e
Λ
n
ð Þ
l, cos mϕ
e
Λ
n
ð Þ
l, sin mϕ
¼ e
Λ
n
ð Þ
l,m
e
Λ
n
ð Þ
l,Àm
À1
ð Þ
m
ffiffi ffi
2
p
À
À1
ð Þ
m i
ffiffi ffi
2
p
1
ffiffi ffi
2
p
i ffiffi ffi
2
p
0
B
B
@
1
C
C
A ,
ð4:61Þ
where we again assume that m is positive. In chemistry and materials science, we
normally use real functions of e
Λ
n
ð Þ
l, cos mϕ and e
Λ
n
ð Þ
l, sin mϕ . In particular, we use the
notations of, e.g., ϕ(2p x ) and ϕ(2p y ) instead of e
Λ
2
ð Þ
1, cos ϕ and e
Λ
2
ð Þ
1, sin ϕ , respectively. In
that case, we explicitly have a following form:
ϕ 2p x
ð Þ ϕ 2p y
À Á
À
Á ¼ e
Λ
2
ð Þ
1,1
e
Λ
2
ð Þ
1,À1
À
1
ffiffi ffi
2
p
i ffiffi ffi
2
p
1
ffiffi ffi
2
p
i ffiffi ffi
2
p
0
B
B
@
1
C
C
A
¼ ϕ 2p xþiy
À
Á ϕ 2p xÀiy
À
Á
À
Á
À
1
ffiffi ffi
2
p
i ffiffi ffi
2
p
1
ffiffi ffi
2
p
i ffiffi ffi
2
p
0
B
B
@
1
C
C
A
¼
1
4
ffiffiffiffiffi
2π
p a
À
3
2
r
a
e
À
r
2a sin θ cos ϕ
1
4
ffiffiffiffiffi
2π
p a
À
3
2
r
a
e
À
r
2a sin θ sin ϕ
:
ð4:62Þ
Thus, the Condon–Shortley phase factor has been removed.
Using this expression, we calculate matrix elements of the electric dipole transition. We have
P
e 1
ð Þ
x,jp x i ¼ e ϕ 1s
ð Þjxjϕ 2p x
ð Þ
h
i
¼
e
4
ffiffi ffi
2
p πa 4
Z 1
0
r
4 e
À3r=2a dr
Z π
0
sin
3
θdθ
Z 2π
0
cos
2
ϕdϕ ¼
2
7
ffiffi ffi
2
p
3
5
ea:
ð4:63Þ
Thus, we obtained the same result as (4.49) apart from the minus sign. Since a square
of an absolute value of the transition moment plays a role, the minus sign is again of
secondary importance. With P
e 2
ð Þ
y , similarly we have
P
e 2
ð Þ
y,jp yi
¼ e ϕ 1s
ð Þjyjϕ 2p y
À Á
¼
e
4
ffiffi ffi
2
p πa 4
Z 1
0
r
4 e
À3r=2a dr
Z π
0
sin
3
θdθ
Z 2π
0
sin
2
ϕdϕ ¼
2
7
ffiffi ffi
2
p
3
5
ea:
ð4:64Þ
Comparing (4.39), (4.63), and (4.64), we have
4.3 Three-Dimensional System
141