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Quantum Physics

  1. Apr 16, 2007 #1
    1. The problem statement, all variables and given/known data
    Two identical particles are descibed by:
    [itex] H(p,x)= H(p_{1},x_{1})+H(p_{2},x_{2})[/itex]
    [itex] H(p,x)=\frac{p^{2}}{2m}+\frac{1}{2}m\omega^2x^2 [/itex]

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

    [itex]H\psi (x_{1},x_{2}) = E\psi (x_{1},x_{2})[/itex]
    [itex] \psi (x_{1},x_{2}) = u_{1}(x_{1})u_{2}(x_{2})[/itex]

    Discuss degeneracy.

    3. The attempt at a solution

    I got the CM hamiltonian to be:

    [itex]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}[/itex]

    where [itex]\mu = \frac{m_{1}m_{2}}{m_{1}+m_{2}}=\frac{m}{2}[/itex]

    and [itex] M=2m [/itex]
    [itex] R=\frac{x_{1}+x_{2}}{2}[/itex]
    [itex] r=(x_{2}-x_{1}) [/itex]

    Not sure how to get the energy spectrum since I don't know the wavefcn.
    Any suggestions?
    Last edited: Apr 16, 2007
  2. jcsd
  3. Apr 18, 2007 #2


    User Avatar
    Science Advisor
    Homework Helper

    Maybe you should take into account [tex]E = \hbar \omega[/tex] or something like that.
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