is the initial state. In the notation, in turn, (e 3 ) denotes the polarization vector and
z represents the electric dipole.
In the case of photon absorption where the transition occurs from jϕ(1s)i to
jϕ(2p z )i, we use the following notation:
P
e 3
ð Þ
z, p z j
h
¼ e ϕ 2p z
À Á
jzjϕ 1s
ð Þ
:
ð4:40Þ
Since all the functions related to the integration are real, we have
P
e 3
ð Þ
z,jp z i
¼ P
e 3
ð Þ
z, p z j
h
:
Meanwhile, if we choose e 1 for ε e to evaluate the matrix element P x , we have
P
e 1
ð Þ
x,jp z i
¼ e ϕ 1s
ð Þjxjϕ 2p z
À Á
¼
e
4
ffiffi ffi
2
p πa 4
Z 1
0
r
4 e
À3r=2a dr
Z π
0
sin
2
θ cos θdθ
Z 2π
0
cos ϕdϕ ¼ 0,
ð4:41Þ
where cos ϕ comes from x ¼ r sin θ cos ϕ and an integration of cos ϕ gives zero. In a
similar manner, we have
P
e 2
ð Þ
y, j p zi
¼ e ϕ 1s
ð Þjyjϕ 2p z
À Á
¼ 0:
ð4:42Þ
Next, we estimate the matrix elements associated with 2p x and 2p y . For this
purpose, it is convenient to introduce the following complex coordinates by a unitary
transformation:
e 1 e 2 e 3
ð
Þ
x
y
z
0
B
@
1
C
A ¼ e 1 e 2 e 3
ð
Þ
1
ffiffi ffi
2
p
1
ffiffi ffi
2
p 0
À
i ffiffi ffi
2
p
0
i ffiffi ffi
2
p 0
0 1
0
B
B
B
B
B
@
1
C
C
C
C
C
A
1
ffiffi ffi
2
p
i ffiffi ffi
2
p 0
1
ffiffi ffi
2
p
0
À
i ffiffi ffi
2
p
0
0
1
0
B
B
B
B
B
@
1
C
C
C
C
C
A
x
y
z
0
B
@
1
C
A
¼
1
ffiffi ffi
2
p e 1 À ie 2
ð
Þ
1
ffiffi ffi
2
p e 1 þ ie 2
ð
Þ e 3
1
ffiffi ffi
2
p x þ iy
ð
Þ
1
ffiffi ffi
2
p x À iy
ð
Þ
z
0
B
B
B
B
@
1
C
C
C
C
A
,
ð4:43Þ
where a unitary transformation is represented by a unitary matrix defined as
4.3 Three-Dimensional System
135