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

The coupled basis consists of the tripletand the singletThese equations list all nonzero Clebsch--Gordan coefficients, each coefficient being $1$, $1/\sqrt{2}$, or $-1/\sqrt{2}$ as displayed, andBecause each irreducible total-spin representation occurs once, an operator commuting with total angular momentum isIn the product basis, the nonzero matrix elements areand