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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

    [itex]J_1\cdot\J_2=\frac{1}{2}((J_1+J_2)^2-J_1^2-J_2^2)[/itex]

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

    Hope this helps.
     
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