Page 233 - Principles and Applications of NanoMEMS Physics

A. QUANTUM MECHANICS PRIMER 223

In the most general case, the one-particle energy operator is given by,
+
ˆ
H = ¦ α K β a α a . (A.45)
β
α β ,
The kinetic energy of a one-particle state, is given by,
+
+
+
ˆ
1 ρ H 1 σ = 0 a ρ H ˆ a 0 = ¦ α K β 0 a ρ a α a β a 0
σ
σ
αβ
. (A.46)
= ¦ α K β δ αρ δ βσ = ρ K σ
αβ
+
where use was made of the identity a β a σ = δ βσ . In the case of the two-
particle potential,
V = ¦ v( i, j) = 1 ¦ v( i, j) , (A.46)
i < j 2 i ≠ j
the second-quantization operator, is given by,

+
+
V = 1 ¦ αβ v γδ a α a β a δ a , (A.47)
γ
2 αβγδ
where the two-particle interaction energy may be evaluated in any basis. One
typical source of confusion in this equation is the nature of the order of the
annihilation elements in the number operator, in particular, the fact that
+ + + +
instead of having a α a β a γ a , we have a α a β a δ a . This is done to make the
δ
γ
expression valid for both bosons, where a γ a = a δ a , and fermions, where
δ
γ
a γ a = − a δ a . Thus, for fermions the concomitant sign reversal will be
δ
γ
automatically present. The matrix element, in configuration space, is given
similarly as for the one-particle case, namely,
G
G
G
+
r′ φ
αβ v γδ = φ + () () rvrr φ β G ( G , G ) γ () () drdrr φ δ G G r′ . (A.48)
³ α
The two-particle interaction energy is given in terms of the wave functions
as,
x φ
+
x
x
1 m 1 V 1 p 1 = 1 ³³ [φ m + () ( ) φφ n + x − n + () ( )]
2
m
n
q
1
1
2
2 , (A.49a)
[
× v ( , xx 1 2 ) ( ) ( ) φφφ p x 1 q x − q ( ) ( )]dxxx φ p 2 1 dx 2
1
2