(a) Define the quantum orbital angular momentum operator $\hat{\boldsymbol{L}}=\left(\hat{L}_{1}, \hat{L}_{2}, \hat{L}_{3}\right)$ in three dimensions, in terms of the position and momentum operators.

(b) Show that $\left[\hat{L}_{1}, \hat{L}_{2}\right]=i \hbar \hat{L}_{3}$. [You may assume that the position and momentum operators satisfy the canonical commutation relations.]

(c) Let $\hat{L}^{2}=\hat{L}_{1}^{2}+\hat{L}_{2}^{2}+\hat{L}_{3}^{2}$. Show that $\hat{L}_{1}$ commutes with $\hat{L}^{2}$.

[In this part of the question you may additionally assume without proof the permuted relations $\left[\hat{L}_{2}, \hat{L}_{3}\right]=i \hbar \hat{L}_{1}$ and $\left.\left[\hat{L}_{3}, \hat{L}_{1}\right]=i \hbar \hat{L}_{2} .\right]$

[Hint: It may be useful to consider the expression $[\hat{A}, \hat{B}] \hat{B}+\hat{B}[\hat{A}, \hat{B}]$ for suitable operators $\hat{A}$ and $\hat{B}$.]

(d) Suppose that $\psi_{1}(x, y, z)$ and $\psi_{2}(x, y, z)$ are normalised eigenstates of $\hat{L}_{1}$ with eigenvalues $\hbar$ and $-\hbar$ respectively. Consider the wavefunction

$$\psi=\frac{1}{2} \psi_{1} \cos \omega t+\frac{\sqrt{3}}{2} \psi_{2} \sin \omega t$$

with $\omega$ being a positive constant. Find the earliest time $t_{0}>0$ such that the expectation value of $\hat{L}_{1}$ in $\psi$ is zero.