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

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 $m$.