208
6 Appendices
In these integrals, θ 12 is a shorthand for cos(θ 12 ). Since these integrals are over the
“2” coordinates, factors of r 1 can be extracted from within them; we will eventually
write r 1 (θ 1 ). For the integration over r 2 we then get simple powers of r 2 , which
become powers of r 1 upon substituting the limits:
U 21 = r
2
1
2π
φ=0
π
θ=0
1
3
P 0 (θ 12 ) +
1
4
P 1 (θ 12 ) +
1
5
P 2 (θ 12 )
dΩ 2 .
(6.50)
We come now to a very important step in the algebra. This is that there exists
an identity known as the Addition Theorem for spherical harmonics. This theorem
lets one write a Legendre polynomial P k (cos θ 12 ) in terms of products of so-called
Associated Legendre polynomials P
m
k (cos θ ) whose arguments are the cosines of the
individual direction angles of the volume elements. If m = 0, the Associated Legendre
polynomials become the “ordinary” Legendre polynomials of (6.26)–(6.28); this will
come to be the case in a moment. The Addition Theorem is
P k (cos θ 12 ) =
k
m=−k
(k − m) !
(k + m) !
P
m
k (cos θ 1 ) P
m
k (cos θ 2 ) exp[ιm(φ 1 − φ 2 )]. (6.51)
This paragraph is important; read it carefully. Imagine (6.51) substituted into
(6.50) where the various P’s appear; also remember that we must eventually integrate over coordinate set “1” in (6.47). When integrating over φ 1 and φ 2 , only m = 0
will give non-zero contributions because of the imaginary exponential in (6.51);
you should convince yourself of the veracity of this statement. The Associated
Legendre polynomials consequently reduce to regular Legendre polynomials, which
I will designate as P k(1) and P k(2) , where P k(1) designates the k’th-order Legendre
polynomial for coordinate set “1”, and P k(2) that for coordinate set “2”.
The value of invoking the Addition Theorem is that we reduce U 21 in (6.50) to
products of individual-angle P’s, which allows us to invoke (6.31) and (6.32). Now
set d 2 = sinθ 2 dθ 2 dφ 2 in (6.50); do not forget the factor of 2π from integrating
over φ. This gives
U 21 = 2πr
2
1
⎧
⎨
⎩
P 0(1)
3
π
θ=0
P 0(2) sin θ 2 dθ 2 +
P 1(1)
4
π
θ=0
P 1(2) sin θ 2 dθ 2
+
P 2(1)
5
π
θ=0
P 2(2) sin θ 2 dθ 2
⎫
⎬
⎭
.
(6.52)
Invoking (6.32) indicates that the last two integrals in (6.52) vanish; recall that
you can insert a factor of P 0(2) = 1 into any integrand as desired. The first integral in
(6.52) is equal to 2 via (6.31), hence
6 Appendices
In these integrals, θ 12 is a shorthand for cos(θ 12 ). Since these integrals are over the
“2” coordinates, factors of r 1 can be extracted from within them; we will eventually
write r 1 (θ 1 ). For the integration over r 2 we then get simple powers of r 2 , which
become powers of r 1 upon substituting the limits:
U 21 = r
2
1
2π
φ=0
π
θ=0
1
3
P 0 (θ 12 ) +
1
4
P 1 (θ 12 ) +
1
5
P 2 (θ 12 )
dΩ 2 .
(6.50)
We come now to a very important step in the algebra. This is that there exists
an identity known as the Addition Theorem for spherical harmonics. This theorem
lets one write a Legendre polynomial P k (cos θ 12 ) in terms of products of so-called
Associated Legendre polynomials P
m
k (cos θ ) whose arguments are the cosines of the
individual direction angles of the volume elements. If m = 0, the Associated Legendre
polynomials become the “ordinary” Legendre polynomials of (6.26)–(6.28); this will
come to be the case in a moment. The Addition Theorem is
P k (cos θ 12 ) =
k
m=−k
(k − m) !
(k + m) !
P
m
k (cos θ 1 ) P
m
k (cos θ 2 ) exp[ιm(φ 1 − φ 2 )]. (6.51)
This paragraph is important; read it carefully. Imagine (6.51) substituted into
(6.50) where the various P’s appear; also remember that we must eventually integrate over coordinate set “1” in (6.47). When integrating over φ 1 and φ 2 , only m = 0
will give non-zero contributions because of the imaginary exponential in (6.51);
you should convince yourself of the veracity of this statement. The Associated
Legendre polynomials consequently reduce to regular Legendre polynomials, which
I will designate as P k(1) and P k(2) , where P k(1) designates the k’th-order Legendre
polynomial for coordinate set “1”, and P k(2) that for coordinate set “2”.
The value of invoking the Addition Theorem is that we reduce U 21 in (6.50) to
products of individual-angle P’s, which allows us to invoke (6.31) and (6.32). Now
set d 2 = sinθ 2 dθ 2 dφ 2 in (6.50); do not forget the factor of 2π from integrating
over φ. This gives
U 21 = 2πr
2
1
⎧
⎨
⎩
P 0(1)
3
π
θ=0
P 0(2) sin θ 2 dθ 2 +
P 1(1)
4
π
θ=0
P 1(2) sin θ 2 dθ 2
+
P 2(1)
5
π
θ=0
P 2(2) sin θ 2 dθ 2
⎫
⎬
⎭
.
(6.52)
Invoking (6.32) indicates that the last two integrals in (6.52) vanish; recall that
you can insert a factor of P 0(2) = 1 into any integrand as desired. The first integral in
(6.52) is equal to 2 via (6.31), hence