[QUE/QM-10012]

In the position representation a function of \(\vec{r}\), \(f(\vec{r})\), can be thought of as an operator \(\hat{f}\):  \[  \hat{f} \psi(\vec{r}) = f(\vec{r}) \psi(\vec{r}). \] Use spherical harmonics, \(Y_{\ell m}(\theta, \phi)\) to define operators \(\widehat{Y}_{\ell m}\) in the above sense. Use properties of spherical harmonics and known expressions for angular momentum operators in polar coordinates to show  \begin{eqnarray}   [L_z, \widehat{Y}_{\ell m}] &=& m\hbar\, \widehat{Y}_{\ell m}, \nonumber\\{}  [L_{\pm}, \widehat{Y}_{\ell m}] &=& \sqrt{\ell(\ell+1) - m(m\pm1)}\,\hbar\,\widehat{Y}_{\ell\pm 1 m}.\nonumber  \end{eqnarray}