Problem 1 (4 marks): Two identical spin fermions move in one dimension under the
influence of the infinite-wall potential V = for x < 0, x > L , and V = 0 for 0 x L .
(a) Write the ground-state coordinate wave function and the ground-state energy
Problem 1: Use the WKB approximation to find the allowed energies of the potential well:
V ( x) = x . Find the expectation value of kinetic energy (in the WKB approximation) for the
nth stationary state.
Solution: Classical turning points can be found by
Problem 2 (3 marks): The exact fine-structure formula for hydrogen (obtained from the Dirac
equation without recourse to perturbation theory) is
= m c2
n ( j + 1 / 2) + ( j + 1 / 2)
Expand to order of 4 (noting that < 1 ), and sh