My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 7. THE ELECTRONIC GROUND-STATE WAVE FUNCTION
FROM CLASSICAL PLASMON NORMAL MODES
where
M kk ′ = ˆk · ˆk ′ ∫
V
V
ω 2 p(r)e i(k−k′ )·r) d 3 r. (7.25)
The reason for choosing opposite signs in the transforms of f and ρ is to ensure that
f k and ρ k are conjugate variables. This conjugate variable requirement implies that
ρ k = ∂L({f k}, { f ˙ k })
∂ ˙
∫
=
V
= 1 √
V
∫
f k
δL[f, f] ˙
δf(r, ˙ t)
V
∂ ˙ f(r, t)
∂ ˙ f k
d 3 r
ρ(r, t)e ik·r d 3 r. (7.26)
The k = 0 terms are excluded; this will be the case for all sums from now on. The
justification for this is that the average charge density fluctuation is constrained to
be zero, and the average potential (which corresponds to ˙ f k=0 ) may be chosen to be
zero.
7.2 Quantisation and diagonalisation of the Hamiltonian
In equation (7.24), the Hamiltonian is expressed in terms of pairs of discrete conjugate
variables.
subject to the commutation relation
Quantisation proceeds by letting f k and ρ k become operators,
[ ]
ρk , f k ′ = −iδkk ′. (7.27)
These are not Hermitian operators; however, they are the Fourier components of
operators corresponding to real, observable fields, and must obey the symmetry
rules
ρ † k = ρ −k, f † k = f −k. (7.28)
The quantum-mechanical Hamiltonian operator is therefore
H = 1 ∑
(
)
ρ † δ kk ′
k
2 ɛ
k,k ′ 0 k ρ 2 k ′ + ɛ 0f k kM kk ′k ′ f † k
. (7.29)
′
111