Adding $\ell=4$ to $\ell=2$

As an example, we count the states for each value of total m (z component quantum number) if we add $\ell_1 =4$ to $\ell_2 =2$.

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m $(m_1,m_2)$

6 (4,2)

5 (3,2) (4,1)

4 (2,2) (3,1) (4,0)

3 (1,2) (2,1) (3,0) (4,-1)

2 (0,2) (1,1) (2,0) (3,-1) (4,-2)

1 (-1,2) (0,1) (1,0) (2,-1) (3,-2)

0 (-2,2) (-1,1) (0,0) (1,-1) (2,-2)

-1 (1,-2) (0,-1) (-1,0) (-2,1) (-3,2)

-2 (0,-2) (-1,-1) (-2,0) (-3,1) (-4,2)

-3 (-1,-2) (-2,-1) (-3,0) (-4,1)

-4 (-2,-2) (-3,-1) (-4,0)

-5 (-3,-2) (-4,-1)

-6 (-4,-2)

Since the highest m value is 6, we expect to have a $j=6$ state which uses up one state for each m value from -6 to +6. Now the highest m value left is 5, so a $j=5$ states uses up a state at each m value between -5 and +5. Similarly we find a $j=4$, $j=3$, and $j=2$ state. This uses up all the states, and uses up the states at each value of m. So we find in this case,

$$\bgroup\color{black} \vert\ell_1 - \ell_2\vert\leq j \leq \vert\ell_1 + \ell_2\vert \egroup$$

and that $j$ takes on every integer value between the limits. This makes sense in the vector model.