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Normalization of Spherical Harmonic Functions 📂Mathematical Physics

Normalization of Spherical Harmonic Functions

Theorem

The standardized spherical harmonics are as follows.

Ylm(θ,ϕ)=2l+14π(l−m)!(l+m)!Plm(cos⁡θ)eimϕ Y_{l}^{m}(\theta,\phi)=\sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}}P_{l}^{m}(\cos\theta)e^{im\phi}

∇2f=1r2∂∂r(r2∂f∂r)+1r2sin⁡θ∂∂θ(sin⁡θ∂f∂θ)+1r2sin⁡2θ∂2f∂2ϕ=0 \nabla ^2 f = \frac{1}{r^2}\frac{\partial}{\partial r} \left( r^2\frac{\partial f}{\partial r} \right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\left( \sin\theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2 f}{\partial^2 \phi}=0

f(r,θ,ϕ)=R(r)Θ(θ)Φ(ϕ) f(r,\theta,\phi)=R(r)\Theta (\theta)\Phi (\phi)

Description

In the Laplace equation on spherical coordinates, solutions for the polar angle θ\theta and the azimuthal angle ϕ\phi are referred to as spherical harmonics. Θ(θ)Φ(ϕ)=Ylm(θ,ϕ)=eimϕPlm(cos⁡θ) \Theta (\theta)\Phi (\phi)=Y_{l}^{m}(\theta,\phi)=e^{im\phi}P_{l}^{m}(\cos \theta) In quantum mechanics, spherical harmonics are dealt with as wave functions and must be standardized. ∭∣R(r)Θ(θ)Φ(ϕ)∣2r2sin⁡θdrdθdϕ=∫0∞∣R(r)∣2r2dr∫02π∫0π∣Ylm(θ,ϕ)∣2sin⁡θdθdϕ=1 \iiint |R(r)\Theta (\theta) \Phi (\phi)|^{2}r^{2}\sin \theta dr d \theta d\phi=\int_{0}^{\infty}|R(r)|^{2}r^{2}dr\int_{0}^{2\pi}\int_{0}^{\pi}|Y_{l}^{m}(\theta,\phi)|^{2}\sin\theta d\theta d\phi=1 Let’s isolate the angular component that is a spherical harmonic and call the standardizing constant CC. Then, ∣C∣2∫02π∫0π∣Ylm(θ,ϕ)∣2sin⁡θdθdϕ=1  ⟹  ∣C∣2∫02π∣eimϕ∣2dϕ∫0π∣Plm(cos⁡θ)∣2sin⁡θdθ=1  ⟹  2π∣C∣2∫0π∣Plm(cos⁡θ)∣2sin⁡θdθ=1 \begin{align*} &&& |C|^{2}\int_{0}^{2\pi}\int_{0}^{\pi}|Y_{l}^{m}(\theta,\phi)|^{2}\sin\theta d\theta d\phi=1 \\ \implies &&&|C|^{2}\int_{0}^{2\pi}|e^{im\phi}|^{2} d\phi\int_{0}^{\pi}|P_{l}^{m}(\cos \theta)|^{2}\sin\theta d\theta =1 \\ \implies &&&2\pi|C|^{2}\int_{0}^{\pi}|P_{l}^{m}(\cos \theta)|^{2}\sin\theta d\theta =1 \end{align*} The integral over θ\theta is due to the orthogonality of the associated Legendre polynomials and hence 22l+1(l+m)!(l−m)!\frac{2}{2l+1} \frac{(l+m)!}{(l-m)!}, C=2l+14π(l−m)!(l+m)! C=\sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}} Therefore, the standardized spherical harmonics are as follows. Ylm(θ,ϕ)=2l+14π(l−m)!(l+m)!Plm(cos⁡θ)eimϕ Y_{l}^{m}(\theta,\phi)=\sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}}P_{l}^{m}(\cos\theta)e^{im\phi} In quantum mechanics, it is usually assumed that spherical harmonics are standardized unless stated otherwise.