(a) Write down the angular momentum operators $L_{1}, L_{2}, L_{3}$ in terms of $x_{i}$ and

$$p_{i}=-i \hbar \frac{\partial}{\partial x_{i}}, i=1,2,3$$

Verify the commutation relation

$$\left[L_{1}, L_{2}\right]=i \hbar L_{3}$$

Show that this result and its cyclic permutations imply

$$\begin{aligned} &{\left[L_{3}, L_{1} \pm i L_{2}\right]=\pm \hbar\left(L_{1} \pm i L_{2}\right)} \\ &{\left[\mathbf{L}^{2}, L_{1} \pm i L_{2}\right]=0} \end{aligned}$$

(b) Consider a wavefunction of the form $\psi=\left(x_{3}^{2}+a r^{2}\right) f(r)$, where $r^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$. Show that for a particular value of $a, \psi$ is an eigenfunction of both $\mathbf{L}^{2}$ and $L_{3}$. What are the corresponding eigenvalues?