The Hamiltonian of a two-dimensional isotropic harmonic oscillator is given by

$$H=\frac{p_{x}^{2}+p_{y}^{2}}{2 m}+\frac{m \omega^{2}}{2}\left(x^{2}+y^{2}\right)$$

where $x$ and $y$ denote position operators and $p_{x}$ and $p_{y}$ the corresponding momentum operators.

State without proof the commutation relations between the operators $x, y, p_{x}, p_{y}$. From these commutation relations, write $\left[x^{2}, p_{x}\right]$ and $\left[x, p_{x}^{2}\right]$ in terms of a single operator. Now consider the observable

$$L=x p_{y}-y p_{x}$$

Ehrenfest's theorem states that, for some observable $\mathrm{Q}$ with expectation value $\langle Q\rangle$,

$$\frac{d\langle Q\rangle}{d t}=\frac{1}{i \hbar}\langle[Q, H]\rangle+\left\langle\frac{\partial Q}{\partial t}\right\rangle$$

Use it to show that the expectation value of $L$ is constant with time.

Given two states

$$\psi_{1}=\alpha x \exp \left(-\beta\left(x^{2}+y^{2}\right)\right) \text { and } \psi_{2}=\alpha y \exp \left(-\beta\left(x^{2}+y^{2}\right)\right)$$

where $\alpha$ and $\beta$ are constants, find a normalised linear combination of $\psi_{1}$ and $\psi_{2}$ that is an eigenstate of $L$, and the corresponding $L$ eigenvalue. [You may assume that $\alpha$ correctly normalises both $\psi_{1}$ and $\psi_{2}$.] If a quantum state is prepared in the linear combination you have found at time $t=0$, what is the expectation value of $L$ at a later time $t ?$