Tensor force operator

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!
 
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Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
 

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