Angular Momentum Diagrams

Starting from the simplest possible building blocks—a ket, a bra, a time-reversed ket, and a time-reversed bra— a diagrammatic formalism is developed for angular momentum coupling problems. The formalism comprises Clebsch–Gordan coefficients as well as 3jm-symbols. The idea of constructing invariants (internal lines) by contracting contragredient pairs of quantities is emphasized throughout. The Clebsch–Gordan series, and its extension to the coupling of more than two angular momenta, is introduced algebraically and diagrammatically. Recoupling between bases obtained in different coupling schemes is introduced and the connection between recoupling coefficients and irreducible 3nj -symbols is derived diagrammatically. The well-known diagrammatic rules due to Jucys and co-workers are derived by group theoretical means and simple rules for their practical exploitation are presented.