Homework Set # 9 - Due November 6

1. In a two-dimensional Hilbert space the density operator is given by its matrix elements:

x R

R* (1-x)

a) Show that the density matrix has the expected value for its trace.

b) Calculate the entropy as a function of x and R.

2. A system is described by a density operator ρ. In this problem, the eigenvalues of this operator are either 0 or 1. The number of particles in the system is N and the volume of the system is V.

a) How many eigenvalues of ρ are equal to 1?

b) What is the entropy of the system?

c) Assume that the particles are independent. If a single particle is described by a density operator ρ_i, how do you construct ρ from ρ_i?

d) What would be the entropy if they eigenvalues of ρ where either 0 or 1/N?

3. Using the density matrix for a free particle in a box show that E=<H>=3kT/2.

4. Problem 5.4

5. Problem 5.5 (Hint: use that P/kT=(∂ln Z_N/∂V)|_N,T)