Ladder recurrence for Clebsch-Gordan coefficients ($C_m^j(m_1,m_2)=⟨j_1,m_1;j_2,m_2\mid j,m⟩$)

Apply a total ladder operator to a coupled state and evaluate against an uncoupled product state. The total operator is the sum of the two factor operators, so its one coupled ladder amplitude times the corresponding one of the Clebsch-Gordan coefficients equals the sum of two factor ladder amplitudes times shifted coefficients. An overall common rescaling of the ladder operators cancels from the recurrence. Nonzero coefficients obey $m_1+m_2=m$, and the positive ladder amplitudes require a consistent phase convention within each irreducible summand.

// Target: lie-theory.bigb