Adding Angular Momenta: The Vector Model

Two angular momenta j1 and j2 combine into a total J restricted to |j1-j2| <= J <= j1+j2. The primary scene is the 3D vector model: J1 and J2 precess on cones about the resultant J, their lengths sqrt(j(j+1)) and their tips circling so the sum stays fixed. The side panel is the Clebsch-Gordan coefficient table for the chosen total, with the allowed-J ladder; these coefficients form a unitary change of basis between the uncoupled |j1 m1>|j2 m2> states and the coupled |J M> states, with the selection rule M = m1 + m2.