\[\label{eq:spharmonicdef} Y_{lm}(\theta, \phi) = \qty[ \frac{2l+1}{4 \pi} \frac{(l-|m|)!}{(l+|m|)!}]^{\frac{1}{2}} P_l^m (\cos(\theta)) e^{im\phi}\]

\[ \begin{align}\begin{aligned} \label{eq:negativespharm} Y_{l, -m} (\theta, \phi) = (-1)^m Y^{*}_{lm} (\theta, \phi)\\and they are orthogonal,\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned} \label{eq:spahrmorthog} \int \dd{\Omega} Y^{*}_{lm} (\theta, \phi) Y_{l^{\prime}m^{\prime}}(\theta, \phi) = \delta_{l l^{\prime}} \delta_{m, m^{\prime}}\\The first few spherical harmonics are\end{aligned}\end{align} \]

\[Y_{00} = \sqrt{\frac{1}{4 \pi}}\]

\[Y_{10}= \sqrt{\frac{3}{4 \pi}} \cos(\theta)\]

\[Y_{1, \pm 1} = \mp \sqrt{\frac{3}{8 \pi}} \sin(\theta) e^{\pm i \phi}\]

\[ \begin{align}\begin{aligned}\begin{split} \label{eq:tisespherical} \begin{split} - \frac{\hbar^2}{2m} \frac{1}{r^2 \sin^2 \theta} \bigg( \pdv{r} \qty[ r^2 \sin(\theta) \pdv{\psi}{r}] + \pdv{\theta} \qty[ \sin \theta \pdv{\psi}{\theta}] \\ + \pdv{\phi} \qty[ \frac{1}{\sin \theta} \pdv{\psi}{\phi}] \bigg) + (V-E)\psi = 0 \end{split}\end{split}\\Now, splitting this into the radial part and an angular part,\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned} \label{eq:sepschrod} \psi(r, \theta, \phi) = R(r) Y(\theta, \phi)\\Then, substituting this in, and setting both sides of the equation\end{aligned}\end{align} \]

\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned} \dv{r} \qty[r^2 \dv[2]{R}{r}] - \frac{2m}{\hbar^2} \qty(V(r)-E) r^2 R - l(l+1) R &= 0 \\\frac{1}{\sin \theta} \pdv{\theta} \qty[ \sin \theta \pdv{Y}{\theta}] + \frac{1}{\sin^2(\theta)} \pdv[2]{Y}{\phi} +l(l+1)Y &= 0\end{aligned}\end{split}\\The angular part is solved by spherical harmonics,\end{aligned}\end{align} \]

\[\label{eq:sphericalharmonics} Y_l^m(\theta, \phi) = N e^{im\phi} P_l^m (\cos \theta)\]