Write down a formula for the orbital angular momentum operator $\hat{\mathbf{L}}$. Show that its components satisfy

$$\left[L_{i}, L_{j}\right]=i \hbar \epsilon_{i j k} L_{k} .$$

If $L_{3} \psi=0$, show that $\left(L_{1} \pm i L_{2}\right) \psi$ are also eigenvectors of $L_{3}$, and find their eigenvalues.