Angular momentum coupling and forbidden transitions in quantum mechanics
Selection Rules Table
Operator rank k: 1
■ Allowed
■ Forbidden
⟨j'm'|T^k_q|jm⟩ ≠ 0 only if:
|j-k| ≤ j' ≤ j+k
m' = m + q
The Wigner-Eckart theorem states that matrix elements of tensor operators factorize into a Clebsch-Gordan coefficient (encoding geometry) and a reduced matrix element (encoding dynamics). This leads to strict selection rules: only transitions where angular momentum changes by at most k (the operator rank) are allowed, and the magnetic quantum number must change by exactly q.