My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 7. THE ELECTRONIC GROUND-STATE WAVE FUNCTION
FROM CLASSICAL PLASMON NORMAL MODES
In what follows, terms of order ρ 2 will be neglected: the system will be linearised.
Classically, the (linearised) kinetic energy of the electron gas is
∫
1
T =
2 m ev 2 (r, t)¯n(r) d 3 r (7.3)
V
where v(r, t) is the electron velocity field. The current density is given by
J(r, t) = −¯n(r)ev(r, t). (7.4)
The kinetic energy may be written in terms of the current density as
T = 1
2ɛ 0
∫V
J 2 (r, t)
ω 2 p(r)
d 3 r (7.5)
where
is the plasma frequency.
ω p (r) =
√
¯n(r)e 2
m e ɛ 0
(7.6)
The potential energy is the electrostatic self-interaction energy of the plasmon
charge density:
V = 1 2
∫
V
∫
d 3 r d 3 r ′ ρ(r, t)ρ(r ′ , t)
V 4πɛ 0 |r − r ′ | . (7.7)
Working in the Coulomb gauge, the charge density is related to the electrostatic
potential φ(r, t) in the usual way:
−∇ 2 φ(r, t) =
ρ(r, t)
ɛ 0
, (7.8)
or equivalently
∫
φ(r, t) =
V
ρ(r ′ , t)
4πɛ 0 |r − r ′ | d3 r ′ . (7.9)
This allows the potential energy to be rewritten in the alternative form:
V = 1 ∫
ɛ 0 [∇φ(r, t)] 2 d 3 r. (7.10)
2
V
The Hamiltonian is the sum of the kinetic and potential energies:
H = 1 ∫ ( )
J 2 (r, t)
2 ɛ 0 ωp(r) + ɛ 2 0 [∇φ(r, t)] 2 d 3 r. (7.11)
V
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