Spherical Harmonics

Spherical Harmonics
Let us consider the eigenfunctions of the orbital angular momenta Lz and L2:
Lz lm = m lm
L2 lm = l (l + 1) 2 lm

If we write
nˆ lm = Ylm (θ , φ ),
with
 ∂
Lz =
i ∂φ
we have
∂
− i nˆ lm = m nˆ lm
∂φ
which suggests that
Ylm (θ , φ ) = Plm (θ )e imφ .

From
2 ⎡ 1 ∂ ⎛
2 ∂ ⎞ 1 ∂ 2 ⎤
L = − ⎢ ⎜ sin θ ⎟ +
⎣ sin θ ∂θ ⎝ ∂θ ⎠ sin 2 θ ∂φ 2 ⎥⎦
we have
⎡ 1 ∂ ⎛ ∂ ⎞ 1 ∂ 2 ⎤
−  2 ⎢ ⎜ sin θ ⎟ + 2 2 ⎥
nˆ lm = l (l + 1) 2 nˆ lm
⎣ sin θ ∂θ ⎝ ∂θ ⎠ sin θ ∂φ ⎦
or
⎡ 1 ∂ ⎛ ∂ ⎞ 1 ∂ 2 ⎤
−  2 ⎢ ⎜ sin θ ⎟ + 2
P (θ )e imφ = l (l + 1) 2 Plm (θ )e imφ
2 ⎥ lm
⎣ sin θ ∂θ ⎝ ∂θ ⎠ sin θ ∂φ ⎦
which leads to the associated Legendre equation
⎡ 1 ∂ ⎛ ∂ ⎞ m 2 ⎤
⎢ ⎜ sin θ ⎟ + l (l + 1) − 2 ⎥ Plm (θ ) = 0
⎣ sin θ ∂θ ⎝ ∂θ ⎠ sin θ ⎦

The associated Legendre polynomials may be evaluated using the ladder operators
L± = Lx ± iLy
Operating on the eigenkets of Lz and L2,
L± lm =  l (l + 1) − m(m ± 1) l , m ± 1 .

Now,
 ⎡ ∂ ∂ ⎤
Lx = − ⎢sin φ + cot θ cos φ ⎥
i ⎣ ∂θ ∂φ ⎦
 ⎡ ∂ ∂ ⎤
L y = ⎢cos φ − cot θ sin φ ⎥
i ⎣ ∂θ ∂φ ⎦

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 Spherical Harmonics

Thus,
 ⎡ ∂ ∂ ⎛ ∂ ∂ ⎞⎤
L± = ⎢− sin φ − cotθ cosφ ± i⎜⎜ cosφ − cotθ sin φ ⎟⎥
i ⎣ ∂θ ∂φ ⎝ ∂θ ∂φ ⎟⎠⎦
⎡ ∂ ∂ ⎤
= −ie ±iφ ⎢± i − cotθ ⎥
⎣ ∂θ ∂φ ⎦
Starting from
L+ l , l = 0 ,
we have
⎡ ∂ ∂ ⎤
− ie iφ ⎢i − cotθ ⎥ Pll (θ )e ilφ = 0
⎣ ∂θ ∂φ ⎦
or
dPll
= l cotθPll
dθ
This yields
ln Pll = l ln sin θ + ln Cl
Thus,
Pll (θ ) = Cl sin l θ .

Normalizing, we have
* 2 π 2π
∫ Ylm (θ ,φ )Ylm (θ ,φ )dΩ = Cl ∫ sin 2l θ sin θdθ ∫ dφ = 1
0 0
Since sine is symmetric about π/2,
π π /2
∫ sin 2l θ sin θdθ =2∫ sin 2l θ sin θdθ .
0 0
Integrating by parts, we have
π /2 π /2 π /2
∫ sin 2l θ sin θdθ = − sin 2l θ cosθ + 2l ∫ sin 2l −1 θ cos 2 θdθ
0 0 0
π /2 π /2
= 2l ⎡∫ sin 2l −1 θdθ − ∫ sin 2l +1 θdθ ⎤
⎢⎣ 0 0 ⎥⎦
or
π /2 π /2
[2l + 1]∫0 sin 2l θ sin θdθ = 2l ∫ sin 2l −1 θdθ
0
Applying the last relation successively, we have
π /2 2l π / 2 2l −1 2l 2l − 2 π / 2 2l −3
∫ sin 2l θ sin θdθ = sin θdθ = sin θdθ
0 2l + 1 ∫0 2l + 1 2l − 1 ∫0
(2l )(2l − 2)2 π / 2 2l!!
= ∫ sin θdθ =
(2l + 1)(2l − 1)3 0 (2l + 1)!!
The double factorials may be recast as follows:
2
2l!!
=
(2l )(2l − 2)2
=
[(2l )(2l − 2)2]
(2l + 1)!! (2l + 1)(2l − 1)(3)(1) (2l + 1)2l (2l − 1)(3)(2)(1)
=
[2 (l )(l − 1)1]
l 2

