logo

Spherical Harmonics: General Solutions for the Polar and Azimuthal Angles in the Spherical Coordinate Laplace's Equation 📂Mathematical Physics

Spherical Harmonics: General Solutions for the Polar and Azimuthal Angles in the Spherical Coordinate Laplace's Equation

Definition

The general solution for the polar and azimuthal angles in the spherical coordinate system for the Laplace equation is as follows, and this is called Spherical harmonics.

Ylm(θ,ϕ)=eimϕPlm(cos⁡θ) Y_{l}^{m}(\theta,\phi)=e^{im\phi}P_{l}^{m}(\cos \theta)

Here, ll is l=0,1,2⋯l=0,1,2\cdots and mm is an integer that satisfies −l≤m≤l -l \le m \le l. Also, Plm(cos⁡θ)P_{l}^{m}(\cos\theta) is as follows.

Plm(cos⁡θ)=(1−cos⁡2θ)∣m∣2d∣m∣dx∣m∣Pl(x)=(1−cos⁡2θ)∣m∣2d∣m∣dx∣m∣[12ll!dldxl(x2−1)l] \begin{align*} P_{l}^{m}(\cos \theta)&= (1-\cos ^{2}\theta)^{\frac{|m|}{2}} \frac{ d^{|m|} }{ dx^{|m|} } P_{l}(x) \\ & =(1-\cos ^{2}\theta)^{\frac{|m|}{2}} \frac{ d^{|m|} }{ dx^{|m|} }\left[ \dfrac{1}{2^l l!} \dfrac{d^l}{dx^l}(x^2-1)^l \right] \end{align*}