P l x
ð Þ
À1
ð Þ
l
2
l l!
d
l
dx l 1 À x
2
À
Á l
h
i
,
ð3:168Þ
where a constant
À1
ð Þ
l
2
l l!
is multiplied according to the custom so that we can explicitly
represent Rodrigues formula of Legendre polynomials. Thus, from (3.164) P l (x)
defined above satisfies Legendre differential equation. Rewriting it, we get
1 À x
2
À
Á d
2 P l x
ð Þ
dx 2 À 2x
dP l x
ð Þ
dx
þ l l þ 1
ð
ÞP l x
ð Þ ¼ 0:
ð3:169Þ
Or equivalently, we have
d
dx
1 À x
2
À
Á dP l x
ð Þ
dx
!
þ l l þ 1
ð
ÞP l x
ð Þ ¼ 0:
ð3:170Þ
Returning to (3.127) and using x as a variable, we rewrite (3.127) as
d
dx
1 À x
2
À
Á dP
m
l x
ð Þ
dx
!
þ l l þ 1
ð
ÞÀ
m
2
1 À x 2
!
P
m
l x
ð Þ ¼ 0,
ð3:171Þ
where l is a non-negative integer and m is an integer that takes following values:
m ¼ l, l À 1, l À 2, Á Á Á1, 0, À 1, Á Á Á À l þ 1, À l:
Deferential equations expressed as
d
dx
p x
ð Þ
dy x
ð Þ
dx
!
þ c x
ð Þy x
ð Þ ¼ 0
are of particular importance. We will come back to this point in Sect. 8.3.
Since m can be either positive or negative, from (3.171) we notice that P
m
l x
ð Þ and
P
Àm
l
x
ð Þ must satisfy the same differential equation (3.171). This implies that P
m
l x
ð Þ
and P
Àm
l
x
ð Þ are connected, i.e., linearly dependent. First, let us assume that m ! 0. In
the case of m < 0, we will examine it later soon.
According to Dennery and Krzywicki [5], we assume
P
m
l x
ð Þ ¼ κ 1 À x
2
À
Á m=2 C x
ð Þ,
ð3:172Þ
where κ is a constant. Inserting (3.172) into (3.171) and rearranging the terms, we
obtain
3.6 Orbital Angular Momentum: Analytic Approach
93
ð Þ
À1
ð Þ
l
2
l l!
d
l
dx l 1 À x
2
À
Á l
h
i
,
ð3:168Þ
where a constant
À1
ð Þ
l
2
l l!
is multiplied according to the custom so that we can explicitly
represent Rodrigues formula of Legendre polynomials. Thus, from (3.164) P l (x)
defined above satisfies Legendre differential equation. Rewriting it, we get
1 À x
2
À
Á d
2 P l x
ð Þ
dx 2 À 2x
dP l x
ð Þ
dx
þ l l þ 1
ð
ÞP l x
ð Þ ¼ 0:
ð3:169Þ
Or equivalently, we have
d
dx
1 À x
2
À
Á dP l x
ð Þ
dx
!
þ l l þ 1
ð
ÞP l x
ð Þ ¼ 0:
ð3:170Þ
Returning to (3.127) and using x as a variable, we rewrite (3.127) as
d
dx
1 À x
2
À
Á dP
m
l x
ð Þ
dx
!
þ l l þ 1
ð
ÞÀ
m
2
1 À x 2
!
P
m
l x
ð Þ ¼ 0,
ð3:171Þ
where l is a non-negative integer and m is an integer that takes following values:
m ¼ l, l À 1, l À 2, Á Á Á1, 0, À 1, Á Á Á À l þ 1, À l:
Deferential equations expressed as
d
dx
p x
ð Þ
dy x
ð Þ
dx
!
þ c x
ð Þy x
ð Þ ¼ 0
are of particular importance. We will come back to this point in Sect. 8.3.
Since m can be either positive or negative, from (3.171) we notice that P
m
l x
ð Þ and
P
Àm
l
x
ð Þ must satisfy the same differential equation (3.171). This implies that P
m
l x
ð Þ
and P
Àm
l
x
ð Þ are connected, i.e., linearly dependent. First, let us assume that m ! 0. In
the case of m < 0, we will examine it later soon.
According to Dennery and Krzywicki [5], we assume
P
m
l x
ð Þ ¼ κ 1 À x
2
À
Á m=2 C x
ð Þ,
ð3:172Þ
where κ is a constant. Inserting (3.172) into (3.171) and rearranging the terms, we
obtain
3.6 Orbital Angular Momentum: Analytic Approach
93