P ¼ e 1 e 2 e 3
ð
Þ
P x
P y
P z
0
B
@
1
C
A ¼ ex ¼ e 1 e 2 e 3
ð
Þ
ex
ey
ez
0
B
@
1
C
A:
Hence, the z-component of the electric dipole moment P z of electron is given by
P z ¼ ez:
ð5:54Þ
The quantum-mechanical analog of (5.54) is given by
P z
h i ¼ e Ψ 0 jzjΨ 0
h
i ,
ð5:55Þ
where jΨ 0 i is taken from (5.53). Substituting (5.53) for (5.55), we get
Ψ 0 jzjΨ 0
h
i
¼ h0 j ÀeF
X
k6 ¼0
hk j
kjzj0
h
i
E
0
ð Þ
0 À E
0
ð Þ
k
h
i jzjj 0i À eF
X
k6 ¼0
j ki
kjzj0
h
i
E
0
ð Þ
0 À E
0
ð Þ
k
h
i
*
+
¼ 0jzj0
h
iÀ eF
X
k6 ¼0
0jzjk
h
i
kjzj0
h
i
E
0
ð Þ
0 À E
0
ð Þ
k
h
iÀ eF
X
k6 ¼0
0jz
{
jk
kjzj0
h
i
E
0
ð Þ
0 À E
0
ð Þ
k
h
i
þ eF
ð Þ
2
X
k6 ¼0
kjzjk
h
i
kjzj0
h
i
j
j
2
E
0
ð Þ
0 À E
0
ð Þ
k
h
i 2 :
ð5:56Þ
In (5.56) readers are referred to the computation rule of inner product such as
(13.20) and (13.64). Since z is Hermitian (i.e., z
{
¼ z), the second and third terms of
RHS of (5.56) equal. Neglecting the second-order perturbation factor on the fourth
term, we have
Ψ 0 jzjΨ 0
h
i% 0jzj0
h
iÀ 2eF
X
k6 ¼0
kjzj0
h
i
j
j
2
E
0
ð Þ
0 À E
0
ð Þ
k
h
i:
ð5:57Þ
Thus, from (5.55) we obtain
P z
h i % e 0jzj0
h
iÀ 2e
2 F
X
k6 ¼0
kjzj0
h
i
j
j
2
E
0
ð Þ
0 À E
0
ð Þ
k
h
i:
Using (3.301), the coordinate representation of h0j zj 0i is described by
5.1 Perturbation Method
165