3.7.3 Associated Laguerre Polynomials
It will be of great importance to compare the functions e
ψ
n
ð Þ
l with conventional wave
functions that are expressed using associated Laguerre polynomials. For this purpose
we define the following functions Φ
n
ð Þ
l
ρ
ð Þ such that
Φ
n
ð Þ
l
ρ
ð Þ
2
n
lþ
3
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n À l À 1
ð
Þ !
2n n þ l
ð
Þ!
s
e
À
ρ
n ρ
lþ1 L
2lþ1
nÀlÀ1
2ρ
n
:
ð3:279Þ
The associated Laguerre polynomials are described as
L
ν
n x
ð Þ ¼
1
n!
x
Àν e
x d
n
dx
n x
nþν e
Àx
ð
Þ , ν > À1
ð
Þ :
ð3:280Þ
In a form of power series expansion, the polynomials are expressed for integer
k ! 0 as
L
k
n x
ð Þ ¼
X n
m¼0
À1
ð Þ
m n þ k
ð
Þ!
n À m
ð
Þ! k þ m
ð
Þ!m!
x
m
:
ð3:281Þ
Notice that “Laguerre polynomials” L n (x) are defined as
L n x
ð Þ L
0
n x
ð Þ:
Hence, instead of (3.280) and (3.281), the Rodrigues formula and power series
expansion of L n (x) are given by [2, 5]
L n x
ð Þ ¼
1
n!
e
x d
n
dx
n x
n e
Àx
ð
Þ,
L n x
ð Þ ¼
X n
m¼0
À1
ð Þ
m n!
n À m
ð
Þ! m!
ð Þ
2
x
m
:
The function Φ
n
ð Þ
l
ρ
ð Þ contains multiplication factors e
À
ρ
n and ρ
l + 1 . The function
L
2lþ1
nÀlÀ1
2ρ
n
À Á
is a polynomial of ρ with the highest order of ρ
n À l À 1 . Therefore, Φ
n
ð Þ
l
ρ
ð Þ
consists of summation of terms containing e
À
ρ
n ρ
t , where t is an integer equal to 1 or
larger. Consequently, Φ
n
ð Þ
l
ρ
ð Þ ! 0 when ρ ! 0 and ρ ! 1 (vide supra). Thus, we
have confirmed that Φ
n
ð Þ
l
ρ
ð Þ certainly satisfies proper BCs mentioned earlier and,
hence, the operator
d
dρ is indeed an anti-Hermitian. To show this more explicitly, we
define D
d
dρ . An inner product between arbitrarily chosen functions f and g is
116
3 Hydrogen-Like Atoms
It will be of great importance to compare the functions e
ψ
n
ð Þ
l with conventional wave
functions that are expressed using associated Laguerre polynomials. For this purpose
we define the following functions Φ
n
ð Þ
l
ρ
ð Þ such that
Φ
n
ð Þ
l
ρ
ð Þ
2
n
lþ
3
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n À l À 1
ð
Þ !
2n n þ l
ð
Þ!
s
e
À
ρ
n ρ
lþ1 L
2lþ1
nÀlÀ1
2ρ
n
:
ð3:279Þ
The associated Laguerre polynomials are described as
L
ν
n x
ð Þ ¼
1
n!
x
Àν e
x d
n
dx
n x
nþν e
Àx
ð
Þ , ν > À1
ð
Þ :
ð3:280Þ
In a form of power series expansion, the polynomials are expressed for integer
k ! 0 as
L
k
n x
ð Þ ¼
X n
m¼0
À1
ð Þ
m n þ k
ð
Þ!
n À m
ð
Þ! k þ m
ð
Þ!m!
x
m
:
ð3:281Þ
Notice that “Laguerre polynomials” L n (x) are defined as
L n x
ð Þ L
0
n x
ð Þ:
Hence, instead of (3.280) and (3.281), the Rodrigues formula and power series
expansion of L n (x) are given by [2, 5]
L n x
ð Þ ¼
1
n!
e
x d
n
dx
n x
n e
Àx
ð
Þ,
L n x
ð Þ ¼
X n
m¼0
À1
ð Þ
m n!
n À m
ð
Þ! m!
ð Þ
2
x
m
:
The function Φ
n
ð Þ
l
ρ
ð Þ contains multiplication factors e
À
ρ
n and ρ
l + 1 . The function
L
2lþ1
nÀlÀ1
2ρ
n
À Á
is a polynomial of ρ with the highest order of ρ
n À l À 1 . Therefore, Φ
n
ð Þ
l
ρ
ð Þ
consists of summation of terms containing e
À
ρ
n ρ
t , where t is an integer equal to 1 or
larger. Consequently, Φ
n
ð Þ
l
ρ
ð Þ ! 0 when ρ ! 0 and ρ ! 1 (vide supra). Thus, we
have confirmed that Φ
n
ð Þ
l
ρ
ð Þ certainly satisfies proper BCs mentioned earlier and,
hence, the operator
d
dρ is indeed an anti-Hermitian. To show this more explicitly, we
define D
d
dρ . An inner product between arbitrarily chosen functions f and g is
116
3 Hydrogen-Like Atoms