=
[2 l!]
l 2

(2l + 1)! (2l + 1)!

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 Spherical Harmonics

The normalization condition thus gives

1 = Cl
2 π 2l
sin θ sin θdθ ∫ dφ = Cl
2π 2
2
[2 l!]
l 2

2π
∫
0 0 (2l + 1)!
or

Cl =
(− 1)l (2l + 1)2l!
2l l! 4π
where the phase factor (-1)l was included to fix Pl0(1) = 1. We then have

l
(− 1) (2l + 1)2l! l ilφ
Yll (θ ,φ ) = l sin θe .
2 l! 4π
We may verify that this is indeed an eigenfunction of L2.

⎡ 1 d ⎛ d (sin θ ) l ⎞ l2 ⎤
L2Yll =  2Cl ⎢− ⎜⎜ sin θ ⎟⎟ + 2
(sin θ ) l ⎥ e ilφ
⎣ sin θ dθ ⎝ dθ ⎠ sin θ ⎦
⎡ 1 d ⎤
= Cl ⎢− ( )
sin θ (l sin l −1 θ ) cosθ + l 2 sin l −2 θ ⎥  2 e ilφ
⎣ sin θ dθ ⎦
⎡ 1 1 ⎤
= Cl ⎢− (
l 2 sin l −1 θ cos 2 θ − ) (
l sin l θ (− sin θ ) + l 2 sin l −2 θ ⎥  2 e ilφ)
⎣ sin θ sin θ ⎦
[ 2 l −2 2 l
= Cl − l sin θ cos θ + l sin θ + l sin θ  e 2 l −2 2 ilφ
]
= C [− l
l
2
sin l −2 2 l l
θ + l sin θ + l sin θ + l sin 2 l −2
θ ] 2 e ilφ
= l (l + 1) 2Cl sin l θe ilφ
= l (l + 1) 2Yll

Other spherical harmonics may be obtained by successive use of the (lowering) ladder operator.
We may facilitate the evaluation of a generator for spherical harmonics by noting that
l (l + 1) − m(m ± 1) = l 2 + l − m 2  m = (l + m)(l − m) + (l  m)
= (l  m)(l ± m + 1)
Thus,
L− lm =  (l + m)(l − m + 1) l , m − 1
This expression allows for a quicker evaluation of the term in the square root.

Letting the ladder operator L- act on Yll, we have

L−Yll (θ ,φ ) =  (l + l )(l − l + 1)Yl ,l −1 (θ ,φ ) =  2lYl ,l −1 (θ ,φ )
Specifically,
⎡ dP ⎤
L−Yll (θ , φ ) = −ie −iφ Cl ⎢− i ll e ilφ − cot θPll (ile ilφ )⎥
⎣ dθ ⎦
⎡ d ⎤
= −Cl e i (l −1)φ ⎢ + l cot θ ⎥ Pll (θ )
⎣ dθ ⎦
Hence,

