Statistical Mechanics - Physics at Oregon State University

6.5. PROBLEMS FOR CHAPTER 6 137
A system is described by a state in a two-dimensional Hilbert space. The
Hamiltonian is given by
∗ ɛ α
H =
α 2ɛ
Assume that ɛ ≫ |α| and that βɛ ≪ 1.
(A) Calculate the partition function up to second order in β.
(B) Calculate the high temperature limit of the heat capacity.
(C) Suppose that α = N
V and that the density is small. Calculate the pressure
in the large volume and high temperature limit.
Problem 4.
A quantum mechanical system is described by a Hamiltonian H = H0 + κV ,
with [H0, V ] = 0. κ is a small constant. The Helmholtz free energy is Fκ(T ).
Calculate the change in Helmholtz free energy, ∆F = Fκ − F0 for this system
up to second order in κ
kBT .
Problem 5.
In a two-dimensional Hilbert space the density operator is given by its matrix
elements:
x R
ρ =
R∗
1 − x
This form is clearly Hermitian and has trace one. Calculate the entropy as
a function of x and R, and find the values of x and R that make the entropy
maximal. Note that you still need to check the condition that the matrix is
positive! Also, show that it is a maximum!
Problem 6.
A quantum mechanical system is described by a simple Hamiltonian H,
which obeys H 2 = 1. Evaluate the partition function for this system. Calculate
the internal energy for T → 0 and T → ∞.
Problem 7.
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.