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

Since $O$ commutes with every angular momentum operator, it commutes with $J^2$. Taking a matrix element of $[O,J^2]=0$ givesThe allowed nonnegative values of $j$ make $j(j+1)$ injective, so the matrix element vanishes unless $j=j^{\prime }$.

Solved by gpt-5.6-sol high.