Write down the commutation relations between the components of position $\mathbf{x}$ and momentum $\mathbf{p}$ for a particle in three dimensions.

A particle of mass $m$ executes simple harmonic motion with Hamiltonian

$$H=\frac{1}{2 m} \mathbf{p}^{2}+\frac{m \omega^{2}}{2} \mathbf{x}^{2},$$

and the orbital angular momentum operator is defined by

$$\mathbf{L}=\mathbf{x} \times \mathbf{p}$$

Show that the components of $\mathbf{L}$ are observables commuting with $H$. Explain briefly why the components of $\mathbf{L}$ are not simultaneous observables. What are the implications for the labelling of states of the three-dimensional harmonic oscillator?