a) Find the closed form expression for the spin-orbit force. Show that the spin-orbit force {\bf LS} gives a zero contribution for \( S \)-waves (orbital angular momentum \( l=0 \)). What is the value of the spin-orbit force for spin-singlet states (\( S=0 \))?

b) Find thereafter the expectation value of \( \mathbf{\sigma}_1\cdot\mathbf{\sigma}_2 \), where \( \mathbf{\sigma}_i \) are so-called Pauli matrices.

c) Add thereafter isospin and find the expectation value of \( \mathbf{\sigma}_1\cdot\mathbf{\sigma}_2\mathbf{\tau}_1\cdot\mathbf{\tau}_2 \), where \( \mathbf{\tau}_i \) are also so-called Pauli matrices. List all the cases with \( S=0,1 \) and \( T=0,1 \).