448
9 The Unification of Structure and Reaction Frameworks
B. One now assumes long wavelength approximation. Show that the electric
operator reads
r
L Y LM L (Ω r ) =
int +L CM =L
B
L CM
int
r
int
int Y int (Ω int ) ⊗ R
L CM
CM Y L CM (Ω CM )
L
M L
,
and that the magnetic operators read
∇
r
L Y LM L (Ω r )
· l
=
int +L CM =L
B
L CM
int
∇ int
r
int
int Y int (Ω int )
· l int
⊗ R
L CM
CM Y L CM (Ω CM )
L
M L
+
m
M P
int +L CM =L
B
L CM
int
∇ int
r
int
int Y int (Ω int )
· L CM
⊗R
L CM
CM Y L CM (Ω CM )
L
M L
+
m
M P
int +L CM =L
B
L CM
int
∇ int
r
int
int Y int (Ω int )
· (r int × P CM )
⊗R
L CM
CM Y L CM (Ω CM )
L
M L
+
int +L CM =L
B
L CM
int
∇ int
r
int
int Y int (Ω int )
· (R CM × p int )
⊗R
L CM
CM Y L CM (Ω CM )
L
M L
∇
r
L Y
M L
L (Ω r )
· s
=
int +L CM =L
B
L CM
int
∇ int
r
int
int Y int (Ω int )
· s
⊗R
L CM
CM Y L CM (Ω CM )
L
M L
,
where (2 int + 1)!! (2L CM + 1)!! B
L CM
int
= (2L + 1)!! A
L CM
int
.
Show that the equations involving ∇ int in A with long wavelength
approximation now read
∇ int
r
int
int Y int (Ω int )
· v =
int (2 int + 1)r
int −1
int
[Y int −1 (Ω r int ) ⊗ v]
int ,
(9.134)
where v is s, L CM , l int , r int × P CM , or R CM × p int .