Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 33B / a / Solution

The states are defined bywhere $j=0,\frac{1}{2},1,\dots$ and $m=-j,-j+1,\dots ,j$. Fromand positivity of the norm,Thus ${\lambda }_1^{+}={\lambda }_1^{-}=-1$, ${\lambda }_2^{+}=1$, and ${\lambda }_2^{-}=-1$.