Description
Problem 1:
A solid is composed of N atoms whose nuclei have angular momentum ¯h/2 and thus a
nuclear magnetic moment of magnitude µ. When the solid is placed in a homogeneous
field along a given axis, the magnetic moment of each of the atoms can have the value µ or
−µ.
Let us assume that the interaction between the magnetic moments and the external
field is much, much stronger than the interaction between neighboring magnetic moments.
(a) Use the canonical ensemble to calculate the internal energy of the solid in the external
field at temperature T.
(b) Use the canonical ensemble to calculate the entropy of the solid in the external field at
temperature T.
Problem 2:
(a) An experimentalist measures the specific heat CV of a gas of non-interacting onedimensional particles of mass m at temperature T. The potential of the j-th particle is
given by
V (xj ) = 0
xj
a
n
, (1)
where 0 and a are constants with units “energy” and “length”, respectively, and n is an
integer, n = 1, 2, · · ·.
From the measurements, can the experimentalist determine the value
of n?
(b) The experimentalist repeats the measurement for a gas of non-interacting two-dimensional particles of mass m at temperature T for which the potential of the j-th particle is
given by
V (xj
, yj ) = 0
ρj
a
n
, (2)
where ρj =
q
x
2
j + y
2
j
. What is CV in this case?
The following integral may be helpful:
Z ∞
0
exp(−z)z
(1−n)/ndz = Γ(1/n). (3)
Problem 3:
For zero-mass particles, the energy-momentum relation is E = c|~p|. This relationship can
also be used to approximate “ordinary” particles when kT mc2
.
A gas described by this
relationship is sometimes referred to as an extremely relativistic classical gas.
Assuming three-dimensional space, calculate the pressure and energy per particle of such
a gas as functions of the density and temperature.
Problem 4:
This problem serves as a review of undergraduate quantum knowledge in preparation for
quantum statistical mechanics.
(a) A one-dimensional harmonic oscillator potential is a potential of the form
V (x) = 1
2
kx2
. (4)
What is the energy and degeneracy of the ground state of a system consisting of five noninteracting particles of mass m that are confined by V (x) in the cases that
(ai) the particles are spin-0 bosons,
(aii) the particles are spin- 1
2
fermions,
(aiii) the particles are spin- 1
2
bosons,
(aiv) the particles are spin-0 fermions, and
(av) the particles are spin- 5
2
fermions?
(b) Repeat part (a) for the isotropic two-dimensional harmonic oscillator potential.
(c) Which of the cases (ai)-(av) is physically possible/impossible? Explain.
