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Tensor force

  1. Nov 17, 2009 #1
    I am trying to prove the identity
    [tex]S_{12} ^ 2 = 4S^2-2S_{12}[/tex]
    where S12 is the tensor operator:
    [tex]S_{12} = 3(\vec{\sigma_1} \vec{r})(\vec{\sigma_2} \vec{r}) / r^2 - (\vec{\sigma_1} \vec{\sigma_2})[/tex]
    where sigmas are vectors made of the Pauli matrices in the space of particle 1 and 2, and
    [tex]\vec{S} = (\vec{\sigma_1} + \vec{\sigma_2})/2[/tex]
    the spin of the two particle system, and I am using the identity:
    [tex](\vec{a} \vec{\sigma})(\vec{b} \vec{\sigma}) = \vec{a}\vec{b} + i \sigma (\vec{a} X \vec{b})[/tex]
    to match the terms in each sides, however, i get a term like:
    [tex](\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}))[/tex]
    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, [tex]\vec{\sigma_1} \vec{\sigma_2}[/tex] and [tex](\vec{\sigma_1} \vec{n}) (\vec{\sigma_2}[/tex]\vec{n}), where n is a [tex]vec{r} / r[/tex]. So, how do I further simplify this? Or is there an easier way to prove this without tedious computation?
  2. jcsd
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