Analyzing the Energy Spectrum of Two Identical Particles

[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.