P
e þ
ð Þ
xÀiy,jp þ i
¼ À
e
8
ffiffi ffi
2
p πa 4
Z 1
0
r
4 e
À3r=2a dr
Z π
0
sin
3
θdθ
Z 2π
0
e
Àiϕ e
iϕ dϕ
¼ À
2
7
ffiffi ffi
2
p
3
5
ea,
ð4:49Þ
where we used
x À iy ¼ r sin θe
Àiϕ
:
ð4:50Þ
In the definite integral of (4.49), e
Àiϕ comes from x À iy, while e
iϕ comes from
ϕ(2p x + iy ). Note that from (3.24) e
iϕ is an eigenfunction corresponding to an angular
momentum eigenvalue h. Notice that in (4.49) exponents e
Àiϕ and e
iϕ cancel out and
that an azimuthal integral is nonvanishing.
If we choose e À for ε e , we have
P
e À
ð Þ
xþiy,jp þi
¼ e ϕ 1s
ð Þj
1
ffiffi ffi
2
p x þ iy
ð
Þjϕ 2p xþiy
À
Á
(
)
ð4:51Þ
¼
e
8
ffiffi ffi
2
p πa 4
Z 1
0
r
4 e
À3r=2a dr
Z π
0
sin
3
θdθ
Z 2π
0
e
2iϕ dϕ ¼ 0,
ð4:52Þ
where we used
x þ iy ¼ r sin θe
iϕ
:
With (4.52), a factor e
2iϕ results from the product ϕ(2p x + iy )(x + iy) which renders the
integral (4.51) vanishing. Note that the only difference between (4.49) and (4.52) is
about the integration of ϕ factor. For the same reason, if we choose e 3 for ε e , the
matrix element vanishes. Thus, with the ϕ(2p x + iy )-related matrix element, only
P
e þ
ð Þ
xÀiy, p þ
j i
survives. Similarly, with the ϕ(2p x À iy )-related matrix element, only
P
e À
ð Þ
xþiy,jp À i survives. Notice that j p À i is a shorthand notation of ϕ(2p x À iy ). That is,
we have, e.g.,
P
e À
ð Þ
xþiy,jp À i ¼ e ϕ 1s
ð Þj
1
ffiffi ffi
2
p x þ iy
ð
Þjϕ 2p xÀiy
À
Á
(
)
¼
2
7
ffiffi ffi
2
p
3
5
ea,
P
e þ
ð Þ
xÀiy, p À
j i ¼ e ϕ 1s
ð Þj
1
ffiffi ffi
2
p x À iy
ð
Þjϕ 2p xÀiy
À
Á
(
)
¼ 0:
ð4:53Þ
Taking complex conjugate of (4.48), we have
4.3 Three-Dimensional System
137