My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 7. THE ELECTRONIC GROUND-STATE WAVE FUNCTION
FROM CLASSICAL PLASMON NORMAL MODES
or, on rearranging,
ω 2 i
∫
V
∫
∇f j · ∇f i d 3 r =
V
ω 2 p∇f j · ∇f i d 3 r. (7.65)
Swapping the indices gives the corresponding expression
∫
∫
∇f i · ∇f j d 3 r = ωp∇f 2 i · ∇f j d 3 r. (7.66)
ω 2 j
V
Taking the difference of the two previous equations shows that
V
( ) ∫
ω
2
i − ωj
2 ∇f i · ∇f j d 3 r = 0. (7.67)
V
In the non-degenerate case, this implies that ∇f i and ∇f j are orthogonal, in the
sense that
∫
V
∇f i · ∇f j d 3 r = 0 (ω i ≠ ω j ). (7.68)
When the modes are degenerate, it is always possible to construct combinations
of the (linearly independent) functions which are orthogonal. If, additionally, the
modes are taken to be normalised, the general result
∫
∇f i · ∇f j d 3 r = δ ij (7.69)
V
is obtained. A secondary consequence is that
∫
ωp∇f 2 i · ∇f j d 3 r = ωi 2 δ ij . (7.70)
V
Any solution of the original problem (equation (7.62)) may therefore be expanded
in terms of the normal modes as follows:
∇f = ∑ i
α i ∇f i . (7.71)
The normal mode amplitudes are determined by the inverse relation
∫
α i = ∇f i · ∇f d 3 r. (7.72)
This converts the classical Hamiltonian of equation (7.24) into the form
H = ɛ 0
2
V
∑ ( )
˙α
2
i + ωi 2 αi
2 . (7.73)
i
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