My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 7. THE ELECTRONIC GROUND-STATE WAVE FUNCTION
FROM CLASSICAL PLASMON NORMAL MODES
where
χ(r i ) =
e2
ɛ 0 √ V
∑
k,k ′ e −ik·r i
(
M
−1/2 ) kk ′
kk ′ ¯n k ′ (7.58)
and
(
u(r i , r j ) =
e2 ∑
) e −ik·r i
M
−1/2
kk ′
e ik′·r j
. (7.59)
ɛ 0 V
kk ′ k,k ′
In this notation, the double sum over i and j is unrestricted and includes the case
i = j; this means that a part of the one-body term is incorporated in the sum over
u.
It is useful to obtain the ground-state wave function in terms of f, rather than
ρ. Instead of solving equation (7.48) for p i , it may be solved for q i , giving
(
ψ i (q i ) = exp − ɛ )
0ω i qi
2
2
The full ground-state wave function is then
(
Ψ({f k }) = exp
7.4 Normal modes
− ɛ 0
2
(7.60)
∑
f k k ( )
M 1/2) k ′ f
kk ′ k ∗ . (7.61)
′
k,k ′
An alternative approach to the analysis of the previous section is to diagonalise
the Hamiltonian before quantising; this is perhaps neater, and emphasises the rôle
played by the normal modes of the classical system, which will be useful later. To
diagonalise the classical Hamiltonian, it is necessary to establish the equation of
motion satisfied by the field f; this is obtained by combining equations (7.18) and
(7.20):
−∇ 2 ¨f = ∇ ·
(
ω
2
p ∇f ) . (7.62)
The defining characteristic of a normal mode is harmonic time dependence; the
normal modes for f therefore satisfy the equation
ω 2 i ∇ 2 f i = ∇ · (ω 2 p∇f i
)
. (7.63)
Multiplying by f j and integrating gives
∫
∫
f j ∇ 2 f i d 3 r =
ω 2 i
V
V
f j ∇ · (ω 2 p∇f i
)
d 3 r, (7.64)
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