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I have some troubles to find eigenfunctions common to [tex] H, L_{z}, L^2 [/tex] in the problem of spherical simmetric harmonic oscillator.

I start with the Hamiltonian [tex] H=\frac{\textbf{p}^2}{2 \mu} - \frac{1}{2}k\textbf{x}^2[/tex] that in spherical coordinates become

[tex] H=\frac{- \hbar ^2}{2 \mu} (\frac{\partial^2}{r \partial r^2} r - \frac{L^2}{r^2 \hbar^2}) - \frac{k r^2}{2} [/tex]

now,the eigenfunctions equation is: [tex] H \psi = E \psi [/tex]

i know the angular part of the problem that are the spherical armonics...so it remain to solve the radial equation

[tex] (\frac{- \hbar ^2}{2 \mu} \frac{\partial^2}{r \partial r^2} r + \frac{\hbar^2 l(l+1)}{2 \mu r^2 } - \frac{k r^2}{2}) R(r) = E R(r) [/tex]

then i don't know if to try with the hydrogen-like method (but potential is different) or with the armonic 1D oscillator (but I don't know whato to do with the centrifugal term).........so, please give me an help!!!