My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 7. THE ELECTRONIC GROUND-STATE WAVE FUNCTION
FROM CLASSICAL PLASMON NORMAL MODES
where
Diagonalisation is now trivial. The usual operators
( )
1 1
p 2 i + λ i ɛ 0 qi 2 ψ i = E i ψ i . (7.44)
2 ɛ 0
√ √ 1 ɛ0 ω i
a i = p i − i
2ɛ 0 ω i 2 q i
√ √ 1
a † i = ɛ0 ω i
p i + i
2ɛ 0 ω i 2 q i
(7.45a)
(7.45b)
are introduced, which obey the commutation relation
[ ]
ai , a † j = δij . (7.46)
Defining ω i = √ λ i , the Hamiltonian becomes
H = ∑ (
ω i a † i a i + i [ ] )
pi , q i
2
i
= ∑ (
ω k a † i a i + 1 )
2
i
= ∑ i
ω i a † i a i + zero-point energy. (7.47)
The zero-point energy, though infinite, is constant; it therefore does not affect the
form of the wave function, and will be omitted from now on. It appears as the
consequence of having an infinite number of oscillators, each of which contributes
its own zero-point energy. This may be clearly seen in equation (7.47).
7.3 The ground-state wave function
The equation which determines the form of ψ i for the ground-state wave function is
a i ψ i = 0. (7.48)
To express a i in terms of q i alone, the representation of p i in the q i basis is required.
The commutation relation for these operators, equation (7.41), implies that
q i (p i ) = i ∂
∂p i
. (7.49)
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