Tensor Force Operator Between Nucleons: Spin & Position

[kelly0303]

Homework Statement

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.

Homework Equations

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!

 

[samalkhaiat]

Homework Statement

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.

Homework Equations

3. The Attempt at a Solution [/B]
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!

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