Let $x, y, z$ be Cartesian coordinates in $\mathbb{R}^{3}$. The angular momentum operators satisfy the commutation relation

$$\left[L_{x}, L_{y}\right]=i \hbar L_{z}$$

and its cyclic permutations. Define the total angular momentum operator $\mathbf{L}^{2}$ and show that $\left[L_{z}, \mathbf{L}^{2}\right]=0$. Write down the explicit form of $L_{z}$.

Show that a function of the form $(x+i y)^{m} z^{n} f(r)$, where $r^{2}=x^{2}+y^{2}+z^{2}$, is an eigenfunction of $L_{z}$ and find the eigenvalue. State the analogous result for $(x-i y)^{m} z^{n} f(r)$.

There is an energy level for a particle in a certain spherically symmetric potential well that is 6-fold degenerate. A basis for the (unnormalized) energy eigenstates is of the form

$$\left(x^{2}-1\right) f(r),\left(y^{2}-1\right) f(r),\left(z^{2}-1\right) f(r), x y f(r), x z f(r), y z f(r) \text {. }$$

Find a new basis that consists of simultaneous eigenstates of $L_{z}$ and $\mathbf{L}^{2}$ and identify their eigenvalues.

[You may quote the range of $L_{z}$ eigenvalues associated with a particular eigenvalue of $\mathbf{L}^{2}$.]