Problem 1: (G. Baym, Lectures on Quantum Mechanics, p. 244,
Estimate the ground state energy of the hydrogen atom using a three-dimensional harmonic oscillator ground state wave function as a trial function. Express your answer in Ry, compare to the exact result, and comment.
Problem 2: Irreversible expansion of a Fermi gas
(adapted from C. Kittel, Thermal Physics, 2nd edition, p.259, #10)
Consider a gas of N free noninteracting spin 1/2 fermions of mass M, initially in a volume Vi, thermally insulated from its surroundings, at temperature T i=0. The density of states is D(E) = (Vi/2pi 2)(2M/hbar2)3/2E1/2. Let the gas expand irreversibly into a vacuum, without doing work, to a final volume Vf .
(a) What is the temperature of the gas after expansion if Vf is sufficiently large for the classical limit to apply?
(b) How large should Vf in fact be for the classical limit to apply?