The Eightfold Way — Question 1

Question 1

Nucleon charges follow from a shared hypercharge

For light u,d,s hadrons use $Y=B+S$ and $Q/e=I_3+Y/2$. The proton and neutron form an isospin doublet with B=1, S=0 and $I_3=\pm1/2$.

1. Determine hypercharge and the charge of each doublet member.

2. Find total isospin I from the number of states and distinguish I from ordinary angular momentum.

Question 1 - Solution

Nucleon charges follow from a shared hypercharge

(a) Method and calculation

$$\boxed{Y=1+0=1}.$$$$\boxed{Q_p/e=1/2+1/2=1,\quad Q_n/e=-1/2+1/2=0}.$$

(b) Interpretation and further calculation

$$2I+1=2\Rightarrow\boxed{I=1/2}.$$$$\boxed{I_3=-I,\ldots,I}\quad\text{labels internal flavor states, not spatial spin}.$$

Physical check

Both nucleons also have spin 1/2, but that equality is not the definition of isospin.