# Tensor force operator

#### kelly0303

1. The problem statement, all variables and given/known data
The tensor force operator between 2 nucleons is defined as $S_{12}=3\sigma_1\cdot r\sigma_2\cdot r - \sigma_1\cdot \sigma_2$. Where r is the distance between the nucleons and $\sigma_1$and $\sigma_2$ are the Pauli matrices acting on each of the 2 nucleons. Rewrite $S_{12}$ only in terms of the spin operator S and relative position r.

2. Relevant equations

3. The attempt at a solution
For the second part of the equation I tried this. Using the fact that $S=\sigma_1+\sigma_2$ we have $\sigma_1\sigma_2=(S^2-\sigma_1-\sigma_2)/2=(S^2-6)/2$. For the first part I was thinking to use this expression $(\sigma\cdot a)(\sigma \cdot b) = a\cdot b + i(a\times b)\sigma$ As in my case a and b are both r, the cross product would vanish and the first term would be just $r^2$. But I am not sure if I can do that, as my $\sigma$ is not the same in both cases. Can someone tell me if I can use that formula or give me some hint or how to approach the first part of the equation? Thank you!

#### PF_Help_Bot

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