Mathematical Methods for Physics and Engineering - Matematica.NET

SPECIAL FUNCTIONSorthonormal set, i.e.∫ 1 ∫ 2π−10[Yml (θ, φ) ] ∗Ym ′l (θ, φ) dφ d(cos θ) =δ ′ ll ′δ mm ′. (18.46)In addition, the spherical harmonics form a complete set in that any reasonablefunction (i.e. one that is likely to be met in a physical situation) of θ and φ canbe expanded as a sum of such functions,f(θ, φ) =∞∑l∑l=0 m=−la lm Yl m (θ, φ), (18.47)the constants a lm being given by∫ 1 ∫ 2π[a lm = Yml (θ, φ) ] ∗f(θ, φ) dφ d(cos θ). (18.48)−10This is in exact analogy with a Fourier series and is a particular example of thegeneral property of Sturm–Liouville solutions.Aside from the orthonormality condition (18.46), the most important relationshipobeyed by the Ylm is the spherical harmonic addition theorem. This readsP l (cos γ) =4π2l +1l∑m=−lY ml (θ, φ)[Y ml (θ ′ ,φ ′ )] ∗ , (18.49)where (θ, φ)and(θ ′ ,φ ′ ) denote two different directions in our spherical polar coordinatesystem that are separated by an angle γ. In general, spherical trigonometry(or vector methods) shows that these angles obey the identitycos γ =cosθ cos θ ′ +sinθ 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 ′ )byΩandΩ ′ , respectively. We will also denote the element of solid angle on the sphere bydΩ =dφ d(cos θ). We begin by deriving the form of the closure relationship obeyed by thespherical harmonics. Using (18.47) and (18.48), and reversing the order of the summationand integration, we may write∫f(Ω) = dΩ ′ f(Ω ′ ) ∑ Ylm∗ (Ω ′ )Yl m (Ω),4πlmwhere ∑ lmis a convenient shorthand for the double summation in (18.47). Thus we maywrite the closure relationship for the spherical harmonics as∑lmY ml (Ω)Y m∗l (Ω ′ )=δ(Ω − Ω ′ ), (18.51)where δ(Ω − Ω ′ ) is a Dirac delta function with the properties that δ(Ω − Ω ′ )=0ifΩ≠Ω ′and ∫ 4π δ(Ω) dΩ =1. 594