[QUE/QM-10002]

The spherical harmonics $Y_{lm}(\theta,\phi)$ are normalized simultaneous eigenfunctions of $L^2$ and $L_z$ operators. Use the co-ordinate space expressions

\begin{eqnarray*}
L_x &=& i\hbar \Big( \sin\phi {\partial\over \partial\theta } + \cot \theta
 \cos\phi{\partial\over \partial \phi} \Big)\\
L_y &=& i\hbar \Big(-\cos\phi {\partial\over \partial\theta } + \cot \theta
 \sin\phi{\partial\over \partial \phi} \Big)\\
L_z &=& -i\hbar {\partial\over\partial \phi}
\end{eqnarray*}

Note : For the orbital angular momentum operators and the properties of the ladder operators, $L^\pm$, and construct expressions for $Y_{lm}(\theta,\phi)$ for
$l=2$ and $m=2,1,0,-1,-2$.