Past exam of the quantum physics course of the University of Oxford / 2016 / 7 / b / Solution

Expanding the total angular momentum and replacing transverse components by ladder operators givesThe particle exchange operator obeys $P_{12}^2=I$ and $P_{12}{s}_{1\alpha }{P}_{12}=s_{2\alpha }$, with the reverse relation for index $2$. These statements also apply to the ladder operators. Exchanging the indices preserves $s_1^2+s_2^2$ and $s_{1z}+s_{2z}$; it interchanges the last two terms in $S^2$. Operators on different tensor factors commute. HenceThe full particle exchange includes the spin coordinates as well as the spatial coordinates.