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

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 ${\lambda }_2^{-}=-1$.

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 $j=j^{\prime }$.

Taking a matrix element of $[O,J_3]=0$ givesThus the matrix element vanishes unless $m=m^{\prime }$.

For $-j\le m<j$, the matrix element of $[O,J_{+}]$ between $|j,m+1⟩$ and $|j,m⟩$ givesThe ladder coefficient is nonzero, so adjacent diagonal entries agree. Induction makes the diagonal entry independent of $m$.