In the non-relativistic quark model the baryon octet is approximated as three-quark states that are totally symmetric in flavor, space, and spin but antisymmetric in color. The total and internal orbital angular momenta are zero.
(a) Show that the flavor-spin part of the proton wave function is
∣p+⟩=321[∣uud⟩×(2∣++−⟩−∣+−+⟩−∣−++⟩)+{∣123⟩→∣231⟩}+{(∣123⟩→∣312⟩}],
where ± denote spin projections of ±21, and the second and third terms are cyclic permutations on the three quarks. The neutron wave function is similar except u↔d.
(b) From (2.395) the proton magnetic moment is μp=gpμNSp=(1+κp)μNσp, where κp∼1.79 is the anomalous magnetic moment. Similarly, the neutron moment is μn=gnμNSn=κnμNσn, with κn∼−1.91. These can be estimated in the quark model by assuming that the quarks are pointlike, i.e., that their magnetic moment operator is μ=Q2Meσ, where Q has eigenvalues eu=32 and ed=−31, and M∼mp/3 is the constituent quark mass. The nucleon moments can then be calculated by taking the expectation value of ∑a=1,2,3μa, where μa acts on the ath quark. Calculate the ratio gp/gn and compare with the experimental value.
(c) Calculate the individual values of gp/2 and gn/2 and compare with experiment.
See (Georgi, 1999) for a more detailed discussion of the magnetic moments of the baryon octet and the relation to SU(6).