Basis representation for isotropic 2D quantum harmonic oscillator

Question

The basis functions of the 2D isotropic quantum harmonic oscillator are of the form

$$ \psi_{n,\ell} (r,\varphi) = A_{n\ell}(r)e^{i\ell\varphi}$$

where $A_{n\ell}(r) = \frac{\sqrt{2 \times p!}}{\sqrt{(p+\vert\ell\vert)!}} e^{-r^2/2} r^{\vert \ell \vert} L_p^{\vert \ell \vert}(r^2)$ where $p=\frac{n-\vert \ell \vert}{2}.$

They are e.g. derived in (2.10), (2.11) here in this reference.

I am now curious if one can derive the matrix elements

$$\langle \psi_{n,\ell}\vert T \vert \psi_{m,\ell'} \rangle$$

for the two operators $T = r^2$ and $T = e^{-i\varphi} (-i\frac{\partial}{\partial r} - \frac{3}{r} \frac{\partial}{\partial {\varphi}}).$ I somehow feel that these two things should be well-known but I could not find this online. Is this somehow easy? The involved integrals seems very delicate.

I would be very interested in getting the ladder operator representation mentioned in the comments to work, but I do not quite know how to set it up in this case for my operators.

Answer 1

The average $\langle r^2\rangle$ can be computed using Hellmann-Feynman theorem. Since the eigen-energies are $E_{p,l}=\hbar\omega(2p+|l|+1)$ with $p=(n-|l|)/2$, then $${\partial E_{p,l}\over\partial\omega} =\langle {\partial H\over\partial\omega}\rangle$$ gives $$\hbar(2p+|l|+1)=m\omega\langle r^2\rangle$$ and finally $$\langle r^2\rangle =\langle\psi_{p,l}| r^2|\psi_{p,l}\rangle ={\hbar\over m\omega}(2p+|l|+1)$$