where e
Λ
n
ð Þ
l,m is expressed as a product of spherical surface harmonics and radial wave
functions. The latter functions are described by associated Laguerre functions. It is
well known [2] that the spherical surface harmonics constitute the CONS on a unit
sphere and that associated Laguerre functions form the CONS in the real onedimensional space. Thus, their product expressed as (3.300), i.e., aforementioned
collection of jki constitutes the CONS in the real three-dimensional space. This is
equivalent to
X
j
j ji j
h j¼ E,
ð5:63Þ
where E is an identity operator.
The implications of (5.63) are as follows: Take any function jf(r)i and operate
(5.63) from the left. Then, we get
Ef r
ð Þ ¼ f r
ð Þ ¼
X
k
j ki kj f r
ð Þ
h
i¼
X
k
f k j ki:
ð5:64Þ
In other words, (5.64) implies that any function f(r) can be expanded into a series
of jki. The coefficients f k are defined as
f k kj f r
ð Þ
h
i:
ð5:65Þ
Those are so-called “Fourier coefficients.” Related discussion is given in Sect.
10.4 as well as Chaps. 14, 18, and 20.
We further impose the conditions on the operator G defined in (5.62).
(i) G commutes with r (¼j rj ). (ii) G has a functional form described by G ¼ G
(r, θ). (iii) G does not contain a differential operator. On those conditions, we have
∂G/∂ϕ ¼ 0. From (3.301), we have ∂ j 0 i /∂ϕ ¼ 0. Thus, we get
GH 0 À H 0 G
ð
Þj0i
¼ À
h
2
2μ
G
1
r 2
∂
∂r
r
2 ∂
∂r
À
1
r 2
∂
∂r
r
2 ∂
∂r
G À
1
r 2
1
sinθ
∂
∂θ
sinθ
∂
∂θ
G
!
j 0i: ð5:66Þ
This calculation is somewhat complicated, and so we calculate (5.66) termwise.
With the first term of (5.66), we have
5.1 Perturbation Method
167
Λ
n
ð Þ
l,m is expressed as a product of spherical surface harmonics and radial wave
functions. The latter functions are described by associated Laguerre functions. It is
well known [2] that the spherical surface harmonics constitute the CONS on a unit
sphere and that associated Laguerre functions form the CONS in the real onedimensional space. Thus, their product expressed as (3.300), i.e., aforementioned
collection of jki constitutes the CONS in the real three-dimensional space. This is
equivalent to
X
j
j ji j
h j¼ E,
ð5:63Þ
where E is an identity operator.
The implications of (5.63) are as follows: Take any function jf(r)i and operate
(5.63) from the left. Then, we get
Ef r
ð Þ ¼ f r
ð Þ ¼
X
k
j ki kj f r
ð Þ
h
i¼
X
k
f k j ki:
ð5:64Þ
In other words, (5.64) implies that any function f(r) can be expanded into a series
of jki. The coefficients f k are defined as
f k kj f r
ð Þ
h
i:
ð5:65Þ
Those are so-called “Fourier coefficients.” Related discussion is given in Sect.
10.4 as well as Chaps. 14, 18, and 20.
We further impose the conditions on the operator G defined in (5.62).
(i) G commutes with r (¼j rj ). (ii) G has a functional form described by G ¼ G
(r, θ). (iii) G does not contain a differential operator. On those conditions, we have
∂G/∂ϕ ¼ 0. From (3.301), we have ∂ j 0 i /∂ϕ ¼ 0. Thus, we get
GH 0 À H 0 G
ð
Þj0i
¼ À
h
2
2μ
G
1
r 2
∂
∂r
r
2 ∂
∂r
À
1
r 2
∂
∂r
r
2 ∂
∂r
G À
1
r 2
1
sinθ
∂
∂θ
sinθ
∂
∂θ
G
!
j 0i: ð5:66Þ
This calculation is somewhat complicated, and so we calculate (5.66) termwise.
With the first term of (5.66), we have
5.1 Perturbation Method
167