Addition of two angular momenta

What you are seeing: the allowed values of total $J$ when coupling two angular momenta $j_1$ and $j_2$ . Range: $|j_1 - j_2| \le J \le j_1 + j_2$ . The (2J+1) substates of each $J$ multiplet sum to $(2j_1+1)(2j_2+1)$ .

Figure 1.

j_1 \otimes j_2

decomposed into total-

J

multiplets.

2 j11/2

2 j21/2

WHAT TO TRY

Vary each control and watch the rail readouts respond.
Compare the diagnostic plot against the live scene.