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 Spherical Harmonics

⎡ d ⎤
− e i (l −1)φ ⎢ + l cotθ ⎥ Pll (θ ) =  2lYl ,l −1 (θ ,φ )
⎣ dθ ⎦
and
1 i (l −1)φ ⎡ d ⎤
Yl ,l −1 (θ , φ ) = − e ⎢ + l cot θ ⎥ Pll (θ )
2l ⎣ dθ ⎦
We now note that
cos θ
l cot θ = l
sin θ
and that
d
l (sin θ) l cot θ = l (sin θ) l −1 cos θ = (sin θ)l
dθ
Thus,
⎡ d ⎤ df d (sin θ) l
(sin θ) l ⎢ + l cot θ⎥ f (θ) = (sin θ) l + f
⎣ dθ ⎦ dθ dθ
d
=
dθ
(sin θ)l f (θ) [ ]
and
⎡ d ⎤ 1 d
⎢⎣ dθ + l cot θ ⎥⎦ Pll (θ ) = (sinθ ) l Pll (θ )
(sinθ ) l dθ
Cl d
= (sinθ ) 2l
(sinθ ) l dθ
Hence,
Cl i (l −1)φ 1 d
Yl ,l −1 (θ , φ ) = − e l
(sin θ ) 2l
2l (sin θ ) dθ
Applying L- a second time, we have
L−Yl ,l −1 (θ ,φ ) =  (l + l − 1)(l − l + 2)Yl ,l −2 (θ ,φ ) =  2(2l − 1)Yl ,l −2 (θ ,φ )
where
⎡ d ⎤ ⎡ C 1 d ⎤
L−Yl ,l −1 (θ ,φ ) = −ie −iφ ⎢− i − i (l − 1) cotθ ⎥ ⎢− l e i (l −1)φ l
(sinθ ) 2l ⎥
⎣ dθ ⎦ ⎣ 2l (sinθ ) dθ ⎦
Cl ⎡ d ⎤ ⎡ 1 d ⎤
= (−1) 2  e i (l −2)φ ⎢ + (l − 1) cotθ ⎥ ⎢ l
(sinθ ) 2l ⎥
2l ⎣ dθ ⎦ ⎣ (sinθ ) dθ ⎦
l −1
Cl 1 d ⎡ (sin θ ) d ⎤
= (−1) 2  e i ( l − 2 )φ l −1 ⎢ l
(sinθ ) 2l ⎥
2l (sinθ ) dθ ⎣ (sinθ ) dθ ⎦
Thus,
Cl 1 d ⎡ 1 d ⎤
Yl ,l −2 (θ ,φ ) = (−1) 2 e i (l −2)φ l −1 ⎢ (sinθ ) 2l ⎥
2(2l )(2l − 1) (sinθ ) dθ ⎣ sin θ dθ ⎦

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 Spherical Harmonics

Applying L- once more, we have

L−Yl ,l −2 (θ ,φ ) =  (l + l − 2)(l − l + 3)Yl ,l −3 (θ ,φ ) =  3(2l − 2)Yl ,l −3 (θ ,φ )
where
L−Yl ,l −2 (θ ,φ )
(−1) 3 Cl ⎡ d ⎤ ⎡ 1 d ⎛ 1 d ⎞⎤
= e −iφ ⎢ + (l − 2) cotθ ⎥ ⎢e i (l −2)φ l −1 ⎜ (sinθ ) 2l ⎟⎥
2(2l )(2l − 1) ⎣ dθ ⎦ ⎣ (sinθ ) dθ ⎝ sin θ dθ ⎠⎦
(−1) 3 Cl i ( l −3)φ 1 d ⎡ (sin θ )l −2 d ⎛ 1 d ⎞⎤
= e ⎢ ⎜ (sinθ ) 2l ⎟⎥
2(2l )(2l − 1) (sinθ ) l −2 dθ l −1
⎣ (sinθ ) dθ ⎝ sin θ dθ ⎠⎦
Thus,
(−1) 3 Cl 1 d ⎡ 1 d ⎛ 1 d ⎞⎤
Yl ,l −3 (θ ,φ ) = e i ( l − 3 )φ l −2 ⎢ ⎜ (sinθ ) 2l ⎟⎥
3 ⋅ 2 ⋅ 2l (2l − 1)(2l − 2) (sinθ ) dθ ⎣ sin θ dθ ⎝ sin θ dθ ⎠⎦

