midterm solutions 2013 (50 marks)
Question 1) (10 marks)
a) A system has 3 states and the temperature is such that the probabilities of occupying each state
are equal. What is the Gibbs entropy for this system?
Solution:
S = kB
Pi ln Pi = kB (P1 ln P1 + P

NE 334 Problem set 1
Solutions (/20)
1. (/5) N = 99 molecules are placed in a container of volume V . Assume ideal gas behaviour at
equilibrium.
(a) What is the probability to nd one third the molecules in a section that corresponds to
33 % of the volume

NE334 midterm solutions
55 marks in total
Question 1) (5 marks)
a) State the fundamental postulate of statistical mechanics.
All states are equally probable for a systems with a fixed total energy.
or
1 mark
principle of equal a priori probability: P =
1

NE334 midterm solutions
Question 1) (5 marks)
a) What is the Boltzmann entropy for a system with only one five-fold degenerate energy level?
=5
S = kB ln
S = kB ln 5 1 mark
b) What is the probability of occupying a particular state in the microcanonical e

midterm solutions 2013 (50 marks)
Question 1) (10 marks)
a) A system has 3 states and the temperature is such that the probabilities of occupying each state
are equal. What is the Gibbs entropy for this system?
Solution:
S = kB
Pi ln Pi = kB (P1 ln P1 + P

midterm solutions 2014 (60 marks)
Question 1) (25 marks)
One mole of an ideal gas of identical, independent, and indistinguishable atoms of mass m is placed in
a closed container of volume V at temperature T . Only one singly-degenerate nuclear level is t

NE334: Lecture 30
July 22, 2014
Simulations of nano-scale
systems
What to do when you
dont know Z!
1
Fundamentals
In order to perform molecular simulations, we need:
Statistical mechanics (because we cannot
specify everything!)
Quantum Mechanics (thats

NE 334 Lecture 34, July 30 2014
Reading: This and next lectures: Chapter IX
FermiDirac Statistics and Intrinsic Semiconductors.
Crystalline Solids:
Each atom/ion brings electrons to assembly of atoms making up solid: some are called atomic
core, some ato

NE 334 Lecture 33, July 29 2014
Reading: This and next lectures: Chapter IX
The StronglyDegenerate FermiDirac Gas.
Low temperatures and/or high density: consider, for example, the freeelectron model of
metals
for electrons, as we have only translational

NE 334 Lecture 32, July 24 2014
Reading: This and next lectures: Chapter IX
A dierent limiting case is of importance for BoseEinstein particles. We know that nk cannot
be negative; moreover, we may write nBE as
k
nBE
k
e k
=
1 e
k
1
=
1
e
k
1
.
We may al

NE 334 Lecture 31, July 23 2014
Reading: This and next lectures: Chapter IX
FermiDirac and BoseEinstein Statistics.
FermiDirac distribution
Two cases :
BoseEinstein distribution
(for fermions)
(for bosons)
These are the only exact distributions.
N molecu

NE 334 Lecture 29, July 22 2014
Reading: This and next lectures: Bose-Einstein and Fermi-Dirac Statistics; Secs. IX.1-IX.4
Quantum statistics: setting the stage.
Until now, quantum restrictions upon wavefunctions have not had to be considered except
when

NE 334 Problem set 2
Solutions
1. (/3) A system has non-degenerate energy eigenvalues En = an2 + bn + c. The system is at nite
temperature.
(a) (1 mark) Obtain an expression for the canonical partition function, Z, of the system.
2 +bn+c)
e(an
Z=
n
(b) (1

NE 334 Tutorial 2
1. You toss a coin six times
(a) What is the probability that tail is up six times?
1
Probability of 6 tails = ( 1 )6 = 64
2
(b) What is the probability that tail is up three times?
!
Number of ways of getting three tails is 3 6 3 ! = 20

midterm solutions 2014 (60 marks)
Question 1) (25 marks)
One mole of an ideal gas of identical, independent, and indistinguishable atoms of mass m is
placed in a closed container of volume V at temperature T . Only one singly-degenerate nuclear
level is t

NE 334 Problem set 3
solutions
Assigned questions (show all your work):
1. (2 marks) Calculate the rotational partition function for
B
=0.244 cm1 ) at 150 K and at 350 K.
(hc)
35
Cl2 (the rotational constant is
zrot =
1
kB T
(0.695 cm1 K 1 )(150 K)
=
=
=

NE 334 Problem set 2
Solutions
1. A system has non-degenerate energy eigenvalues Em = m2 . The system is at nite temperature.
(a) (1 mark) Obtain an expression for the canonical partition function, Z, of the system.
em
Z=
2
m
(b) (1 mark) Obtain an expres

NE 334 Problem set 1
Solutions
1. You randomly pick a letter from a set of 26 characters, the roman alphabet, 3 times (you put
the letter back in the set after each try)
(a) (3 marks) What is the probability that you will pick the letter A only once?
The

midterm solutions 2015 (50 marks)
Question 1 (10 marks)
a) (/2) Provide an expression for the characteristic function of the microcanonical ensemble and show that
it is equivalent to the Gibbs entropy formula.
Solution The characteristic function of the m

NE 334 Tutorial 3 (May 23, 2014)
1. Consider two systems, A and B, each with the same ground state energy. Both systems have only one
excited energy level lying 300 cm?1 above the ground state. The excited energy level for system A
is nondegenerate, while

NE 334 Tutorial 1
1. What is the probability of throwing a total of six points or less with two honest dice?
2. If in a factory producing bolts, there is a probability 0.05 that a defective bolt will be produced,
what is the average number of defective bo

NE 334 Lecture 28, July 18 2014
Reading for this lecture: Chapter VIII Next lectures: Bose-Einstein and Fermi-Dirac Statistics;
Secs. IX.1-IX.4
nal exam info
problem set 4 (simulations)
Examples:
(a) Dissociation of I2 .
Let us consider the gas phase diss

NE 334 Lecture 27, July 11 2014
Reading for this lecture: Chapter VIII Next lecture: Bose-Einstein and Fermi-Dirac Statistics;
Secs. IX.1-IX.4
KP =
(zC /N0 )C (zD /N0 )D U /RT
0
.
e
(zA /N0 )A (zB /N0 )B
(1)
Aside : Some practical matters:
We shall examin

NE 334 Lecture 13, May 30, 2014
This lecture: Section III.1 in text.
Next Lecture: Section III.2 in text
Entropy and its statistical mechanical interpretation (continued).
Lets rewrite the expression that we obtained previously for the entropy, i.e.,
S =

NE 334 Lecture 8, May 21, 2014
The Canonical Ensemble.
Let us examine in some detail what happens when two systems can exchange energy between
them, i.e., when two systems are not individually isolated systems.
A
A
A*
Can we make any quantitative statemen

NE 334 Lecture 12, May 29, 2014
Section III.1 in textbook (this lecture and upcoming lectures)
Swap Lecture < > Tutorial:
Lecture tomorrow at 16:30 instead of tutorial
Tutorial Tuesday June 2nd at 14:30 instead of lecture
macroscopic result:
W = X dx ,
Er