Consider a quantum mechanical particle moving in two dimensions with Cartesian coordinates $x, y$. Show that, for wavefunctions with suitable decay as $x^{2}+y^{2} \rightarrow \infty$, the operators

$$x \quad \text { and } \quad-i \hbar \frac{\partial}{\partial x}$$

are Hermitian, and similarly

$$y \text { and }-i \hbar \frac{\partial}{\partial y}$$

are Hermitian.

Show that if $F$ and $G$ are Hermitian operators, then

$$\frac{1}{2}(F G+G F)$$

is Hermitian. Deduce that

$$L=-i \hbar\left(x \frac{\partial}{\partial y}-y \frac{\partial}{\partial x}\right) \quad \text { and } \quad D=-i \hbar\left(x \frac{\partial}{\partial x}+y \frac{\partial}{\partial y}+1\right)$$

are Hermitian. Show that

$$[L, D]=0 .$$