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Chapter 9 - Symmetry and Quark Model - Coggle Diagram

- - - - Total isospin \(T^2 = T_1^2 + T_2^2 + T_3^2\) commutes with \(T_i\) and is Hermitian -> observable.
      - \([T_i, T_j] = i\epsilon_{ijk}T_k\); components cannot be simultaneously known.
      - Therefore, denote a state by \(\phi(I, I_3)\)
        
        \(T^2\phi(I, I_3) = I(I+1)\phi\)
        
        \(T_3\phi(I, I_3) = I_3\phi\)
        
        \(T_{\pm}\phi(I, I_3) = \sqrt{I(I+1) -I_3(I_3\pm1)}\phi(I, I_3\pm 1)\)
- - - - Use totally symmetric isospin/spin states: isospin-3/2 and spin-3/2. \(\Delta\) baryon!
      - combination of mixed-symmetry states: \(\frac{1}{\sqrt{2}}(\phi_S\chi_S + \phi_A\chi_A)\)
  - - - Triplet (I=1): total sym. Singlet (I=0): total anti-sym.
    - - Mixed symmetry: S or A under exchanging first two quarks. Respective isospin: 3/2, 1/2, 1/2
    - - Respective isospin: 1, 0 (singlet)