An introduction to the quark model

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Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook Permutation symmetry for (qqq) Again Ψ = ψ(x, y) ψs ψi ψc , For the ground state of ∆ ++ = (uuu) or Ω − = (sss), this is easy, each factor is either S or A, where S now means “fully symmetric” and A “fully antisymmetric” For the nucleon, one has to introduce the concept of “mixed symmetry” The prototype is given by the Jacobi coordinates x = r 2 − r 1 , y = 2 r 3 − r 1 − r 2 √ 3 Odd or even under P12, but ( j = exp(2 i π/3)) P→[y + i x] = j [y + i x] , JMR Quark Model ,
Few-charge systems History of the quark model Mesons Baryons Multiquarks and other exotics Outlook Permutation symmetry for (qqq) The Clebsch–Gordan rules for two mixed-symmetry doublets z = v + i u and Z = V + i U are ℜe[z Z ∗ ] = u U + v V , ℑm[z Z ∗ ] = v U − u V , [z Z ] ∗ = (u U − v V ) − i (u V + v U) . So SM × SM → S, A, or SM. In particular, the coupling of three spins 1/2 to spin 1/2, with, say S3 = +1/2 is Sx = 1 √ 2 [↑↓↑ − ↓↑↑] , Sy = 1 √ 6 [2 ↑↑↓ − ↑↓↑ − ↓↑↑] , is completely analogous to (x, y) and form a SM doublet, So do the isospin wave function for (1/2) × (1/2) × (1/2) → (1/2) In the nucleon, the spin–isospin wave function is symmetric JMR Quark Model
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An introduction to the quark model

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