I am trying to learn how to solve three dimensional Schrödinger Equation in Spherical Coordinates. I was reading a text book and I found that there is a missed step in the solution, seen below:
The $\theta$ equation,$$\sin\theta\frac{\mathrm d}{\mathrm d\theta}\left(\sin\theta\frac{\mathrm d\Theta}{\mathrm d\theta}\right)+\left[\ell\left(\ell+1\right)\sin^2\theta-m^2\right]\Theta=0.\tag{4.25}$$is not so simple. The solution is $$\Theta(\theta)=AP_\ell^m\left(\cos\theta\right).\tag{4.26}$$where $P^m_\ell$ is the associated Legendre function, defined by $$P_\ell^m\left(x\right)\equiv\left(1-x^2\right)^{\vert m\vert/2}\left(\frac{\mathrm d}{\mathrm dx}\right)^{\vert m\vert}P_\ell(x).\tag{4.27}$$
It says that the solution of equation (4.25) is not simple and gives directly as equation (4.26). Can you help me to learn how to solve such differential equations?
You'll have to apply a variable change: let $x=\cos(\theta)$. That will lead you to the associated Legendre Differential Equation
\begin{equation} (1-x^{2})\frac{\mathrm{d}^{2}\Theta}{\mathrm{d}x^{2}}-\left(2x\frac{\mathrm{d}\Theta}{\mathrm{d}x}+\ell\left(\ell+1\right)-\frac{m^{2}}{1-x^{2}}\right)\Theta=0 \end{equation}
This is satisfied for values $x\in [-1,1]$ using Legendre Polynomials given by Rodrigues' formula:
\begin{equation} P_{\ell m}(x)=\frac{(-1)^{m}}{2^{\ell}\ell!}(1-x^{2})^{m/2}\frac{\mathrm{d}^{m+\ell}}{\mathrm{d}x^{m+\ell}}(x^{2}-1) \end{equation}
where $-\ell\leq m \leq \ell$
Here you can see a detailed solution.
I may recommend Arfken & Weber's Mathematical Methods for Physicists text.