If we let u = cosθ,
d d dθ d 1 d
= = =−
du d cos θ d cos θ dθ sin θ dθ
we then have
Cl i (l −1)φ 1 d
Yl ,l −1 (θ , φ ) = e l −1 (1 − u 2 ) l
2 2 du
2l (1 − u )
Cl 1 d2 l
Yl ,l −2 (θ ,φ ) = e i ( l − 2 )φ 2 l −2
(1 − u ) 2 du 2
(
1− u 2 )
2(2l )(2l − 1)
Cl 1 d3 l
Yl ,l −3 (θ ,φ ) = e i (l −3)φ 3
(
1− u2 )l −3
3!2l (2l − 1)(2l − 2) (1 − u 2 ) du 2

and generally for m ≥ 0,

Ylm (θ ,φ ) =
(− 1)l (2l + 1) (l + m)! imφ
e
1 d l −m
1− u2 ( l
) (1)
l 2 m/2 l −m
2 l! 4π (l − m)! (1 − u ) du
We note that

Yl 0 (θ ,φ ) =
(− 1)l (2l + 1) dl
(
1− u 2
l
)
l
2 l! 4π du l
For m < 0, we may apply the lowering operator on Yl0:
L−Yl 0 (θ ,φ ) =  l (l + 1)Yl , −1 (θ ,φ )
where

L−Yl 0 (θ ,φ ) = −
(− 1)l (2l + 1)e −iφ d dl
(
1 − cos 2 θ
l
)
l l
2 l! 4π dθ d (cosθ )
l
(− 1) (2l + 1)e −iφ sin θ d dl l
= l
2 l! 4π d cosθ d (cosθ )l
(
1 − cos 2 θ )
In terms of u = cosθ,

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 Spherical Harmonics

Yl , −1 (θ , φ ) =
(− 1)l (2l + 1) (l − 1)! −iφ
e 1− u2 ( 1/ 2
) d l +1
1− u2 ( )l
l
2 l! 4π (l + 1)! du l +1
Applying the lowering operator further, we have
L−Yl , −1 (θ ,φ ) =  (l − 1)(l + 2)Yl , −2 (θ ,φ )
where
l
⎡ ∂ ∂ ⎤ (− 1) (2l + 1) 1/ 2 d l +1 l
L−Yl , −1 (θ ,φ ) = −ie −iφ ⎢− i − cotθ
∂φ ⎥⎦ 2l l! 4πl (l + 1)
(
e −iφ 1 − u 2 ) du l +1
(
1− u 2 )
⎣ ∂θ

= −
(− 1)l (2l + 1) ⎡ d ⎤
e −2iφ ⎢ − cotθ ⎥ 1 − u 2 ( 1/ 2
) d l +1
1− u 2( ) l
l
2 l! 4πl (l + 1) ⎣ dθ ⎦ du l +1

We now note that

d
(sinθ )−k = −k (sinθ ) −k −1 cosθ = −k (sinθ ) −k cotθ
dθ
and
⎡ d ⎤ df d (sinθ ) − k
(sinθ ) −k ⎢ − k cotθ ⎥ f (θ ) = (sinθ ) −k + f (θ )
⎣ dθ ⎦ dθ dθ
d
=
dθ
[
(sin θ )−k f (θ ) ]
Thus,

L−Yl , −1 (θ ,φ ) = −
(− 1)l (2l + 1) e −2iφ sin θ
d ⎡
(sin θ )−1
sin θ
d l +1 ⎤
(sin θ )2l ⎥
⎢
2l l! 4πl (l + 1) dθ ⎣ d (cosθ )
l +1
⎦

=
(− 1)l (2l + 1) (
e −2iφ 1 − u 2
l +2
) dud (1 − u ) 2 l
l l +2
2 l! 4πl (l + 1)
and

Yl , −2 (θ ,φ ) =
(− 1)l (2l + 1) (l − 2)! −2iφ
e (
1− u 2
d l +2
)
1− u2 ( ) l
l l +2
2 l! 4π (l + 2)! du

Applying the lowering operator once more, we have

L−Yl , −2 (θ ,φ ) =  (l − 2)(l + 3)Yl , −3 (θ ,φ )
where

l l +2
L−Yl , −2 (θ ,φ ) = −
(− 1) (2l + 1) (l − 2)! −3iφ ⎡ ∂ ⎤
e ⎢ − 2 cotθ ⎥ 1 − u 2 d
1− u2 ( ) ( l
)
l l +2
2 l! 4π (l + 2)! ⎣ ∂θ ⎦ du

