The nuclear shell model is characterized by a set of single particle states with quantum numbers (j, l, s) ,where l = 0, 1, 2 ... is the orbital angular momentum quantum number, s =1/2 for a nucleon, and j is the total angular momentum quantum number, obtained from angular momentum coupling: j = l + s. Each single particle shell model state can accommodate up to (2 j +1) protons and (2 j + 1) neutrons, corresponding to a distinct set of quantum numbers (mj, ml) for the identical nucleons, consistent with the Pauli exclusion principle. Keeping careful track of the ‘magnetic’ quantum numbers allows one to deduce the total angular momentum for a collection of nucleons in a particular shell model state. The following exercises should be done with this approach:
a) Consider a shell model state j that is completely filled with protons, and show that the total angular momentum of this maximally occupied state is J = 0 , where
J= j1 + j2 + j3+ ... +jN (N = 2j+1).
b) Consider a shell model state j = 5/2 that contains 2 protons; work out the allowed total angular momentum quantum numbers J for this system.
c) Consider a shell model state j = 5 / 2 that contains 1 proton and 1 neutron; work out the allowed total angular momentum quantum numbers J for this system.