Problem Set #1, due Monday, September 13, 2004

**Problem 1: **(G. Baym, Lectures on Quantum Mechanics, p. 244,
#7)

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 V_{i}, thermally insulated from its
surroundings, at temperature T_{ i}=0.
The density of states is D(E) = (V_{i}/2pi^{ 2})(2M/hbar^{2})^{3/2}E^{1/2}.
Let the gas expand irreversibly into a vacuum, without doing work, to a
final
volume V_{f} .

(a) What is the temperature of the gas after expansion if V_{f}
is sufficiently large for the classical limit to apply?

(b) How large should V_{f} in fact be for the classical limit
to apply?