## Lecture: Spherical Harmonics

Central Forces 2023

$\boxed{\begin{array}{lll} \ell& m & \quad\quad\quad\; Y_\ell^m(\theta,\phi) \\[.35cm] \hline \\[.03cm] 0 & 0 & \quad\;\; Y_0^0=\sqrt{\frac{1}{4\pi}} \\[.35cm] 1 & 0 & \quad\;\; Y_1^0=\sqrt{\frac{3}{4\pi}}\cos\theta \\[.35cm] & \pm1 & \quad Y_1^{\pm1}=\mp\sqrt{\frac{3}{8\pi}}\sin\theta e^{\pm i\phi} \\[.35cm] 2 & 0 & \quad\;\;Y_2^0=\sqrt{\frac{5}{16\pi}}\left(3\cos^2\theta-1 \right) \\[.35cm] & \pm1 & \quad Y_2^{\pm1}=\mp\sqrt{\frac{15}{8\pi}}\sin\theta\cos \theta e^{\pm i\phi} \\[.35cm] & \pm2 & \quad Y_2^{\pm2}=\sqrt{\frac{15}{32\pi}}\sin^2\theta e^{\pm2i\phi} \\[.35cm] 3 & 0 & \quad\;\;Y_3^0=\sqrt{\frac{7}{16\pi}}\left(5\cos^3\theta-3 \cos\theta\right) \\[.35cm] & \pm1 & \quad Y_3^{\pm1}=\mp\sqrt{\frac{21}{64\pi}}\sin\theta \left(5\cos^2\theta-1\right)e^{\pm i\phi} \\[.35cm] & \pm2 & \quad Y_3^{\pm2}=\sqrt{\frac{105}{32\pi}} \sin^2\theta\cos\theta e^{\pm2i\phi} \\[.35cm] & \pm3 & \quad Y_3^{\pm3}=\sqrt{\frac{35}{64\pi}}\sin^3\theta e^{\pm3i\phi} \\[.001cm] \end{array}}$

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