Question

Explain which of the following combination of particles can or cannot exist in I = 1 state

(a) π⁰π⁰ 

(b) π⁺π⁺ 

(c) π⁺π⁻ 

(d) Λπ⁰ 

(e) Σ⁰π⁰ 

(f) π⁻π⁻?

Answer 1

Pions obey Bose statistics and the system of pions must be symmetric upon interchange of space and isospin coordinates of any two pions. For each pion T = 1 and for a system of two pions, total isospin, I = 2, 1, 0. The corresponding states are

Now ψ(total) = ψ (space)ψ (isospin) 

Thus for the symmetric states like |2, ±1 or |0, 0 , ψ (space) must be symmetric because of Bose symmetry. But for two pions the interchange of the spatial coordinates introduces a factor (−1)^l so that only even l states are allowed. Thus, (π⁰, π⁰), (π⁺, π⁺), (π⁻, π⁻) in (a), (b) and (f) will be found in the states L = 0, 2,... However, the states |1, ±1 , |1, 0 are antisymmetric and this requires, L = odd, that is L = 1, 3,... (c) being antisymmetric can have odd values for L, that is L = 1, 3,... (d) and (e) can exist in I = 1 regardless of the L values because of two particles in these two cases are fermion and boson, so that the previous considerations are inconsequential.