Tensor force operator

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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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samalkhaiat

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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!
[tex]S = \frac{1}{2} (\sigma_{1} + \sigma_{2}) \ \ \Rightarrow \ \ S^{2} = \frac{1}{2} ( 3 + \sigma_{1} \cdot \sigma_{2} ) . \ \ \ \ (1)[/tex][tex]S \cdot r = \frac{1}{2} \left( ( \sigma_{1} \cdot r ) + ( \sigma_{2} \cdot r ) \right) \ \ \Rightarrow \ \ (S \cdot r)^{2} = \frac{1}{2} \left( r^{2} + ( \sigma_{1} \cdot r )(\sigma_{2} \cdot r ) \right) . \ \ \ \ (2)[/tex] Substitute (1), (2) in [tex]S_{12} = \frac{3}{r^{2}} ( \sigma_{1} \cdot r )( \sigma_{2} \cdot r ) - \sigma_{1} \cdot \sigma_{2} ,[/tex] to find [tex]S_{12} = \frac{6}{r^{2}} ( S \cdot r )^{2} - 2S^{2} .[/tex]
 
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