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Unformatted text preview: c� (1.8) �
r = � c
k −q �
�k �
k Equation (1.7) is the ﬁrst quantized form of ρ (�), and equation (1.8) is the second quantized
ˆq
†
form with c� the creation operator for an electron with momentum1 � − � and c� the destruction
kq
k
k −q
�
�.
operator for an electron with momentum k
ˆ
Returning to equation (1.1), we’d like to think about V as a perturbation on the external
ˆ . This leads us naturally to consider H as the unperturbed Hamil
ˆ
ﬁeldfree system Hamiltonian H
ˆ
tonian within the interaction picture representation. Recall that this H is a very complicated
beast with all of the electronelectron repulsions included, but for our purposes we just take as a
given that there are a set of eigenstates and energies that diagonalize this Hamiltonian.
Recall the formulation of the interaction representation:
ih
¯ ∂
ˆ
ˆ
φ (t)� = (H + V )φ (t)�
∂t (1.9) ˆ
We can “unwind” the natural time dependence due to H from the state ket φ (t)� to form an
˜ (t)�I by
interaction representation state ket φ
ˆ ˜
φ (t)�I = eiHt φ (t)�
ˆ˜
φ (t)�I = e−iHt φ (t)� (1.10)
(1.11) ˆ
Note that in the absence of V , these interaction picture state kets are actually the Heisenberg
h
picture state kets of the system. Also, we have now oﬃcially set ¯ = 1. After substituting (1.11)
into (1.9), we obtain
i¯
h 1 ∂
ˆˆ
ˆ˜
= eiHt V e−iHt φ (t)�
∂t
ˆ˜
= VI φ (t)� (1.12)
(1.13) � and q are actually wavevectors, which diﬀer from momenta by a factor of ¯ . When in doubt, assume h = 1.
k
�
h
¯ Response Functions and the Interaction Representation 3 where we have set
ˆˆ
ˆ
ˆ
VI = eiH t V e−iHt (1.14) Thus the interaction picture state ket evolves simply according to the dynamics governed solely
ˆ
by the interaction picture perturbing potential VI .
More generally, we can write any observable (operator) in the interaction picture as
ˆˆ
ˆ
ˆ
AI = eiHt Ae−iHt We can integrate equation (1.12) with respect to t to get
�t
˜ (t)� = φ0 � − i
˜
ˆ
φ
dt� VI (t� ) φ (t� )� (1.15) (1.16) −∞ At ﬁrst it seems like we have not done much to beneﬁt ourselves, since all we have done is
ˆ
to convert the ordinary Schrodinger equation, a PDE, into an integral equation. However, if VI
is small, then we can iterate equation (1.16):
�t
˜
ˆ
φ (t)� ≈ φ0 � − i
dt� VI (t� ) φ0 � + · · ·
(1.17)
−∞ ˆ
The essence of linear response theory is that we focus ourselves on cases where VI is suﬃ
ciently weak that the perturbation series represented by equation (1.17) has essentially converged
ˆ
after including just the ﬁrst nontrivial term listed above. This term is linear in VI .
Throughout this discussion, we will be working at T = 0, so φ0 � is simply the ground state
ˆ
of the nonperturbed total system Hamiltonian H . Note that we have taken our initial time,
i.e. the lower limit of integration in equation (1.16), to be −∞. This is because we want to
ˆ
imagine turning on the prob...
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This note was uploaded on 02/07/2014 for the course PHYS 8.512 taught by Professor Patricklee during the Fall '09 term at MIT.
 Fall '09
 PatrickLee
 Charge, Polarization

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