Question

Question 1 (8 marks in total) The deuteron is a bound state of a proton and a neutron. Treating nucleons as identical particles with spin and isospin degrees of freedom, the total state of the deuteron can be writ- ten space Ψ spin Ψ isospin. The deuteron has a total angular momentum quantum number J - 1 and a total spin S -1. Our goal is to determine the parity of the deuteron Q1-1 (1 mark) Show that the possible values of the total orbital angular momentum quantum number L are 0, 1, and 2. Q1-2 (2 marks) Using the fact that the strong nuclear interaction is charge independent (.e, it does not matter if the nucleon is a proton or a neutron), and the fact that the di-neutron is not bound, show that uIisospin is antisyn Q1-3 (2 marks) Using the results in the above questions, determine the parity of the deuteron and show that L = 1 is not allowed. Hint: Exchanging r and r is equivalent to a paritytransformation. e.q., a positive parity corresponds to a wave-function which is symmetic under the exchange of the particles According to quantum field theory, if the strong nuclear interaction occurs from the exchange of one meson only with spin-parity 0, then the interaction between two nucleons in the non- relativistic limit would be given by a Yukawa potential -Mr where r(1) (2)l, M is the mass of the meson and as, is a constant. Q1-4 (2 marks) Write down the static Hartree-Fock equation with this interaction. Solving HF equations requires numerical methods due to the self-consistency of the mean- field Hamiltonian. However, we can easily guess that, as in the atomic case, the lowest single particle level is a ls state, ie, with orbital angular momentum 1 0. Having a proton and a neutron in these states would then produce a L = 0 state. In reality, the deuteron wave-function has a small component (about 4%) of L = 2 state as well, indicating that other mesons with other spin-parities contribute to the nuclear interaction. These mesons produce a tensor term to the interaction which, in the non-relativistic limit, can be written as where s1,2 are the spins of the nucleons. Q1-5 (1 mark) Show that the tensor interaction is invariant under parity transformation P and time reversal transformation T.

Answer 1

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