Consider collisions of pions with nucleons. View the pions as a \(T=1\) isospin triplet \(\pi=\left(\pi^{+}, \pi^{0}, \pi^{-}\right)\), and the nucleon as a \(T=\frac{1}{2}\) isospin doublet, \(N=(p, n)\). The combined system may be coupled to a total isospin \(T=\frac{3}{2}\) or \(\frac{1}{2}\). Write the wavefunctions for all possible states \(\left|T M_{T}\rightangle\) of the coupled \(\pi-N\) system in terms of the uncoupled products \(|\pi Nangle \equiv|\piangle|Nangle\). Use unitarity of the Clebsch-Gordan coefficients to invert these expressions and show that

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|np)= = 3 for the uncoupled wavefunction |)|p)= |p). MT =