My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 7. THE ELECTRONIC GROUND-STATE WAVE FUNCTION
FROM CLASSICAL PLASMON NORMAL MODES
the Hamiltonian may be recast as
(
H = 1 ∑
2
= 1 2
i
1 ∑
ɛ 0
k
U ki ρ † k
k
∑
∑ ∑
+ɛ 0 λ i kU ki f k
∑
The new operators correspond to the normal coordinates:
i
k
Uk ∗ ′ i ρ k ′
(7.37)
k ′ k ′ )
k ′ Uk ∗ if † ′ k ′
k ′
( 1
ɛ 0
p † i p i + ɛ 0 λ i q i q † i
)
. (7.38)
p † i = ∑ k
U ki ρ † k
k
ρ † k = ∑ i
U ∗ kikp † i
(7.39a)
q † i = ∑ k
U ∗ kikf † k
f † k = ∑ i
U ki q † i
k . (7.39b)
One can show that p † i
is an Hermitian operator; beginning with the complex conjugate
of equation (7.39a), and using equations (7.28) and (7.35), it follows that
p i = ∑ k
U ∗ ki ρ k
k
= ∑ k
U (−k)i ρ k
k
= ∑ (7.40)
U k ′ iρ † k ′
k ′
k ′
= p † i .
A similar proof demonstrates that q i is also Hermitian. The commutation relation
for these operators is
[
pi , q j
]
= −iδij . (7.41)
The Hamiltonian is now
H = 1 2
∑
i
( 1
ɛ 0
p 2 i + λ i ɛ 0 q 2 i
)
. (7.42)
In terms of the coordinates {p i } and {q i }, the diagonalised Hamiltonian is completely
separable. The eigenfunctions are products of the form
Ψ({p i }) = ∏ i
ψ i (p i ) (7.43)
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