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

Set 2j1 and 2j2: the total J runs over |j1 - j2| to j1 + j2 in integer steps, and each allowed J appears as one rung. Two spin-halves give J = 0 and 1, the singlet and triplet.
Check the dimension count in the readout: the (2J+1) substates summed over all allowed J always equal (2j1+1)(2j2+1). Coupling rearranges the states, it never creates or destroys them.
Couple a large j1 to a small j2: you get a tight cluster of J multiplets around j1, the near-classical regime where adding a small spin barely tips the total.