Past exam of the theoretical physics II course of the University of Cambridge / 2013 / Paper 1 / 5 / b / Solution

The bosonic commutator is important in $S_{+}{S}_{-}={\hslash }^2{n}_{\uparrow }(n_{\downarrow }+1)$ and $S_{-}{S}_{+}={\hslash }^2{n}_{\downarrow }(n_{\uparrow }+1)$. ThereforeOn the $N$-particle sector,For example $N=0,1,2$ gives spin $0,\frac{1}{2},1$. The total number operator can be retained in the same identity before restricting to a sector.