Spherical harmonics

18.3 SPHERICAL HARMONICS
be derived in a number of ways, such as using the generating function (18.40) or
by differentiation of the recurrence relations for the Legendre polynomials P (x).
Use the recurrence relation (2n + 1)Pn = Pn+1
− Pn−1
for Legendre polynomials to derive
the result (18.43).
Differentiating the recurrence relation for the Legendre polynomials m times, we have
(2n + 1)
d m Pn
dm+1 Pn+1
dm+1 Pn−1
=
−
.
m
m+1
dx
dx
dxm+1
Multiplying through by (1 − x2 )(m+1)/2 and using the definition (18.32) immediately gives
the result (18.43). 18.3 Spherical harmonics
The associated Legendre functions discussed in the previous section occur most
commonly when obtaining solutions in spherical polar coordinates of Laplace’s
equation ∇2 u = 0 (see section 21.3.1). In particular, one finds that, for solutions
that are finite on the polar axis, the angular part of the solution is given by
Θ(θ)Φ(φ) = Pm (cos θ)(C cos mφ + D sin mφ),
where and m are integers with − ≤ m ≤ . This general form is sufficiently
common that particular functions of θ and φ called spherical harmonics are
defined and tabulated. The spherical harmonics Ym (θ, φ) are defined by
Ym (θ, φ) = (−1)m
2 + 1 ( − m)!
4π ( + m)!
1/2
Pm (cos θ) exp(imφ).
(18.45)
Using (18.33), we note that
∗
Y−m (θ, φ) = (−1)m Ym (θ, φ) ,
where the asterisk denotes complex conjugation. The first few spherical harmonics
Ym (θ, φ) ≡ Ym are as follows:
1
3
Y00 =
Y10 =
4π ,
4π cos θ,
3
5
2
Y1±1 = ∓ 8π
sin θ exp(±iφ),
Y20 =
16π (3 cos θ − 1),
2
15
15
Y2±1 = ∓ 8π
sin θ cos θ exp(±iφ), Y2±2 =
32π sin θ exp(±2iφ).
Since they contain as their θ-dependent part the solution Pm to the associated
Legendre equation, the Ym are mutually orthogonal when integrated from −1 to
+1 over d(cos θ). Their mutual orthogonality with respect to φ (0 ≤ φ ≤ 2π) is
even more obvious. The numerical factor in (18.45) is chosen to make the Ym an
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SPECIAL FUNCTIONS
orthonormal set, i.e.
1 2π
−1
∗
Ym (θ, φ) Ym (θ, φ) dφ d(cos θ) = δ δmm .
(18.46)
0
In addition, the spherical harmonics form a complete set in that any reasonable
function (i.e. one that is likely to be met in a physical situation) of θ and φ can
be expanded as a sum of such functions,
f(θ, φ) =
∞ am Ym (θ, φ),
(18.47)
=0 m=−
the constants am being given by
1 2π
m
∗
Y (θ, φ) f(θ, φ) dφ d(cos θ).
am =
−1
(18.48)
0
This is in exact analogy with a Fourier series and is a particular example of the
general property of Sturm–Liouville solutions.
Aside from the orthonormality condition (18.46), the most important relationship obeyed by the Ym is the spherical harmonic addition theorem. This reads
P (cos γ) =
4π m
Y (θ, φ)[Ym (θ , φ )]∗ ,
2 + 1 m=− (18.49)
where (θ, φ) and (θ , φ ) denote two different directions in our spherical polar coordinate system that are separated by an angle γ. In general, spherical trigonometry
(or vector methods) shows that these angles obey the identity
cos γ = cos θ cos θ + sin θ sin θ cos(φ − φ ).
(18.50)
Prove the spherical harmonic addition theorem (18.49).
For the sake of brevity, it will be useful to denote the directions (θ, φ) and (θ , φ ) by
Ω and Ω , respectively. We will also denote the element of solid angle on the sphere by
dΩ = dφ d(cos θ). We begin by deriving the form of the closure relationship obeyed by the
spherical harmonics. Using (18.47) and (18.48), and reversing the order of the summation
and integration, we may write
f(Ω) =
dΩ f(Ω )
Ym∗ (Ω )Ym (Ω),
4π
m
where m is a convenient shorthand for the double summation in (18.47). Thus we may
write the closure relationship for the spherical harmonics as
Ym (Ω)Ym∗ (Ω ) = δ(Ω − Ω ),
(18.51)
m
where
δ(Ω − Ω ) is a Dirac delta function with the properties that δ(Ω − Ω ) = 0 if Ω = Ω
and 4π δ(Ω) dΩ = 1.
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