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HW 4 problems/solutions

# HW 4 problems/solutions - Homework 4 1 Homework IV and...

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Homework 4 1 11/1/2005 Homework IV and Solutions Problems: IV.1 Give arguments based on overall L =0, S = 3 / 2 of the three quarks inside the baryon (lowest-lying state), that the overall symmetry of the combined spin and spatial wave function must be symmet- ric under interchange of any pair of quarks. Why is this an additional argument for the existence of the color quantum number? Solution: The lowest lying state ( d d d ), all spins up (i.e. a symmetric spin wavefunction), has l =0 for any pair of quarks (i.e. a symmetric spatial wavefunction). The spin J = 3 / 2 . All three d -quarks are thus in an S -state, clearly violating the Pauli exclusion principle for identical fermions. Oth- erwise stated: the overall tri-quark wavefunction is symmetric under interchange, which violates Fermi-Dirac statistics. The way out of this dilemma is to “invent” a new quantum number that distinguishes quarks: col- or, adding yet another part to the overall wavefunction ψ color = (6) ½ (rgb+gbr+brg rbg bgr grb). IV.2 Give arguments based on the assignment I J PC = 1 1 for the ρ -meson, to explain that the decay π 0 0 is not observed. Hint: see notes, or e.g. Perkins. Solution: a. Parity conservation would be OK, assuming the two pions have relative orbital angular mo- mentum l =1: P (2 0 ) = ( ) l ( P ) 2 = ( ) 1 ( ) 2 = 1 = P . l =1 is required for angular momentum conservation because J =1 and J =0. b. Because l =1, we have two identical spin-zero bosons in the final state, with an overall anti- symmetry of the bi-pion wavefunction; this is not what is required for a final state of identical bosons and violates Bose-Einstein statistics! c. Charge parity (Charge Conjugation) would be violated , whereas strong interactions conserve C : C (2 0 ) = ( C ) 2 = ( ) 2 = +1 C = 1 Note that the above arguments do not apply to the decay + , ( G -parity conservation only allows even numbers of s!). + is therefore allowed!

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HW 4 problems/solutions - Homework 4 1 Homework IV and...

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