%--------------------------------------------------------------
\begin{frame}
\frametitle{Radial Wave Equation}
\begin{block}{}
By induction,
$$\color{ec}
\Psi_{\ell} = -\Big(\frac{\partial}{\partial r} - \frac{\ell}{r}\Big)\Psi_{\ell -1}, \hspace{2em} D^+_\ell = -\Big(\frac{\partial}{\partial r} - \frac{\ell}{r}\Big) 
$$
If
$$\color{ec}
-\partial^2_t\Psi_{\ell-1} + \partial^2_r\Psi_{\ell-1} - \frac{\ell(\ell-1)}{r^2}\Psi_{\ell-1} = 0,
$$
Then
$$\color{ec}
-\partial^2_t(D^+_\ell \Psi_{\ell -1}) +\partial^2_r(D^+_\ell \Psi_{\ell-1}) - \frac{\ell(\ell +1)}{r^2}D^+_\ell \Psi_{\ell-1} = 0.
$$
\end{block}
\end{frame}
