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.
WHAT TO TRY
- Watch J1 and J2 precess on cones about the resultant J: only the total J is sharp, the individual projections wander. The total ranges over |j1 - j2| to j1 + j2 in integer steps.
- Pick a different allowed total J: the opening angle between J1 and J2 changes to match, from nearly aligned at the maximum J to nearly opposed at the minimum.
- Read the Clebsch-Gordan table: each row sums to 1 (the coupling is unitary), and the cell shading shows which (m1, m2) combinations build a given total-J, M state.