Physics 4523, Statistical Physics, Spring 2016
March 11, 2016
_________________________
EXAM 2
Name and Signature
Please do all of the problems. The problems are not all worth equal points. Remember to work each
problem as fully as possible. SHOW ALL WORK as partial credit will be given.
You are allowed one formula sheet only for this test, IT MUST BE IN YOUR OWN HANDWRITING
and you can have items on both sides. You are also allowed a non-programmable calculator. Your name
and signature above indicates that you have obeyed the honor system on this test and have not received or
given aid to or from anyone on this test. Show All Your Work!!!!
2.9979 × 108 m / s
c=
1.38065 ×10−23 J / K =
8.6173 ×10−5 eV / K
kB =
h = 6.6261×10−34 J − s
N avogadro = 6.0221 × 1023 / mole
=
 1.05457148 ×10−34 J − s
R
=
−19
1 eV =
1.6022 ×10 Joules
8.3145 J / mol / K
9.8 m / s 2
g=
1. (5 points) Suppose that a student (in frustration over a homework problem) drops their “Introductory
Statistical Mechanics” textbook (0.2 kg) from a height of 1.5 m. Assume that when the book hits the
ground, all the excess kinetic energy is dissipated as heat into the atmosphere (at T = 300 K). By
what factor does the number of accessible microstates (of the atmosphere) increase as a result of this
event? (You may give the natural logarithm of the factor, rather than the factor itself.)
2. (5 points) A system has two non-degenerate energy levels with an energy gap of 0.015 eV. What is
the probability of the system being in the lower level if it is in thermal contact with a heat bath at a
temperature of 300 K?
3. (10 points) The partition function of a system is given by the equation
Z = eαT V
2
where α is a constant. Calculate : a) the pressure, b) the entropy and c) the internal energy of the
system.
4.
(5 points) A three-dimensional isotropic harmonic oscillator has energy levels
ε n ,n =
ω (n1 + n2 + n3 + 3 / 2),
,n
1
2
3
where each of the n1 , n2, and n3 can be 0, 1, 2, 3, ……
a. Find the degeneracies of the levels of energy 5ω / 2 and 7 ω / 2 .
b. How large must T be for the 7 ω / 2 level to be more highly occupied that the
5ω / 2 level?
5. (5 points) Calculate the average internal energy U of a one-dimensional quantum harmonic
oscillator. Hint: In class we showed that U =
∂ ( β F ) / ∂β where β = 1/ (k BT ).
6. (10 points) Consider a system of two atoms, each having only three, non-degenerate single-particle
states of energy 0, ε , 2ε . The system is in contact with a heat bath at temperature T. Write down
the energy levels and partition function given that the particles obey:
a. Classical statistics because the particles are indistinguishable;
b. Fermi-Dirac statistics because they are indistinguishable Fermi particles;
c. Bose-Einstein statistics because they are indistinguishable Bose particles.
Do not worry about spin or spin degeneracy in this problem.