4. Consider two indistinguishable spin-1/2 fermions in a one-dimensional infinite square well of length L. Construct the state vector and determine the energy for the ground-state of the two-particle system assuming the particles are non-interacting. i. ii. Determine the first excited state energy of the two-particle system and give all its state vectors, again assuming the particles are non-interacting. What is the degeneracy of the first excited state? ji. Suppose the particles interact via the potential V(x, x2) = k(x1 – x2)² where k is positive and small. Discuss quantitatively how the ground and first-excited state energies differ from the case for non-interacting particles. 5. A one-electron atom has atomic number Z, mass number M and a spherical nucleus of radius ry. Assume electric charge +Ze is uniformly distributed throughout the volume of the nucleus. Ignoring spin, use first order non-degenerate perturbation theory and the hydrogenic wave functions adapted to the one-electron atom to determine the dependence of the ground state energy of the atom on ry.

4. Consider two indistinguishable spin-1/2 fermions in a one-dimensional infinite square well of length L. Construct the state vector and determine the energy for the ground-state of the two-particle system assuming the particles are non-interacting. i. ii. Determine the first excited state energy of the two-particle system and give all its state vectors, again assuming the particles are non-interacting. What is the degeneracy of the first excited state? ji. Suppose the particles interact via the potential V(x, x2) = k(x1 – x2)² where k is positive and small. Discuss quantitatively how the ground and first-excited state energies differ from the case for non-interacting particles. 5. A one-electron atom has atomic number Z, mass number M and a spherical nucleus of radius ry. Assume electric charge +Ze is uniformly distributed throughout the volume of the nucleus. Ignoring spin, use first order non-degenerate perturbation theory and the hydrogenic wave functions adapted to the one-electron atom to determine the dependence of the ground state energy of the atom on ry.