Finally, substitution of Equation (B.10) into Equation (B.9) gives 1 Θsinθ ∂ ∂ θ(sinθΘ0)− m2 sin2θ =−l(l+1). (B.13)
To solve this Equation, the substituion µ = cosθ can be made. From the chain-rule this implies, d dθ = dµ dθ d dµ=−sinθ d dµ. (B.14) Equation (B.13) becomes ∂ ∂ µ (1−µ2) ∂ ∂ µ − m 2 1−µ2 Θ(µ) =−l(l+1)Θ(µ). (B.15) The solution to this Equation is of the form
Θ(µ) =
∞
X
i=1
aiµi. (B.16)
Ifm=0, then the coefficients in Equation (B.16) are given by the Legendre polynomials. The general form of the Legendre polynoimals is given by Rodrigues’ formula:
Pl(µ) = 1 2ll! ∂l ∂ µl(µ 2 −1)l. (B.17) The first few Legendre polynomials are:
P0=1 P1=cosθ P2= 1 2 3 cos2θ−1 P3= 1 2 5 cos3θ−3 cosθ.
In the casem6=0, for the series to terminate,m≤l. The functional form of the series is then given by the associated Legendre polynomials,
Plm(µ) = (−1)m(1−µ2)m/2 ∂
m
∂ µmPl(µ). (B.18)
The spherical harmonics are a combination of both the Legendre polynomials and the associated Legendre polynomials with the addition of a normalisation factor. They are given as
where the normalisation is chosen such that integration over the solid angledΩ =sinθdθdφ is unity, i.e., Z φ=2π φ=0 Z θ=π θ=0 Yl m∗(θ,φ)Yl0m0sinθdθdφ=δl l0δmm0. (B.20)
Combining Equations (B.12), and (B.19) into Equation (B.4) gives the final solution to the Laplacian∇2Ψ =0, Ψ(r,θ,φ) = ∞ X l=0 l X m=−l Al mrl+Bl mr−(l+1) Plm(cosθ)eimφ. (B.21)
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