If we limit ourselves to nucleons only with \( s=1/2 \) we find that $$ |\hat{J}|=\hbar\sqrt{j(j+1)}\ge |\hbar\sqrt{l(l+1)}- \hbar\sqrt{\frac{1}{2}(\frac{1}{2}+1)}|. $$ It is then easy to show that for nucleons there are only two possible values of \( j \) which satisfy the inequality, namely $$ j=l+\frac{1}{2}\hspace{0.1cm} \mathrm{or} \hspace{0.1cm}j=l-\frac{1}{2}, $$ and with \( l=0 \) we get $$ j=\frac{1}{2}. $$