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**1. The problem statement, all variables and given/known data**

I have a question asking me to find the expectation value of [itex] S_{12} [/itex] for a system of two nucleons in a state with total spin [itex] S = 1 [/itex] and [itex] M_s = +1 [/itex], where [itex] S_{12} [/itex] is the tensor operator inside the one-pion exchange nuclear potential operator, equal to

[itex] S_{12} = \frac{3}{r^2}(\sigma^{(1)}\cdot r)(\sigma^{(2)}\cdot r) - \sigma^{(1)} \cdot \sigma^{(2)} [/itex]

Where the sigma are the pauli spin matrices.

**2. Relevant equations**

**3. The attempt at a solution**

What I personally would've done would note that the system is in a state [itex] \Psi = \psi_{space}\chi_{spin} [/itex], where [itex] \chi_{spin} = \alpha(1)\alpha(2) [/itex] and then just have done

[itex] \int \Psi^* S_{12} \Psi dV [/itex] = [itex] \int |\psi_{space}|^2 \chi_{spin}^* S_{12} \chi_{spin} dV [/itex]

So my problem would lie in calculating [itex] \chi_{spin}^* S_{12} \chi_{spin} [/itex] and then evaluating the integral. However, my answer says that the expectation value is simply given by

[itex] \chi_{spin}^* S_{12} \chi_{spin} [/itex]

Why is there no integral over the whole of space? The answer doesn't make sense to me because it comes out as;

[itex] \chi_{spin}^* S_{12} \chi_{spin} \propto Y_{20}(\theta,\phi) [/itex]

So my expectation value depends on the angles between the two nucleons? How can that make any sense? Has my answer forgotten to neglect the then integral over all space?

Lastly, my answer says that because the expectation value has a [itex] Y_{20} [/itex] term that the operator can transfer 2 units of angular momentum to the orbital motion of the particles. My quantum mechanics knowledge must be lacking because I don't exactly see why this is the case, can anyone link my to a place where I can read up more about what expectation value results mean?

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