hamilto-
nian
H
:
Af
(
t
)
H
1
,
(3
.
1
.
2)
where
A
is a constant operator, as for example
i
J
zi
, associated with
the Zeeman term when
f
(
t
)=
gµ
B
H
z
(
t
) (the circumflex over
A
or
B
indicates that these quantities are quantum mechanical operators). As
a consequence of this perturbation, the density operator
ρ
(
t
) becomes
time-dependent, and so also does the ensemble average of the operator
B
:
H
1
=
−
;
H
=
H
0
+
B
(
t
)
ρ
(
t
)
B
=Tr
{
}
.
(3
.
1
.
3)
The linear relation between this quantity and the external force has the
form
=
t
−∞
B
(
t
)
B
t
)
f
(
t
)
dt
,
−
φ
BA
(
t
−
(3
.
1
.
4)
B
B
(
t
=
ρ
0
B
;here
f
(
t
) is assumed to vanish
for
t →−∞
. This equation expresses the condition that the differential
change of
−∞
{
}
where
=
)
=Tr
B
(
t
)
is proportional to the external disturbance
f
(
t
)and
the duration of the perturbation
δt
, and further that disturbances at
different times act independently of each other. The latter condition
implies that the
response function
φ
BA
mayonlydependonthetime
t
. In (3.1.4), the response is independent of any future per-
turbations. This causal behaviour may be incorporated in the response
function by the requirement
difference
t
−
t
)=0
for
t
>t,
φ
BA
(
t
−
(3
.
1
.
5)
in which case the integration in eqn (3.1.4) can be extended from
t
to
+
.
Because
φ
BA
depends only on the time difference, eqn (3.1.4) takes
a simple form if we introduce the Fourier transform
f
(
ω
)=
∞
−∞
∞
f
(
t
)
e
iωt
dt,
(3
.
1
.
6
a
)
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