the first factor of LHS of 4:80
ð
Þ
½
¼ l
0
þ l
ð
Þ
2 l
0
À l
ð
Þ
2 þ 2 l
0
þ l
ð
Þ l
0 2 À l
0 l þ l
2
À 2l
0 l l
0
þ l
ð
ÞÀ2 l
0
þ l
ð
ÞÀ l
0
þ l
ð
Þ
2
¼ l
0
þ l
ð
Þ l
0
þ l
ð
Þ l
0
À l
ð
Þ
2 þ 2 l
0 2 À l
0 l þ l
2
À 2l
0 l À 2 À l
0
þ l
ð
Þ
h
i
¼ l
0
þ l
ð
Þ l
0
þ l
ð
Þ l
0
À l
ð
Þ
2 þ 2 l
0 2 À 2l
0 l þ l
2
À l
0
þ l þ 2
ð
Þ
h
i
¼ l
0
þ l
ð
Þ l
0
þ l
ð
Þ l
0
À l
ð
Þ
2 þ 2 l
0
À l
ð
Þ
2 À l
0
þ l þ 2
ð
Þ
h
i
¼ l
0
þ l
ð
Þ l
0
À l
ð
Þ
2 l
0
þ l
ð
Þþ2
½
À l
0
þ l þ 2
ð
Þ
n
o
¼ l
0
þ l
ð
Þ l
0
þ l þ 2
ð
Þl
0
À l þ 1
ð
Þl
0
À l À 1
ð
Þ :
ð4:81Þ
Thus rewriting (4.80), we get
l
0
þ l
ð
Þ l
0
þ l þ 2
ð
Þl
0
À l þ 1
ð
Þl
0
À l À 1
ð
Þl
0
jzjl
h
i ¼ 0:
ð4:82Þ
We have similar relations with respect to hl
0
j xj li and hl
0
j yj li because of (4.76) and
(4.77). For the electric dipole transition to be allowed, among hl
0
j xj li, hl
0
j yj li, and
hl
0
j zj li at least one term must be nonvanishing. For this, at least one of the four
factors of (4.81) should be zero. Since l
0 + l + 2 > 0, this factor is excluded.
For l
0 + l to vanish, we should have l
0
¼ l ¼ 0; notice that both l
0 and l are nonnegative integers. We must then examine this condition. This condition is equivalent
to that the spherical harmonics related to the angular variables θ and ϕ take the form
of Y
0
0 θ, ϕ
ð
Þ ¼ 1=
ffiffiffiffiffi
4π
p
, i.e., a constant. Therefore, the θ-related integral for the matrix
element hl
0
j zj li only consists of a following factor:
Z π
0
cos θ sin θdθ ¼
1
2
Z π
0
sin 2θdθ ¼ À
1
4
cos 2θ
½
π
0 ¼ 0,
where cosθ comes from a polar coordinate z ¼ r cos θ; sinθ is due to an infinitesimal
volume of space, i.e., r
2 sin θdrdθdϕ. Thus, we find that hl
0
j zj li vanishes on
condition that l
0
¼ l ¼ 0. As a polar coordinate representation, x ¼ r sin θ cos ϕ
and y ¼ r sin θ sin ϕ, and so the ϕ-related integral hl
0
j xj li and hl
0
j yj li vanishes as
well. That is,
Z 2π
0
cos ϕdϕ ¼
Z 2π
0
sin ϕdϕ ¼ 0:
Therefore, the matrix elements relevant to l
0
¼ l ¼ 0 vanish with all the coordinates;
i.e., we have
146
4 Optical Transition and Selection Rules
ð
Þ
½
¼ l
0
þ l
ð
Þ
2 l
0
À l
ð
Þ
2 þ 2 l
0
þ l
ð
Þ l
0 2 À l
0 l þ l
2
À 2l
0 l l
0
þ l
ð
ÞÀ2 l
0
þ l
ð
ÞÀ l
0
þ l
ð
Þ
2
¼ l
0
þ l
ð
Þ l
0
þ l
ð
Þ l
0
À l
ð
Þ
2 þ 2 l
0 2 À l
0 l þ l
2
À 2l
0 l À 2 À l
0
þ l
ð
Þ
h
i
¼ l
0
þ l
ð
Þ l
0
þ l
ð
Þ l
0
À l
ð
Þ
2 þ 2 l
0 2 À 2l
0 l þ l
2
À l
0
þ l þ 2
ð
Þ
h
i
¼ l
0
þ l
ð
Þ l
0
þ l
ð
Þ l
0
À l
ð
Þ
2 þ 2 l
0
À l
ð
Þ
2 À l
0
þ l þ 2
ð
Þ
h
i
¼ l
0
þ l
ð
Þ l
0
À l
ð
Þ
2 l
0
þ l
ð
Þþ2
½
À l
0
þ l þ 2
ð
Þ
n
o
¼ l
0
þ l
ð
Þ l
0
þ l þ 2
ð
Þl
0
À l þ 1
ð
Þl
0
À l À 1
ð
Þ :
ð4:81Þ
Thus rewriting (4.80), we get
l
0
þ l
ð
Þ l
0
þ l þ 2
ð
Þl
0
À l þ 1
ð
Þl
0
À l À 1
ð
Þl
0
jzjl
h
i ¼ 0:
ð4:82Þ
We have similar relations with respect to hl
0
j xj li and hl
0
j yj li because of (4.76) and
(4.77). For the electric dipole transition to be allowed, among hl
0
j xj li, hl
0
j yj li, and
hl
0
j zj li at least one term must be nonvanishing. For this, at least one of the four
factors of (4.81) should be zero. Since l
0 + l + 2 > 0, this factor is excluded.
For l
0 + l to vanish, we should have l
0
¼ l ¼ 0; notice that both l
0 and l are nonnegative integers. We must then examine this condition. This condition is equivalent
to that the spherical harmonics related to the angular variables θ and ϕ take the form
of Y
0
0 θ, ϕ
ð
Þ ¼ 1=
ffiffiffiffiffi
4π
p
, i.e., a constant. Therefore, the θ-related integral for the matrix
element hl
0
j zj li only consists of a following factor:
Z π
0
cos θ sin θdθ ¼
1
2
Z π
0
sin 2θdθ ¼ À
1
4
cos 2θ
½
π
0 ¼ 0,
where cosθ comes from a polar coordinate z ¼ r cos θ; sinθ is due to an infinitesimal
volume of space, i.e., r
2 sin θdrdθdϕ. Thus, we find that hl
0
j zj li vanishes on
condition that l
0
¼ l ¼ 0. As a polar coordinate representation, x ¼ r sin θ cos ϕ
and y ¼ r sin θ sin ϕ, and so the ϕ-related integral hl
0
j xj li and hl
0
j yj li vanishes as
well. That is,
Z 2π
0
cos ϕdϕ ¼
Z 2π
0
sin ϕdϕ ¼ 0:
Therefore, the matrix elements relevant to l
0
¼ l ¼ 0 vanish with all the coordinates;
i.e., we have
146
4 Optical Transition and Selection Rules