Statistical Mechanics - Physics at Oregon State University

7.9. PROBLEMS FOR CHAPTER 7 155
C. Show that the dielectric constant ɛ in the high temperature limit is given
by ɛ = 1 + 4πNd2
3V kT .
Problem 3.
The Hamiltonian for a classical system of two identical particles is given by
H(r1, r2, p1, p2) = 1
2m (p2 1 + p 2 2) + 1
2 mω2 (r1 − r2) 2
Evaluate the Helmholtz free energy and the internal energy of this system
as a function of temperature.
Problem 4.
The positions of the atoms in a one-dimensional solid are given by rn =
na + xn , n = 1 · · · N. Neighboring atoms are connected by springs, and the
Hamiltonian is
H(x1, · · · , xN , p1, · · · , pN) = 1
2m
N
i=1
p 2 i + K
2
N
i=2
(xi−1 − xi) 2
Evaluate the internal energy and heat capacity of this solid.
Problem 5.
Consider a system of N identical particles in one dimension. The positions
xn of the particles are connected via |xn − xn+1| = a. This is a model of a
polymer chain with N − 1 links. The total length of the chain is L = |x1 − xN|.
The potential energy of the n-th particle is mgxn. Calculate the coefficient of
linear thermal expansion for this system, and show that it is negative, like for
a rubber band. Ignore kinetic energy terms in the energy.
Problem 6.
The Hamiltonian for a classical single molecule consisting of two atoms is
given by:
H(r1, r2, p1, p2) = 1
2m (p2 1 + p 2 2) + K
2 (|r1 − r2| − d) 2
Evaluate the Helmholtz free energy and the internal energy of a system of
N independent molecules of this type as a function of temperature.
Problem 7.