Appendix
313
Here,
ˆ
B I (t) = e
i ˆ
Ht ˆ
Be
−i ˆ
Ht
(A.7.7)
is the perturbation operator in the (time-dependent) interaction picture.
Now we consider a property associated with a one-particle operator, say ˆ
A =
A rs c
†
r c s . For t ≥ t 0 , the expectation value of ˆ
A, being
¯
A 0 = = 0 | ˆ
A| 0
(A.7.8)
at t 0 , evolves in time according to
¯
A(t) = =(t)| ˆ
A|(t) = = x (t)| ˆ
A I (t)| x (t)
(A.7.9)
where
ˆ
A I (t) = e
i ˆ
Ht ˆ
Ae
−i ˆ
Ht
(A.7.10)
is representation of ˆ
A in the interaction picture. The difference
A(t) = ¯
A(t) − ¯
A 0
(A.7.11)
is referred to as the response of the property A to the time-dependent perturbation
ˆ
H x (t). Using Eqs. (A.7.5), (A.7.6), the function A(t) can be expanded in a PT
series in the perturbing potential, beginning in first order (linear response):
A
(1)
(t) ==
(1)
x (t)| ˆ
A(t)| 0 ) + + 0 | ˆ
A(t)|
(1)
x (t)
=
t
t 0
i 0 |[ ˆ
B(t
), ˆ
A(t)]| 0 f (t
) dt
(A.7.12)
Introducing the so-called linear response function R AB (t, t
),
R AB (t, t
) = iθ(t − t
) 0 |[ ˆ
B(t
), ˆ
A(t)]| 0
(A.7.13)
the response can be written in the compact form
A
(1)
(t) =
∞
t 0
R AB (t, t
) f (t
) dt
(A.7.14)
Here, R AB (t, t
) represents the generic part of the response, while the particular
time-dependence of the respective perturbation enters via f (t
).
The energy representation of the response function is obtained via the Fourier
transformation
R AB (ω) =
∞
−∞
e
i(ω+iη)t R AB (t, 0) dt
(A.7.15)