If we take the composite angular momentum of the three valence quarks, we should have a proton/neutron spin angular momentum of, $$S=\{ \frac{1}{2},\frac{3}{2} \}$$ using the general rule for adding angular momentum in quantum mechanics. Evidently, we only ever observe a proton spin of $\frac{1}{2}$. So my question is, why do we not observe a spin of $\frac{3}{2}$?
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4$\begingroup$ Are you asking why the spin 1/2 state, p, is lighter than the spin 3/2 one, $\Delta^+$, made up of the very same valence quarks? $\endgroup$– Cosmas ZachosCommented Jan 6, 2023 at 22:53
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$\begingroup$ I suppose that is what my question reduces to after realising that these are made of the same valence quarks @CosmasZachos $\endgroup$– Adrien AmourCommented Jan 7, 2023 at 11:17
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$\begingroup$ It's a subtle question answered in quark model reviews: the energetics "favor" the spin 1/2 state making it lighter, so the heavier states rearrange to that one, in time... $\endgroup$– Cosmas ZachosCommented Jan 7, 2023 at 14:50
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1$\begingroup$ @CosmasZachos makes sense, thanks $\endgroup$– Adrien AmourCommented Jan 7, 2023 at 14:57
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2 Answers
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So my question is, why do we not observe a spin of $\frac{3}{2}$?
Actually we did observe such particles. But we don't call them proton and neutron, because they behave differently in some other ways. See at Table of Baryons.
- The $\Delta^+$ particle.
It has quark composition $uud$ (i.e. same as the proton) and spin $\frac{3}{2}$ (unlike the proton). - The $\Delta^0$ particle.
It has quark composition $udd$ (i.e. same as the neutron) and spin $\frac{3}{2}$ (unlike the neutron).
The $\Delta$ particles are instable (with very short lifetime) and have masses around $30$% larger than proton/neutron. (See this and this question about why the mass is larger.)
answered Jan 6, 2023 at 23:06
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$\begingroup$ Interesting, why would the $\Delta^{+}$ and $\Delta^{0}$ be more massive? Is it due to the magnetic interactions between the quark spins? I assume that we observe more protons and neutrons because they decay to the lowest mass state. $\endgroup$ Commented Jan 7, 2023 at 11:16
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We don't observe spin 3/2 protons because the proton is, by definition, the spin 1/2 ground state of $uud$. Both $\Delta$ baryons and $N^*$ baryons have/can have higher spins.
