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Unformatted text preview: c� (1.8) �
r = � c
k −q �
k Equation (1.7) is the ﬁrst quantized form of ρ (�), and equation (1.8) is the second quantized
form with c� the creation operator for an electron with momentum1 � − � and c� the destruction
operator for an electron with momentum k
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
ﬁeldfree system Hamiltonian H
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:
|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
|φ (t)�I = eiHt |φ (t)�
|φ (t)�I = e−iHt |φ (t)� (1.10)
Note that in the absence of V , these interaction picture state kets are actually the Heisenberg
picture state kets of the system. Also, we have now oﬃcially set ¯ = 1. After substituting (1.11)
into (1.9), we obtain
h 1 ∂
= eiHt V e−iHt |φ (t)�
= 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.
¯ Response Functions and the Interaction Representation 3 where we have set
VI = eiH t V e−iHt (1.14) Thus the interaction picture state ket evolves simply according to the dynamics governed solely
by the interaction picture perturbing potential VI .
More generally, we can write any observable (operator) in the interaction picture as
AI = eiHt Ae−iHt We can integrate equation (1.12) with respect to t to get
˜ (t)� = |φ0 � − i
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
to convert the ordinary Schrodinger equation, a PDE, into an integral equation. However, if VI
is small, then we can iterate equation (1.16):
|φ (t)� ≈ |φ0 � − i
dt� VI (t� ) |φ0 � + · · ·
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
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
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
imagine turning on the prob...
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