Subatomic Physics

15.10. References 497
where
J± = Jx ± iJy, J ′ ± = J ′ x ± iJ ′ y
are raising and lowering operators with properties as given in Eq. (8.27).
(b) Consider the two quark states
|α〉 = |u ↑〉|d ↓〉
|β〉 = |u ↓〉|d ↑〉
where, for instance, |u↑〉 denotes an up quark with spin up (Jz =1/2),
and |d ↓〉 denotes a down-quark with spin down (J ′ z = −1/2). Use the
result of part (a) to find the linear combinations of the states |α〉 and
|β〉 that correspond to values Jtot =1andJtot = 0 of the total angular
momentum quantum number of the two quarks.
15.8. Assume the u and d quarks to be massless and to belong to an isomultiplet
with total isospin of 1/2.
(a) Show that the spin and isospin assignments of the proton, neutron and
delta are those that can be reached with 3 quarks.
(b) Show that the lowest mass mesons that can be made with u and d quarks
(and their antiquarks) have isospin 0 and 1.
15.9. Verify Eqs. (15.14) and (15.15). Why should the various particle states be
orthogonal to each other?
15.10. Apply the argument that leads to Eq. (15.15) to the neutral pseudoscalar
mesons. Try to find possible explanations why the agreement with experiment
is much less satisfactory than for the vector mesons.
15.11. Instead of the assignments made in Table 15.2, one could choose
J A S I I3
u 1/2 1 0 1/2 1/2
d 1/2 1 0 1/2 −1/2
s 1/2 1 −1 0 0
(a) What is q/e for each quark in this case?
(b) Mesons would be constructed from q ′ q ′ ,whereq ′ is an antiquark. Would
this assignment work? Explain any difficulties that are encountered.
(c) Why is this model not used? [S. Sakata, Prog. Theor. Phys. 16, 686
(1956)].