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We can transform the left hand side as follows \begin{align} \sum_{l=x,y,z}\langle J_l^2\rangle &= \sum_{l=x,y,z}\langle\psi|J_l^2|\psi\rangle \\ &= \langle\psi|\left(\sum_{l=x,y,z}J_l^2\right)|\psi\rangle \\ &= \langle\psi|\left(\sum_{l=x,y,z}\left(\sum_{i=1}^N\frac12\sigma_l^i\right)^2\right)|\psi\rangle \\ &= \langle\psi|\left(\sum_{l=x,... 4 Quick answer to point you in the right direction. That's the expression for an infinitesimal rotation operator (near the identity). To get a general rotation operator you exponentiate it so that R(\hat{n},{\theta}) = e^{-i \theta \vec n \cdot \vec J}. This is eqn 2.26 of the lecture notes you linked. The note doesn't provide motivation for this equation ... 2 These are also known as SU(2)-coherent states; one original reference is https://doi.org/10.1103/PhysRevA.6.2211. In a spin system, with states labeled by the eigenvalue of the total angular momentum operator \mathbf{J}^2 and the z-projection of the angular momentum operator J_z,\mathbf{J}^2||J,m\rangle=J(J+1)||J,m\rangle\quad J_z ||J,m\rangle=m||J,m\...