(a) Given the position and momentum operators $\hat{x}_{i}=x_{i}$ and $\hat{p}_{i}=-i \hbar \partial / \partial x_{i}$ (for $i=1,2,3)$ in three dimensions, define the angular momentum operators $\hat{L}_{i}$ and the total angular momentum $\hat{L}^{2}$.

Show that $\hat{L}_{3}$ is Hermitian.

(b) Derive the generalised uncertainty relation for the observables $\hat{L}_{3}$ and $\hat{x}_{1}$ in the form

$$\Delta_{\psi} \hat{L}_{3} \Delta_{\psi} \hat{x}_{1} \geqslant M$$

for any state $\psi$ and a suitable expression $M$ that you should determine. [Hint: It may be useful to consider the operator $\hat{L}_{3}+i \lambda \hat{x}_{1}$.]

(c) Consider a particle with wavefunction

$$\psi=K\left(x_{1}+x_{2}+2 x_{3}\right) e^{-\alpha r}$$

where $r=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}$ and $K$ and $\alpha$ are real positive constants.

Show that $\psi$ is an eigenstate of total angular momentum $\hat{L}^{2}$ and find the corresponding angular momentum quantum number $l$. Find also the expectation value of a measurement of $\hat{L}_{3}$ on the state $\psi$.