Physics 4523, Statistical Physics, Spring 2016
April 25, 2016
_________________________
FINAL EXAM
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 nonprogrammable 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!!!!
c=
2.9979 × 108 m / s
kB =
1.38065 ×10-23 J / K =
8.6173 ×10-5 eV / K
h = 6.6261×10-34 J - s
N avogadro = 6.0221 × 1023 / mole
 1.05457148 ×10-34 J - s
=
1 eV =
1.6022 ×10-19 Joules
=
R
8.3145 J / mol / K
g=
9.8 m / s 2
s = 5.670367 (13) ×10-8 W m -2 K -4
Useful Integrals:
∞
x n e ax n n −−
n!
n ax
x
e
dx
=
− ∫ x 1e ax dx ; ∫ x n e bx dx =
(n =
integer)
∫
a
a
b n +1
0
∞
2 n +1 − x /a
=
∫ x e dx
2
2
0
∞
2 n − x /a
=
∫ x e dx
2
0
2
n! 2n+2
=
a
[ n 0, 1, 2, .....]
2
2n − 1) !!
(=
π a 2 n +1
2
n +1
1 ⋅ 3 ⋅ 5 ⋅ ⋅ ( 2n − 1)
=
π a 2 n +1 [ n 0, 1, 2, .....]
2n +1
1. (15 pts) Fermi and Bose statistics. You are given a system of two identical particles which
may occupy any of the three energy levels=
ε , n 0, 1, 2. The lowest energy state,
En n=
ε 0 = 0 , is doubly degenerate (as shown in the figure below). The system is in thermal
equilibrium at temperature T.
2ε
ε
0
For the two following cases:
(i) the particles obey Fermi statistics
(ii) the particles obey Bose statistics
a. (5 pts) enumerate all possible occupation configurations;
b. (5 pts) determine the partition function Z;
c. (5 pts) calculate the internal energy U;
2. (5pt) How many ways can two indistinguishable particles obeying Bose-Einstein statistics be
arranged among three states?
3. (5pt) To a fair approximation, the sun can be described as a gigantic spherical blackbody at
5800 Kelvin, with radius 695,500 km, located 150 million km from the earth. Calculate the
sun’s irradiance at the earth’s orbit. That is, find the flux of solar energy that is incident on a
square meter that is located above the earth’s equator. Give your answer in W/m2.
4. (15 pts) Two Dimensional Fermi Gas. Consider a non-relativistic Fermi gas at absolute
zero temperature, ε (k ) =  2 k 2 / 2m. The gas consists of N particles (spin-1/2) moving in a twodimensional area A. For T = 0:
a. Find expressions for both the Fermi wavevector kF and the Fermi energy EF for this gas,
as functions of N and A.
b. Find the average kinetic energy per particle, u = E/N and express it in terms of EF.
c. What is the density of states (in energy) at the Fermi level D( EF ) ? Give your answer in
terms of EF .
5. (10 pts) Relativistic Gas. According to the theory of relativity, the energy E of a spin-1/2
particle with mass m and momentum p = k is =
E 2 p 2 c 2 + m 2 c 4 . Suppose that the
temperature is very high, so that k BT 2 mc 2 . Find the partition function Z for one such
particle in a volume V (i.e. three dimensions).
6. (15 pts) Consider a protein molecule that has two identical sites for binding oxygen: When in
contact with a reservoir of oxygen, each molecule of the protein will therefore bind either zero,
one, or two molecules (one on each site) of oxygen. Suppose that when one of the sites has no
oxygen molecule, the site has zero energy. When the site has an oxygen molecule, the site has
energy ε .
a.
Write down the grand partition function Ξ for the protein molecule, in terms of ε , µ and
kBT. Watch out for any possible degeneracy.
b.
Note that Ξ can be written as the square of a simpler expression. Now find the grand
potential Φ G .
c.
Use Φ G . to calculate the average oxygen saturation of the protein ( N , the average
number of oxygen molecules N bound). Give
N in terms of n (the oxygen number
density), ε, nQ (quantum density) , and basic constants. Hint: write µ in terms of n, nQ etc.
7. (5 pts) A large isolated system of N non-interacting particles is in thermal equilibrium. Each
particle has only three possible non-degenerate states of energies 0, ε , and 3ε . In the high
temperature limit k BT  ε , what is the average energy of each particle?
8. (10 pts) Consider a low density gas of hydrogen atoms at very high temperature T. Then the
hydrogen can dissociate (or ionize) to give a proton and electron:
p + + e− ↔ H
where n p = number density of protons
g=2
ne = number density of electrons
nH = number density of hydrogen atoms
proton mass = 1.67 × 10-27 kg
electron mass = 9.11 × 10-31 kg
H-atom mass ≈ proton mass
g=2
g=4
We will ignore the bound state spectrum of the H atom: Assume that every H atom is in its
ground state, which has ionization energy ∆ = 13.6 eV = 2.18 × 10-18 J.
(a) Find the condition that must hold between the number densities n p , ne and nH at thermal
equilibrium. Your answer should include the quantum density nQ ' = g(mkBT/2πħ2)3/2 of each
species.
(b) Suppose that at temperature T = 5000 K, exactly half of the hydrogen is ionized:
n=
n=
nH . What is the number density of electrons under this condition? Give your answer
p
e
-3
in m .