= −
(− 1)l (2l + 1) (l − 2)! −3iφ
e (sin θ )
2 d ⎡
(sin θ )−2
(sin θ )2 d l +2 ⎤
(sin θ )2l ⎥
⎢
Thus, 2 l l! 4π (l + 2)! dθ ⎣ d (cosθ )
l +2
⎦

=
(− 1)l (2l + 1) (l − 2)! −3iφ
e (
1− u 2
3/ 2
) d l +3
1− u2 ( l
)
l 6
2 l! 4π (l + 2)! du l +3
 Spherical Harmonics

Yl , −3 (θ , φ ) =
(− 1)l (2l + 1) (l − 3)! −3iφ
e 1− u2 ( 3/ 2
) d l +3
( 1− u2
l
)
l
2 l! 4π (l + 3)! du l +3

and in general for m < 0,

l
(− 1) (2l + 1) (l + m)! imφ −m / 2 d l −m l
Yl ,m (θ , φ ) = l
2 l! 4π (l − m)!
e 1− u2 ( ) du l −m
(
1− u2 )
which is identical to the expression for m≥ 0.

Let us also note that

∂
LzYlm (θ ,φ ) = −i Ylm (θ ,φ ) = mYlm (θ ,φ )
∂φ
Taking the complex conjugate,
∂ *
i Y lm (θ ,φ ) = −mY *lm (θ ,φ )
∂φ
suggesting the Y*lm is an eigenfunction of Lz with eigenvalue -m. Likewise,

2 ⎡ 1 ∂ ⎛
2 ∂ ⎞ 1 ∂ 2 ⎤
L Ylm (θ , φ ) = − ⎢ ⎜ sin θ ⎟ + 2
Y (θ , φ ) = l (l + 1) 2Ylm (θ , φ )
2 ⎥ lm
⎣ sin θ ∂θ ⎝ ∂θ ⎠ sin θ ∂φ ⎦
and its complex conjugate is
⎡ 1 ∂ ⎛ ∂ ⎞ 1 ∂ 2 ⎤ *
−  2 ⎢ ⎜ sin θ ⎟ + 2 2 ⎥
Y lm (θ ,φ ) = l (l + 1) 2Y *lm (θ ,φ )
⎣ sin θ ∂θ ⎝ ∂θ ⎠ sin θ ∂φ ⎦
suggesting that Y*lm is an eigenfunction of L2 with eigenvalue l(l+1)2. Together, these imply that
Y*lm is at least proportional to Yl,-m. That is,
Y *lm (θ , φ ) = CmYl , − m (θ , φ )
Applying the ladder operator on the spherical harmonics, we note that
⎡ ∂ ∂ ⎤
L+Ylm (θ ,φ ) = −ie iφ ⎢i − cotθ ⎥Ylm (θ ,φ ) = (l − m)(l + m + 1)Yl ,m+1 (θ ,φ )
⎣ ∂θ ∂φ ⎦
and its complex conjugate is
⎡ ∂ ∂ ⎤
ie −iφ ⎢− i − cotθ ⎥Y *lm (θ ,φ ) = (l − m)(l + m + 1)Y *l ,m+1 (θ ,φ )
⎣ ∂θ ∂φ ⎦
or
⎡ ∂ ∂ ⎤
ie −iφ ⎢− i − cotθ ⎥CmYl , − m (θ ,φ ) = (l − m)(l + m + 1)Y *l ,m+1 (θ ,φ )
⎣ ∂θ ∂φ ⎦
On the other hand
⎡ ∂ ∂ ⎤
L−Yl , − m (θ ,φ ) = −ie −iφ ⎢− i − cotθ ⎥Yl , − m (θ ,φ ) = (l − m)(l + m + 1)Yl ,− m−1 (θ ,φ )
⎣ ∂θ ∂φ ⎦
Thus,
Y *l ,m+1 (θ ,φ ) = −CmYl , −( m+1) (θ ,φ )
Since we also have
Y *l ,m+1 (θ ,φ ) = Cm+1Yl , −( m+1) (θ ,φ )
we then find that in general,

