Two nuclei, each of spin , are combined, and someone argues: “Each nucleus carries…
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Symmetry, angular momentum and spin
Invariance, Ch Representations of Lie algebras: spin, products and SU(3)
Problem
Two 7Li nuclei, each of spin 23, are combined, and someone argues: “Each nucleus carries angular momentum j=23. Combining them is like adding two classical vectors of length 23: the total must be their sum, j=23+23=3. So the pair has only total spin 3, made of seven states, and the other nine of the sixteen joint states have nowhere to go.”
((a)) Find the error in this argument.
((b)) Count the joint weights M=m1+m2, peel off ladders to find the correct decomposition of 23⊗23, and check that the dimensions sum to sixteen.