- #1

silmaril89

- 86

- 0

## Homework Statement

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.

## Homework Equations

Not really any specific to put here.

## 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: