My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 7. THE ELECTRONIC GROUND-STATE WAVE FUNCTION
FROM CLASSICAL PLASMON NORMAL MODES
The Lagrangian becomes
L = ɛ 0
2
∑ ( )
˙α
2
i − ωi 2 αi
2 , (7.74)
i
leading to the set of conjugate variables
β i = ∂L = ɛ 0 ˙α i . (7.75)
∂α˙
i
In terms of the new variables, the Hamiltonian is
H = 1 ∑
( )
1
βi 2 + ɛ 0 ωi 2 αi 2 . (7.76)
2 ɛ 0
i
As before, the Hamiltonian may be quantised; this time, the commutation relation
is between α i and β j :
[
βj , α i
]
= −iδij . (7.77)
If the normal modes are chosen to be real, the operators {α i , β i } are Hermitian. The
Hamiltonian operator is therefore
Ĥ = ∑ i
= ∑ i
[ ( √ √ )(√ √ )
1 ɛ0 ω i 1
ω i β i + i
2ɛ 0 ω i 2 α ɛ0 ω i
i
β i − i
2ɛ 0 ω i 2 α i − i [ ] ]
αi , β i
2
(
ω i a † i a i + 1 )
2
= ∑ i
ω i a † i a i + zero-point energy (7.78)
where the operators a † i and a i are defined in equation (7.45); they obey the commutation
relation (7.46).
as
Comparison with the results of section 7.3 gives the ground-state wave function
Ψ({β i }) = exp
(
− 1
2ɛ 0
∑
i
)
1
βi
2 , (7.79)
ω i
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