State the condition for a linear operator $\hat{O}$ to be Hermitian.

Given the position and momentum operators $\hat{x}_{i}$ and $\hat{p}_{i}=-i \hbar \frac{\partial}{\partial x_{i}}$, define the angular momentum operators $\hat{L}_{i}$. Establish the commutation relations

$$\left[\hat{L}_{i}, \hat{L}_{j}\right]=i \hbar \epsilon_{i j k} \hat{L}_{k}$$

and use these relations to show that $\hat{L}_{3}$ is Hermitian assuming $\hat{L}_{1}$ and $\hat{L}_{2}$ are.

Consider a wavefunction of the form

$$\chi(\mathbf{x})=x_{3}\left(x_{1}+k x_{2}\right) e^{-r}$$

where $r=|\mathbf{x}|$ and $k$ is some constant. Show that $\chi(\mathbf{x})$ is an eigenstate of the total angular momentum operator $\hat{\mathbf{L}}^{2}$ for all $k$, and calculate the corresponding eigenvalue. For what values of $k$ is $\chi(\mathbf{x})$ an eigenstate of $\hat{L}_{3}$ ? What are the corresponding eigenvalues?