Write down the angular momentum operators $L_{1}, L_{2}, L_{3}$ in terms of the position and momentum operators, $\mathbf{x}$ and $\mathbf{p}$, and the commutation relations satisfied by $\mathbf{x}$ and $\mathbf{p}$.

Verify the commutation relations

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

Further, show that

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

A wave-function $\Psi_{0}(r)$ is spherically symmetric. Verify that

$$\mathbf{L} \Psi_{0}(r)=0$$

Consider the vector function $\boldsymbol{\Phi}=\nabla \Psi_{0}(r)$. Show that $\Phi_{3}$ and $\Phi_{1} \pm i \Phi_{2}$ are eigenfunctions of $L_{3}$ with eigenvalues $0, \pm \hbar$ respectively.