192
7 Quadrupole Contributions from Interface and Bulk
(E) =
1
2 ¯
h 2
g,m,n
(ρ
(0)
g − ρ
(0)
m )
ω mg
ω 1
g| ˆ
μ p |nn| ˆ
μ r |mm| ˆ
μ s |g
(ω 1 − ω mg )(( − ω ng )
−
ω mg
ω 1
g| ˆ
μ r |nn| ˆ
μ p |mm| ˆ
μ s |g
(ω 1 − ω mg )(( − ω mn )
+
ω nm
ω 1
g| ˆ
μ p |nn| ˆ
μ s |mm| ˆ
μ r |g
(ω 2 − ω mg )(( − ω ng )
−
ω gn
ω 1
g| ˆ
μ s |nn| ˆ
μ p |mm| ˆ
μ r |g
(ω 2 − ω mg )(( − ω mn )
r q .
(7.131)
In what follows, we prove that (B) = (E) and (C) = (D).
In the term (D) of Eq. (7.130), we apply the three relations,
ω mg
ω 1
1
(ω 1 − ω mg )
=
1
ω 1 − ω mg
−
1
ω 1
,
ω nm
ω 1
1
(ω 2 − ω mg )(( − ω ng )
=
1
(ω 2 − ω mg )(( − ω ng )
−
1
ω 1
1
ω 2 − ω mg
−
1
− ω ng
,
ω gn
ω 1
1
(ω 2 − ω mg )(( − ω mn )
=
1
(ω 2 − ω mg )(( − ω mn )
−
1
ω 1
1
ω 2 − ω mg
−
1
− ω mn
,
and thereby obtain
(D) =
1
2
α
D0
pqr ((, ω 1 , ω 2 ))r s
+
1
2 ¯
h 2
r s
ω 1
g,m,n
(ρ
(0)
g − ρ
(0)
m )
−
g| ˆ
μ p |nn| ˆ
μ r |mm| ˆ
μ q |g
(( − ω ng )
+
g| ˆ
μ r |nn| ˆ
μ p |mm| ˆ
μ q |g
(( − ω mn )
−
g| ˆ
μ p |nn| ˆ
μ q |mm| ˆ
μ r |g
(ω 2 − ω mg )
+
g| ˆ
μ p |nn| ˆ
μ q |mm| ˆ
μ r |g
(( − ω ng )
+
g| ˆ
μ q |nn| ˆ
μ p |mm| ˆ
μ r |g
(ω 2 − ω mg )
−
g| ˆ
μ q |nn| ˆ
μ p |mm| ˆ
μ r |g
(( − ω mn )
.
In the above expression of (D), we can show that six terms in the square
bracket cancel each other and vanish. By adopting the relation of completeness,
m |m m| = 1, and the commutation relation, ˆ
μ p ˆ
μ q = ˆ
μ q ˆ
μ p , we find that some
terms are canceled and consequently obtain the following expression of (D),