Solve the energy eigenvalue problem for the two-dimensional isotropic harmonic oscillator. Assume that the eigenfunctions are of the form
$$
\psi=\rho^{\ell} \exp \left(-\frac{m \omega}{2 \hbar} \rho^2\right) f(\rho) \exp ( \pm i \ell \varphi)
$$
where $\rho$ and $\varphi$ are plane polar coordinates and $\ell$ is a nonnegative integer. Show that $f(\rho)$ can be expressed as an associated Laguerre polynomial of the variable $m \omega \rho^2 / \hbar$ and determine the eigenvalues. Solve the same problem in Cartesian coordinates, and establish the correspondence between the two methods. Discuss a few simple eigenfunctions.