Using the pion triplet ,

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Particle Physics & Quantum Fields · Symmetry, groups and the quark model

Symmetry and Flavour, Ch Isospin as a group: SU(2), Clebsch-Gordan ratios and G-parity

Problem

Using the pion triplet ∣1,1⟩=π+|1,1\rangle=\pi^+, ∣1,0⟩=π0|1,0\rangle=\pi^0, ∣1,−1⟩=π−|1,-1\rangle=\pi^- and the nucleon doublet ∣12,12⟩=p|\tfrac12,\tfrac12\rangle=p, ∣12,−12⟩=n|\tfrac12,-\tfrac12\rangle=n, the 1⊗121\otimes\tfrac12 table gives ∣32,12⟩=23 ∣π0p⟩+13 ∣π+n⟩,∣12,12⟩=−13 ∣π0p⟩+23 ∣π+n⟩.|\tfrac32,\tfrac12\rangle=\sqrt{\tfrac23}\,|\pi^0p\rangle+\sqrt{\tfrac13}\,|\pi^+n\rangle,\qquad |\tfrac12,\tfrac12\rangle=-\sqrt{\tfrac13}\,|\pi^0p\rangle+\sqrt{\tfrac23}\,|\pi^+n\rangle. Invert these two relations to express the physical state ∣π+n⟩|\pi^+n\rangle as a combination c1∣32,12⟩+c2∣12,12⟩c_1|\tfrac32,\tfrac12\rangle+c_2|\tfrac12,\tfrac12\rangle, giving the coefficients c1c_1 and c2c_2, and check that c12+c22=1c_1^2+c_2^2=1.

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