Expected value of the spin tensor operator

[Montejo]

Hello everyone,
I'm evaluating the one-gluon-exchange tensor part of a phenomenological potential between two particles, and it involves a term like this:
$$S_{ij}=3(\vec{\sigma_i}\hat{r_{ij}})(\vec{\sigma_j}\hat{r_{ij}})-\vec{\sigma_i}\vec{\sigma_j}$$

With $$r_{ij}$$ the unit vector in the direction along the axis from the first to the second particle

The second term $$\vec{\sigma_i}\vec{\sigma_j}$$ is very easy to evaluate, it yields -3 for S=0 and 1 for S=1
But I can't solve the first term, whatever I try I always end up with terms including $$r^{z}_{ij}$$ the proyection of the unit vector along de Z-axis. I suppose that implies that it doesn't only depend on S but also on Sz, which doesn't sound right to me.

Can anyone help me? or even better guide me in the right direction?
Thanks

 

[Meir Achuz]

If the wave function is spherically symmetric, then the average over angle is
$$<(\vec{\sigma_i}\hat{r_{ij}})(\vec{\sigma_j}\hat{r_{ij}})> =(1/3)\vec{\sigma}\cdot\vec{\sigma}$$, so S_ij=0.
If the wave function is not spherically symmetric, the average is harder, and expansion in spherical harmonics may be necessary.

 

[Montejo]

Thanks Meir, in fact it is not spherically symetric.
I worked out an expansion in spherical harmonics:
$$\sqrt{\frac{2\pi}{15}}(\sigma_{1-}\sigma_{2-}Y_{22}-(\sigma_{1-}\sigma_{2z}+\sigma_{1z}\sigma_{2-})Y_{21}-\frac{1}{\sqrt{6}}(\sigma_{1+}\sigma_{2-}-4\sigma_{1z}\sigma_{2z}+\sigma_{1-}\sigma_{2+})Y_{20}+(\sigma_{1+}\sigma_{2z}+\sigma_{1z}\sigma_{2+})Y_{2-1}+\sigma_{1+}\sigma_{2+}Y_{2-2})$$
Is it right?

Now, how do I evaluate this? I need to solve it for S=1, L=1, J=1 and for S=1, L=2, J=1
I'm trying to reproduce some calculations and the results show that for L=1 the energy is raised by adding this term whereas it is lowered for the L=2 state, so L definitely does play a part.
In addition, there are three posible spin-functions with S=1 but the projection is not given, so how can we evaluate the terms that depend on the spin of one of the quarks?

Pfiu, lots of questions, as I said in the previous post, I would greatly appreciate a hint in the right direction more than a straightforward answer (which I wouldn't regret either, but learning is more important in my opinion)