Calculating Energy Spectrum of Angular Momentum Interaction Between Two Particles

1. Jan 20, 2013

silmaril89

1. The problem statement, all variables and given/known data

The Hamiltonian for two particles with angular momentum $j_1$ and $j_2$ is given by:
$$\hat{H} = \epsilon [ \hat{\bf{j}}_1 \times \hat{\bf{j}}_2 ]^2,$$
where $\epsilon$ 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:
$$\hat{H} = \epsilon [( \hat{\bf{j}}_1 \cdot \hat{\bf{j}}_1) ( \hat{\bf{j}}_2 \cdot \hat{\bf{j}}_2) - \hat{j}_{1 i} ( \hat{\bf{j}}_2 \cdot \hat{\bf{j}}_1) \hat{j}_{2 i}].$$

Now I have the problem that since $\hat{j}_{2i}$ doesn't commute with $( \hat{\bf{j}}_2 \cdot \hat{\bf{j}}_1)$, I can't simplify the Hamiltonian further. I'm not sure what my next steps should be.

Last edited: Jan 20, 2013
2. Jan 22, 2013

CFede

There is quite a common trick for this kinds of things, you can always express the dot product of two angular momentums as

$J_1\cdot\J_2=\frac{1}{2}((J_1+J_2)^2-J_1^2-J_2^2)$

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

Hope this helps.