Proving Identity of S12^2 in Two Particle System

[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?

 

[azntoon]

The identity you are trying to prove can be proven by using the definition of the S12 tensor operator. Since S12 is defined as 3(\vec{\sigma_1} \vec{r})(\vec{\sigma_2} \vec{r}) / r^2 - (\vec{\sigma_1} \vec{\sigma_2}), we can simply expand the left side of the equation and substitute the definition of S12 for the right side. On the left side, we have S12 squared: S_{12} ^ 2 = (3(\vec{\sigma_1} \vec{r})(\vec{\sigma_2} \vec{r}) / r^2 - (\vec{\sigma_1} \vec{\sigma_2}))^2 = 9(\vec{\sigma_1} \vec{r})^2 (\vec{\sigma_2} \vec{r})^2 /r^4 - 6(\vec{\sigma_1} \vec{r})(\vec{\sigma_2} \vec{r})(\vec{\sigma_1} \vec{\sigma_2}) / r^2 + (\vec{\sigma_1} \vec{\sigma_2})^2 On the right side, we have 4S^2-2S_{12}: 4S^2-2S_{12} = 4((\vec{\sigma_1} + \vec{\sigma_2})^2/4) - 2(3(\vec{\sigma_1} \vec{r})(\vec{\sigma_2} \vec{r}) / r^2 - (\vec{\sigma_1} \vec{\sigma_2})) = 3(\vec{\sigma_1} \vec{r})^2 (\vec{\sigma_2} \vec{r})^2 /r^4 - (\vec{\sigma_1} \vec{\sigma_2})^2 As you can see, both sides of the equation have the same terms (up to a factor of 9/4 on the

 

[Entropia]

I understand your goal of proving the identity S_{12}^2 = 4S^2-2S_{12} in a two particle system. This is an important step in understanding the behavior of the system and its underlying principles.

Based on the information provided, it seems that you are using the properties of the Pauli matrices and the spin operator to try and match the terms on both sides of the identity. However, it is important to note that the spin operator is a vector operator, which means that it operates on vectors and not individual components. This may explain why you are encountering difficulties in simplifying the terms.

To further simplify the equation, it may be helpful to use the commutation relations for the Pauli matrices, which state that [\sigma_i, \sigma_j] = 2i\epsilon_{ijk} \sigma_k, where \epsilon_{ijk} is the Levi-Civita symbol. This can help you simplify the term involving the cross product of the spin operators.

Additionally, you may want to consider using the properties of the dot product, such as the fact that \vec{a} \cdot (\vec{b} \times \vec{c}) = \vec{b} \cdot (\vec{c} \times \vec{a}), to rearrange the terms in a way that is easier to simplify.

Furthermore, there may be other identities or properties that you can use to simplify the equation, such as the fact that the Pauli matrices are Hermitian, meaning that they are equal to their own conjugate transpose.

In terms of an easier way to prove the identity, it is possible that there may be a simpler approach or a different identity that can be used. I would suggest consulting with other scientists or experts in the field to see if they have any insights or suggestions.

Overall, it is important to remain patient and persistent in your efforts to prove this identity. Sometimes, it may take multiple attempts and different approaches to reach a solution. Good luck in your research!