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 Spherical Harmonics

Cm+1 = −Cm .
We now note that for m = 0,
Y *l ,0 (θ ,φ ) = C0Yl ,0 (θ ,φ )
Since the Associated Legendre Polynomials are real,
C0 = 1.
It then follows that
C1 = −C0 = −1; C2 = −C1 = 1; C3 = −C2 = −1; and so on …
and that in general,
Cm = (−1) m .
We thus obtain the conjugation relation for spherical harmonics
*
Ylm (θ , φ ) = (−1) m Yl , − m (θ , φ )

The spherical harmonics also obey the completeness condition

∞ l
*
∑ ∑Y
l =0 m = − l
lm (θ ' ,φ ')Ylm (θ ,φ ) = δ (φ − φ ')δ (cosθ − cosθ ' )
and the orthonormality condition
2π π *
∫φ ∫θ Yl 'm' (θ , φ )Ylm (θ , φ )sin θdθdφ = δ ll 'δ mm'
=0 =0
The latter suggests that the associated Legendre functions has the following normalization
π 2 (l + m)!
∫ Pl 'm (cosθ )Plm (cosθ )sin θdθ = δ l 'l
0 2l + 1 (l − m)!
Let us now look at the possible values of l. Angular momenta quantum numbers are either
integral or half-integral. On the other hand, orbital angular momenta are the generators of
rotation. If an angular momentum quantum number is half integral (2n+1)/2, so will the
corresponding magnetic quantum number. Looking at the expression for spherical harmonics, a
rotation of φ by 2π will yield
e i ( 2 n+1)π = −1 .
The wave functions will therefore not be single-valued. Since the expansion of a state ket in terms
of position eigenkets is unique, wave functions must be single-valued. This therefore suggests
that quantum numbers of orbital angular momenta must be integral.

Alternatively, we may start with Yl,-1 and apply the raising operator to generate the other spherical
harmonics. In particular, since
L− l ,−l = 0 ,
we have
⎡ ∂ ⎤
− ie −iφ ⎢− i + il cotθ ⎥ Pl , −l (θ )e −ilφ = 0
⎣ ∂θ ⎦
which gives
dPl , − l
= l cotθPl , − l
dθ
and
Pl , −l (θ ) = C 'l sin l θ

Here, we choose a phase factor of (+1) for C'l. Other spherical harmonics may be obtained by
applying the raising ladder operator on Yl,-l. Thus,
8
 Spherical Harmonics

L+Yl , −l (θ ,φ ) =  (l + l )(l − l + 1)Yl , −l +1 (θ ,φ ) =  2lYl , −l +1 (θ ,φ )

where
⎡ d ⎤
L+Yl , −l (θ ,φ ) = e iφ C 'l ⎢ + l cotθ ⎥ Pl , −l e −ilφ
⎣ dθ ⎦
Thus,
C 'l i ( −l +1)φ 1 d
Yl ,l −1 (θ , φ ) = e l
(sin θ ) 2l
2l (sin θ ) dθ
Applying L+ successively, we find that,
C 'l 1 d ⎡ 1 d ⎤
Yl , −l + 2 (θ ,φ ) = e i ( −l +2)φ l −1 ⎢ (sinθ ) 2l ⎥
2(2l )(2l − 1) (sinθ ) dθ ⎣ sin θ dθ ⎦

C 'l 1 d ⎡ 1 d ⎛ 1 d ⎞⎤
Yl , −l +3 (θ ,φ ) = e i ( −l +3)φ l −2 ⎢ ⎜ (sinθ ) 2l ⎟⎥
3 ⋅ 2 ⋅ 2l (2l − 1)(2l − 2)
In terms of u = cosθ, (sinθ ) dθ ⎣ sin θ dθ ⎝ sin θ dθ ⎠⎦
C 'l 1 d
Yl , −l +1 (θ ,φ ) = − e i ( −l +1)φ l −1 (1 − u 2 ) l
2 2 du
2l (1 − u )
C 'l 1 d2 l
Yl , −l + 2 (θ ,φ ) = (−1) 2
e i ( − l + 2 )φ
2 l −2
(1 − u ) 2 du 2
1− u2 ( )
2(2l )(2l − 1)
C 'l 1 d3 l
Yl , −l +3 (θ ,φ ) = (−1) 3 e i ( −l +3)φ l −3
du 3
(
1− u 2 )
3!2l (2l − 1)(2l − 2) 2
(1 − u ) 2

