The quantum mechanical angular momentum operators are

$$L_{i}=-i \hbar \epsilon_{i j k} x_{j} \frac{\partial}{\partial x_{k}} \quad(i=1,2,3)$$

Show that each of these is hermitian.

The total angular momentum operator is defined as $\mathbf{L}^{2}=L_{1}^{2}+L_{2}^{2}+L_{3}^{2}$. Show that $\left\langle\mathbf{L}^{2}\right\rangle \geqslant\left\langle L_{3}^{2}\right\rangle$ in any state, and show that the only states where $\left\langle\mathbf{L}^{2}\right\rangle=\left\langle L_{3}^{2}\right\rangle$ are those with no angular dependence. Verify that the eigenvalues of the operators $\mathbf{L}^{2}$ and $L_{3}^{2}$ (whose values you may quote without proof) are consistent with these results.