Spherical Harmonics

The spherical harmonics, $Y_{l,m}(\theta,\varphi)$ , are the angular portions of the global solutions to Laplace's equation in standard spherical coordinates, $r$ , $\theta$ , $\varphi$ . Here, $l$ is a non-negative integer (known as the degree), and $m$ is an integer (known as the order) lying in the range $-l\leq m\leq l$ . The spherical harmonics are well behaved and single valued functions that satisfy the differential equation

$$r^{\,2}\,\nabla^{\,2} Y_{l,m} +l\,(l+1)\,Y_{l,m}=0,$$ (307)

and take the form

$$Y_{l,m}(\theta,\varphi)= \left[\frac{(2\,l+1)\,(l-m)!}{4\pi\,(l+m)!}\right]^{1/2}\,P_l^{\,m}(\cos\theta)\,{\rm e}^{\,{\rm i}\,m\,\varphi}.$$ (308)

It follows from Equation (295) that

$$Y_{l,-m} = (-1)^{\,m}\,Y_{l,m}^{\,\ast}.$$ (309)

The $Y_{l,m}(\theta,\varphi)$ satisfy the orthonormality constraint

$$\oint Y_{l,m}(\theta,\varphi)\,Y^{\,\ast}_{l',m'}(\theta,\varphi)\,d{\mit\Omega}= \delta_{ll'}\,\delta_{mm'},$$ (310)

where $d{\mit\Omega}=\sin\theta\,d\theta\,d\varphi$ is a an element of solid angle, and the integral is taken over all solid angle. Note that the spherical harmonics form a complete set of angular functions.

All of the spherical harmonics of degree less than 3 are listed below:

$$Y_{0,0}(\theta,\varphi) = \left(\frac{1}{4\pi}\right)^{1/2},$$ (311)

$$Y_{1,-1}(\theta,\varphi) = \left(\frac{3}{8\pi}\right)^{1/2}\sin\theta\,\,{\rm e}^{-{\rm i}\,\varphi},$$ (312)

$$\ Y_{1,0}(\theta,\varphi) = \left(\frac{3}{4\pi}\right)^{1/2}\cos\theta,$$ (313)

$$Y_{1,+1}(\theta,\varphi) =- \left(\frac{3}{8\pi}\right)^{1/2}\sin\theta\,\,{\rm e}^{+{\rm i}\,\varphi},$$ (314)

$$Y_{2,-2}(\theta,\varphi) =\left(\frac{15}{32\pi}\right)^{1/2}\sin^2\theta\,{\rm e}^{-2\,{\rm i}\,\varphi},$$ (315)

$$Y_{2,-1}(\theta,\varphi) =\left(\frac{15}{8\pi}\right)^{1/2}\sin\theta\,\cos\theta\,{\rm e}^{-{\rm i}\,\varphi},$$ (316)

$$Y_{2,0}(\theta,\varphi) =\left(\frac{5}{16\pi}\right)^{1/2}(3\,\cos^2\theta-1)$$ (317)

$$Y_{2,+1}(\theta,\varphi) =-\left(\frac{15}{8\pi}\right)^{1/2}\sin\theta\,\cos\theta\,{\rm e}^{+{\rm i}\,\varphi},$$ (318)

$$Y_{2,+2}(\theta,\varphi) =\left(\frac{15}{32\pi}\right)^{1/2}\sin^2\theta\,{\rm e}^{+2\,{\rm i}\,\varphi}.$$ (319)

Consider two spherical coordinate systems, $r$ , $\theta$ , $\varphi$ and $r'$ , $\theta'$ , $\varphi'$ , whose origins coincide, but whose polar axes subtend an angle $\gamma$ with respect to one another. It follows that

$$\cos\gamma = \cos\theta\,\cos\theta'+\sin\theta\,\sin\theta'\,\cos(\varphi-\varphi').$$ (320)

Moreover, the so-called addition theorem for spherical harmonics states that

$$P_l(\cos\gamma) =\frac{4\pi}{2\,l+1}\sum_{m=-l,+l}Y^{\,\ast}_{l,m}(\theta',\varphi')\,Y_{l,m}(\theta,\varphi).$$ (321)