1. The deuteron, comprising a proton and a neutron, may be modelled using a three-dimensional spherically symmetric well which extends to radius r = a. The potential V = -Vo, for r < a and V = 0, for r > a. (a) Starting from the radial equation (derived from the Schrödinger equation): dR dr d dr (2d) - 2µr² [V (r) – E] R = 1(1 + 1)R, where is the reduced mass of the deuteron, find the appropriate equations μ for u(r) = rR(r), both inside and outside of the well. (b) For a state with angular momentum 1 = 0, show that the solution inside the well is of the form u(r) = Bsin(kr) and find k. Justify any boundary conditions that you use to arrive at this solution. (c) The = 0 state of the deuteron is bound and has a negative energy (E). Find the exponential solution, in terms of a constant K, outside of the well. Find and justify any boundary conditions that you use to arrive at this solution. (d) Show that the energies of states with quantum number = 0 are determined by the condition k cot(ka) = -K. (e) The binding energy of the ground state of the deuteron is -2.23MeV and the size of the well is a = 2.0 × 10-15 m, find the corresponding value of Vo. To determine the reduced mass of the deuteron system you can take the masses of the proton and the neutron to be the same. Hint: x 1.82 is a solution to the equation x cot x = -0.46. =

1. The deuteron, comprising a proton and a neutron, may be modelled using a three-dimensional spherically symmetric well which extends to radius r = a. The potential V = -Vo, for r < a and V = 0, for r > a. (a) Starting from the radial equation (derived from the Schrödinger equation): dR dr d dr (2d) - 2µr² [V (r) – E] R = 1(1 + 1)R, where is the reduced mass of the deuteron, find the appropriate equations μ for u(r) = rR(r), both inside and outside of the well. (b) For a state with angular momentum 1 = 0, show that the solution inside the well is of the form u(r) = Bsin(kr) and find k. Justify any boundary conditions that you use to arrive at this solution. (c) The = 0 state of the deuteron is bound and has a negative energy (E). Find the exponential solution, in terms of a constant K, outside of the well. Find and justify any boundary conditions that you use to arrive at this solution. (d) Show that the energies of states with quantum number = 0 are determined by the condition k cot(ka) = -K. (e) The binding energy of the ground state of the deuteron is -2.23MeV and the size of the well is a = 2.0 × 10-15 m, find the corresponding value of Vo. To determine the reduced mass of the deuteron system you can take the masses of the proton and the neutron to be the same. Hint: x 1.82 is a solution to the equation x cot x = -0.46. =