Summary: Physics 607 Exam 1
Please be well-organized, and show all significant steps clearly in all problems.
You are graded on your work, so please do not just write down answers with no explanation!
The variables have their usual meanings: E = energy, S = entropy, V =volume, N =number of particles,
T = temperature, P pressure, u = chemical potential, B = applied magnetic field, C = heat capacity at
constant volume, F = Helmholtz free energy, k = Boltzmann constant. Also, (·.·) represents an average.
1. Here we will deal with an ideal quantum gas of particles which is generalized in two ways:
(i) The gas is in D dimensions. (ii) The grand partition function has the form
where k labels a single-particle state, but all we know about Zk is that
and is the energy of state k. Recall that the Landau free energy (or grand potential) is defined by
(a) (5) Starting with the expression for dE, obtain dQ in terms of dT, dV, and du.
(b) (5) Using Q = --kT in Z, obtain a simple general expression for the average number of particles (ak) in the
single-particle state labeled by k, in terms of Zk and Zk.
(c) (5) Using Euler's theorem, obtain an expression for PV in terms of 2, and then in terms of the Zk.
(d) (5) For our ideal gas (which we will take to consist of spinless particles), k just corresponds to the
momentum p. Replace the sum for PV by an integral over p, using the density of states in momentum space
p(p). [You do not need to obtain an expression for p(p) ; just leave the integral in terms of p(p)dp .j