# Tensor force

1. Nov 17, 2009

### evilcman

I am trying to prove the identity
$$S_{12} ^ 2 = 4S^2-2S_{12}$$
where S12 is the tensor operator:
$$S_{12} = 3(\vec{\sigma_1} \vec{r})(\vec{\sigma_2} \vec{r}) / r^2 - (\vec{\sigma_1} \vec{\sigma_2})$$
where sigmas are vectors made of the Pauli matrices in the space of particle 1 and 2, and
$$\vec{S} = (\vec{\sigma_1} + \vec{\sigma_2})/2$$
the spin of the two particle system, and I am using the identity:
$$(\vec{a} \vec{\sigma})(\vec{b} \vec{\sigma}) = \vec{a}\vec{b} + i \sigma (\vec{a} X \vec{b})$$
to match the terms in each sides, however, i get a term like:
$$(\vec{\sigma_1} \vec{n})(\vec{\sigma_2} \vec{n})(\vec{\sigma_1} \vec{\sigma_2}) = 1 + i (\vec{\sigma_1} \vec{n}) (\vec{n}(\vec{\sigma_1} X \vec{\sigma_2}))$$
and I don't know how to further simplify this, but if the identity really holds, then this should be a linear combination of 1, $$\vec{\sigma_1} \vec{\sigma_2}$$ and $$(\vec{\sigma_1} \vec{n}) (\vec{\sigma_2}$$\vec{n}), where n is a $$vec{r} / r$$. So, how do I further simplify this? Or is there an easier way to prove this without tedious computation?