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Homework Help: Calculating Energy Spectrum of Angular Momentum Interaction Between Two Particles

  1. Jan 20, 2013 #1
    1. The problem statement, all variables and given/known data

    The Hamiltonian for two particles with angular momentum [itex]j_1[/itex] and [itex]j_2[/itex] is given by:
    [tex] \hat{H} = \epsilon [ \hat{\bf{j}}_1 \times \hat{\bf{j}}_2 ]^2, [/tex]
    where [itex]\epsilon[/itex] is a constant. Show that the Hamiltonian is a Hermitian scalar and find the energy spectrum.

    2. Relevant equations

    Not really any specific to put here.

    3. The attempt at a solution

    I tried simplifying the Hamiltonian using suffix notation with the Einstein summation convention. I was able to get the following:
    [tex] \hat{H} = \epsilon [( \hat{\bf{j}}_1 \cdot \hat{\bf{j}}_1) ( \hat{\bf{j}}_2 \cdot \hat{\bf{j}}_2) - \hat{j}_{1 i} ( \hat{\bf{j}}_2 \cdot \hat{\bf{j}}_1) \hat{j}_{2 i}]. [/tex]

    Now I have the problem that since [itex] \hat{j}_{2i} [/itex] doesn't commute with [itex] ( \hat{\bf{j}}_2 \cdot \hat{\bf{j}}_1) [/itex], I can't simplify the Hamiltonian further. I'm not sure what my next steps should be.
    Last edited: Jan 20, 2013
  2. jcsd
  3. Jan 22, 2013 #2
    There is quite a common trick for this kinds of things, you can always express the dot product of two angular momentums as


    The operators on the right hand side are easy to handle.

    Hope this helps.
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