Spherical harmonics - Aetherwisp

The spherical harmonics $Y_{l}^{m}(\theta,\phi)$ are the angular part of the solution to Laplace's equation in spherical coordinates. They are

Y_{l}^{m}(\theta,\phi)=\sqrt{\frac{2l+1}{4\pi} \frac{(l-m)!}{(l+m)!}}e^{im\phi}P_{l}^{m}(\cos\theta)

where $\theta\in[0,\pi]$ is the polar angle, $\phi\in[0,2\pi[$ is the azimuthal angle and $P_{l}^{m}(\cos\theta)$ is the associated Legendre polynomial.

The normalization is chosen such that the spherical harmonics are pairwise orthogonal, that is

\int_{0}^{2\pi}\int_{0}^{\pi}Y_{l}^{m}(\theta,\phi)(Y_{l'}^{m'})^{*}(\theta,\phi)\sin\theta d\theta d\phi=\delta_{mm'}\delta_{ll'}

with $^{*}$ being the complex conjugate and $\delta$ the Kronecker delta.

The first spherical harmonics are

\begin{align} Y_{0}^{0}&=\sqrt{\frac{1}{4\pi}} \\ Y_{1}^{0}&=\sqrt{\frac{3}{4\pi}}\cos\theta,\quad Y_{1}^{\pm1}=\mp\sqrt{\frac{3}{8\pi}}\sin\theta e^{\pm i\phi} \\ Y_{2}^{0}&=\sqrt{\frac{5}{16\pi}}(3\cos^{2}\theta-1),\quad Y_{2}^{\pm1}=\mp\sqrt{\frac{15}{8\pi}}\sin\theta\cos\theta e^{\pm i\phi},\quad Y_{2}^{\pm2}=\sqrt{\frac{15}{32\pi}}\sin^{2}\theta e^{\pm2i\phi} \end{align}

Symmetry#

In quantum mechanics, a rather useful property of the harmonics is the following: the square modulus over all $m$ for a given $l$

\sum_{m=-l}^{l} \lvert Y_{l}^{m}(\theta,\phi) \rvert ^{2}

is spherically symmetrical. This is quite useful because any quantity whose shape is given by the harmonics is automatically also spherically symmetrical. For instance, the charge distribution in the hydrogen atom.

Another useful property is that the harmonics have alternating parity $(-1)^{l}$ . This means that the harmonic inherits the evenness of $l$ :

if $l$ is even, the harmonic is even, $Y_{l}^{m}(\pi-\theta,\phi+\pi)=Y_{l}^{m}(\theta,\phi)$
if $l$ is odd, the harmonic is odd, $Y_{l}^{m}(\pi-\theta,\phi+\pi)=-Y_{l}^{m}(\theta,\phi)$