[QUE/QM-08007]

Let $Y_{\ell m}(\theta,\phi)$ denote the  simultaneous normalized eigenfunctions of $L^2$ and $L_z$ operators. Use the properties of the ladder operators, $L_\pm$, and construct the expressions for $Y_{lm}(\theta,\phi)$ for $l=2$ and $m=2,1,0,-1,-2$.

1. Note that $Y_{\ell\ell}(\theta,\phi)$ satisfies\begin{eqnarray} L_z Y_{\ell\ell}(\theta,\phi) &=& \ell \hbar Y_{\ell\ell}(\theta, \phi)\nonumber\\ L_+ Y_{\ell\ell}(\theta,\phi) &=& 0. \nonumber\end{eqnarray}  Set up the above differential equations and solve them using separation of  variables and find (normalized) $Y_{2,2}$.
2. Next apply $L_-$ repeatedly on \(Y_{2,2}\) and use $$L_- Y_{\ell, m} = \sqrt{\ell(\ell + 1)-m(m - 1)}\,  \hbar\,Y_{\ell(m-1)}$$ to successively construct $Y_{2,m}$ for other values $m =  1, 0,-1,-2$.
3. Normalize your answers and compare them with known expressions of $Y_{2m}(\theta,\phi), m=-2,-1,0,1,2.$

Hint:
Use coordinate space representation  for angular mormentum operators.