The angular momentum operators are $\mathbf{L}=\left(L_{1}, L_{2}, L_{3}\right)$. Write down their commutation relations and show that $\left[L_{i}, \mathbf{L}^{2}\right]=0$. Let

$$L_{\pm}=L_{1} \pm i L_{2},$$

and show that

$$\mathbf{L}^{2}=L_{-} L_{+}+L_{3}^{2}+\hbar L_{3} .$$

Verify that $\mathbf{L} f(r)=0$, where $r^{2}=x_{i} x_{i}$, for any function $f$. Show that

$$L_{3}\left(x_{1}+i x_{2}\right)^{n} f(r)=n \hbar\left(x_{1}+i x_{2}\right)^{n} f(r), \quad L_{+}\left(x_{1}+i x_{2}\right)^{n} f(r)=0$$

for any integer $n$. Show that $\left(x_{1}+i x_{2}\right)^{n} f(r)$ is an eigenfunction of $\mathbf{L}^{2}$ and determine its eigenvalue. Why must $L_{-}\left(x_{1}+i x_{2}\right)^{n} f(r)$ be an eigenfunction of $\mathbf{L}^{2}$ ? What is its eigenvalue?