and generally,
1 (2l + 1) (l − m)! imφ d l +m l
Yl ,m (θ ,φ ) = (−1) l + m
2 l l! 4π (l + m)!
e (1 − u 2 ) m / 2 l + m 1 − u 2
du
( ) (2)

or
1 (2l + 1) (l − m)! imφ d l +m l
Yl ,m (θ ,φ ) = (−1) m
2 l l! 4π (l + m)!
e (1 − u 2 ) m / 2 l + m u 2 − 1
du
( )
The last expression may be compared with the Rodrigues' formula for the associated Legendre
polynomials
1 m/2 d m +l 2 l
Plm (u ) = l
2 l!
(
1− u 2 ) dx m +l
u −1 ; ( ) −l ≤ m ≤ l
giving us,
m 2l + 1 (l − m )!
nˆ lm = Ylm (θ ,φ ) = (− 1) Plm (cosθ )e imφ
4π (l + m )!
The factor (-1)m is a phase factor called the Condon-Shortley phase.

We note that in this alternative expression,

(−1) l (2l + 1) dl l
Yl ,0 (θ ,φ ) =
2l l! 4π du l
(
1− u 2 )
showing that our choice of phase factors for Cl and C'l are consistent with each other.

The conjugation relation for spherical harmonics may be evaluated by comparing the two
alternative expressions. From the latter expression, we note that

9
 Spherical Harmonics

1 (2l + 1) (l + m)! −imφ d l −m l

Yl , − m (θ ,φ ) = (−1) −m
2 l l! 4π (l − m)!
e (1 − u 2 ) −m / 2 l −m u 2 − 1
du
( )
and from the former,
1 (2l + 1) (l + m)! −imφ 1 d l −m 2 l
*
Y lm (θ ,φ ) = l
2 l! 4π (l − m)!
e 2 m/2
(1 − u ) du l −m
u −1 ( )
Thus,
*
Yl , − m (θ , φ ) = (−1) m Ylm (θ , φ )

l m Ylm (θ , φ ) rYlm ( x, y, z )
1 1
0 0
4π 4π
3 3
1 0 cosθ z
4π 4π
3 3
1 ±1  sin θe ±iφ  (x ± iy )
8π 8π
5 1 5 1 2
2 0
4π 4
(
3 cos 2 θ − 1 ) 4π 4
(3z − r 2 )
5 3 5 3
2 ±1  sin θ cos θe ±iφ  z (x ± iy )
4π 2 4π 2
5 3 5 3
2 ±2 sin 2 θe ± 2iφ (x ± iy )2
4π 8 4π 8
7 1 7 1
3 0
4π 4
(2 cos3 θ − 3 cosθ sin 2 θ ) 4π 4
(
z 5 z 2 − 3r 2 )
7 3 7 3
3 ±1 
4π 16
(
4 cos 2 θ sin θ − sin 3 θ e ±iφ ) 
4π 16
(
5 z 2 − r 2 (x ± iy ))
7 15 7 15 2
3 ±2 cos θ sin 2 θe ± 2iφ z (x ± iy )
4π 8 4π 8
7 5 7 5
3 ±3  sin 3 θe ±3iφ  (x ± iy )3
4π 16 4π 16
9 1 9 1
4 0
4π 64
(
35 cos 4 θ − 30 cos 2 θ + 3 ) 4π 64
(
35 z 4 − 30 z 2 r 2 + 3r 4 )
9 5 9 5
4 ±1 
4π 16
(
sin θ 7 cos3 θ − 3 cos θ e ±iφ ) 
4π 16
(
7 z 3 − 3zr 2 (x ± iy ) )
9 5 9 5
4 ±2
4π 32
(
sin 2 θ 7 cos 2 θ − 1 e ± 2iφ ) 4π 32
(
7 z 2 − r 2 (x ± iy ))2

9 35 9 35 3
4 ±3  sin 3 θ cosθe ±3iφ  z (x ± iy )
4π 16 4π 16
9 35 9 35
4 ±4 sin 4 θe ± 4iφ (x ± iy )4
4π 128 4π 128

Table 4.1 Some Spherical Harmonics

10