A Pause for Reflection — Question 9

Question 9

A flavor label alone does not reveal all internal angular momentum

Two mesons have the same valence content $u\bar d$. Quark and antiquark each have spin 1/2. For orbital angular momentum L=0, total J equals the combined spin S.

1. Derive allowed J values and their magnetic-substate counts.

2. For L=1 and S=1, list the allowed J values and check total state counting.

Question 9 - Solution

A flavor label alone does not reveal all internal angular momentum

(a) Method and calculation

$$\tfrac12\otimes\tfrac12=0\oplus1\Rightarrow\boxed{J=0,1}.$$$$\boxed{(2J+1)=1,3,\quad1+3=4}.$$

(b) Interpretation and further calculation

$$|L-S|\le J\le L+S\Rightarrow\boxed{J=0,1,2}.$$$$\boxed{1+3+5=9=(2L+1)(2S+1)}.$$

Physical check

Same flavor and charge do not imply the same particle mass or spin. Spin-orbit coupling reorganizes a product basis without changing its dimension.