81. The bag model of hadron structure has colored quarks moving as independent spin-half Dirac…

81. The bag model of hadron structure has colored quarks moving as independent spin-half Dirac particles within a spherical cavity of radius R. To obtain wave functions for particular hadron states, the individual quark “orbitals” must be combined to produce states of zero total color and the appropriate values of the spin and flavor (isospin, charge, strangeness) quantum numbers. In the very good approximation that the “up” and “down” quarks are
View complete question »                          81. The bag model of hadron structure has colored quarks moving as independent spin-half Dirac particles within a spherical cavity of radius R. To obtain wave functions for particular hadron states, the individual quark “orbitals” must be combined to produce states of zero total color and the appropriate values of the spin and flavor (isospin, charge, strangeness) quantum numbers. In the very good approximation that the “up” and “down” quarks are massless one can easily obtain the lowest energy (s-wave) bag orbitals. These are given by the Dirac spinor where x is a 2-component spinor, k = 2.04/R, jl =spherical Bessel function. (a) The lowest-lying baryons (proton and neutron) are obtained by putting three quarks in this orbital. How would you construct the wave function for the proton and for the neutron, i.e., which quarks would be combined and what is the structure of the spin wave function consistent with the quantum numbers of proton and neutron and Pauli’s principle? (b) The magnetic moment operator is defined as  JEM, where JEM is the usual Dirac electric current operator. Find an expression for this operator in terms of the spin operators of the constituent quarks. (You may leave integrals over Bessel functions undone.) (c) Show that _n/_p = _2/3. You may need the following Clebsch–Gordon coefficients:              

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