A photon carrying angular momentum is emitted from an excited nuclear state of spin

PHYSICS · Subjective · 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

A photon carrying angular momentum j=1j=1 is emitted from an excited nuclear state of spin j=32j=\tfrac32. Model the coupling as the tensor product 1⊗321\otimes\tfrac32 of su(2)\mathfrak{su}(2) representations. The spins in 1⊗321\otimes\tfrac32 are the possible spins jfj_f of the nucleus after the emission, since the final nucleus and the photon must together carry the initial 32\tfrac32. Using the Clebsch-Gordan series j1⊗j2=∣j1−j2∣⊕(∣j1−j2∣+1)⊕⋯⊕(j1+j2)j_1\otimes j_2=|j_1-j_2|\oplus(|j_1-j_2|+1)\oplus\cdots\oplus(j_1+j_2), derived from the product of characters, decompose 1⊗321\otimes\tfrac32, list the possible jfj_f, and check that the total dimension is 1212.

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