Page 234 - Principles and Applications of NanoMEMS Physics

224 Appendix A

and may be expressed in second-quantization noation as,
+
+
1 ρ 1 σ V 11 τ ν = 1 ¦ αβ v γδ 0 a σ a ρ a α a + β a δ a γ a τ + a 0
ν
2 αβγδ
= 1 ¦ αβ v γδ ( [δ δ ± δ δ )(δ δ ± δ δ )]
2 αβγδ αρ βσ ασ βρ αρ βσ ασ βρ . (A.49b)

= 1 [ ρσ v τν + σρ v ντ ± ( σρ v τν + ρσ v ντ )]
2
= ρσ v τν ± ρσ v ντ

A.5.1 Field Operators

A common practice in the application of the number representation
formalism in interacting (many-body) systems is to express the Hamiltonians
in terms of so-called field operators, () x and ψ + () x , which are defined
ψ
by,

ψ () =x ¦ φ i ()cx i , (A.50)
i
and
ψ + () =x ¦ φ * i ()cx i + . (A.51)
i
The field operators obey the commutation relations,
′
{ ψ () ( )}=xx ,ψ ′ ¦ φ i ( ) () ′ ccxx φ j i j + ¦ φ j () ( ) ccxx φ i j i
i , j i , j
, (A.52)
= ¦ φ i () ( ){ ,ccxx φ j ′ i j } 0=
i , j
x′
+
{ ψ () x ψ , + ( )}= ¦ φ i ( ) () ccxx φ * j ′ i + j ¦ φ * j () ( ) ccxx′ φ i * j i
i, j i, j
= ¦ φ i () ( ){ ccxx φ * j ′ i , + j }= 0 , (A.53)
i, j
= ¦ φ i () ( ) = δφ i * x′ (x − x′ )
x
i