Derivation of 3D simple harmonic oscillator energies in spherical coordinates I'm trying to show the permitted energies of the 3D simple harmonic oscillator (which is spherically symmetrical) are: $E_n = \hbar \omega(N + \dfrac{3}{2})$
In particular, $V(x) = \dfrac{1}{2} m \omega x^2 +  \dfrac{\hbar^2l(l+1)}{2mr^2}$
The steps I'm using are: 


*

*Rearrange schrodinger's time independent equation and solve for an approximation to $u$ for large $r$. This gave me $u(\rho) \approx Ae^{\rho^2/2}$. Where $\rho = \sqrt{\dfrac{m\omega}{\hbar}} r$

*Assume $u(\rho) = P(\rho) e^{\rho^2/2}$, I plug this into the original differential equation to get:
$$\dfrac{-\hbar\omega}{2}(P'' - 2\rho P' + \rho^2P)e^{\rho^2/2} = (\rho^2 +\dfrac{m\omega}{\hbar\rho^2}l(l+1))Pe^{\rho^2/2}$$


*I understand that the next step is to assume a taylor expansion for $P(\rho)$ and thence derive a recurrence relationship which will ultimately yield the energies


My problem is that whenever I plug in taylor expansions for $P$ and compare coefficients I end up with a 3 term recurrence relationship. I have no idea where I could have gone wrong, any help would be greatly appreciated.
 A: 
  
*
  
*Rearrange Schrödinger's time independent equation and solve for an
   approximation to $u$ for large $r$. This gave me
   $u(\rho) \approx Ae^{\rho^2/2}$.
  

Your step 1 is basically OK. But you need to choose the decreasing
solution $u(\rho) \approx Ae^{-\rho^2/2}$ instead of the increasing one.
In addition to the above step you also need to solve for an approximation
to $u$ for small $r$. This will give you something like
$u(\rho) \approx A \rho^{l+1}$.


*Then, using the two approximations of the steps above,
you can then assume $u(\rho) = P(\rho) \rho^{l+1} e^{-\rho^2/2}$,
plug this into the original differential equation,
and get a differential equation for $P(\rho)$.




  
*I understand that the next step is to assume a Taylor expansion
   for $P(\rho)$ and thence derive a recurrence relationship which will
   ultimately yield the energies.
  

Your step 3 is fine.
Now, when you plug in the Taylor expansion for $P(\rho)$ and compare
coefficients, you should end up with a 2 term recurrence relationship
(instead of your 3 term recurrence relationship).
