Energy spectrum of two identical harmonic oscillators in CM coordinates

[CaptainQuaser]

Homework Statement

Two identical particles are descibed by:
$H(p,x)= H(p_{1},x_{1})+H(p_{2},x_{2})$
where
$H(p,x)=\frac{p^{2}}{2m}+\frac{1}{2}m\omega^2x^2$

Separate to CM, obtain energy Spectrum. Show it agrees with:

$H\psi (x_{1},x_{2}) = E\psi (x_{1},x_{2})$
with
$\psi (x_{1},x_{2}) = u_{1}(x_{1})u_{2}(x_{2})$

Discuss degeneracy.

The Attempt at a Solution

I got the CM hamiltonian to be:

$H(R,r) = \frac{P_{R}^{2}}{2M} + \frac{P_{r}^{2}}{2\mu}+\frac{M\omega^{2} R^{2}}{2}+ \frac{\mu \omega^{2} r^{2}}{2}$

where $\mu = \frac{m_{1}m_{2}}{m_{1}+m_{2}}=\frac{m}{2}$

and $M=2m$
$R=\frac{x_{1}+x_{2}}{2}$
$r=(x_{2}-x_{1})$

Not sure how to get the energy spectrum since I don't know the wavefcn.
Any suggestions?

 

[CarlB]

Maybe you should take into account $$E = \hbar \omega$$ or something like that.