Define the commutator $[A, B]$ of two operators, $A$ and $B$. In three dimensions angular momentum is defined by a vector operator $\mathbf{L}$ with components

$$L_{x}=y p_{z}-z p_{y} \quad L_{y}=z p_{x}-x p_{z} \quad L_{z}=x p_{y}-y p_{x}$$

Show that $\left[L_{x}, L_{y}\right]=i \hbar L_{z}$ and use this, together with permutations, to show that $\left[\mathbf{L}^{2}, L_{w}\right]=0$, where $w$ denotes any of the directions $x, y, z$.

At a given time the wave function of a particle is given by

$$\psi=(x+y+z) \exp \left(-\sqrt{x^{2}+y^{2}+z^{2}}\right)$$

Show that this is an eigenstate of $\mathbf{L}^{2}$ with eigenvalue equal to $2 \hbar^{2}$.