# Expected value of the spin tensor operator

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
Homework Helper
Gold Member
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.

Last edited:
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)