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FINAL EXAM, PHYSICS 5305, Spring, 2010, Dr. Charles W. Myles
Take Home Exam: Distributed, Friday, April 30
DUE, 5pm, TUESDAY, May 11!! NO EXCEPTIONS!
Bring it to my office or put it in my mailbox. (I prefer it in my mailbox! Put it in a sealed envelope!)
RULE: You may use almost any resources (library, internet, etc.) to solve these problems.
EXCEPTION: You MAY NOT COLLABORATE WITH ANY OTHER PERSON!
For questions/difficulties, please consult with me, not with other students (whether or not they are in this class!), with
people who had this course previously, with other faculty, with post-docs, or with anyone else I may have forgotten
to list here. You are bound by the TTU Code of Student Conduct not to violate this! Anyone caught violating
this will, at a minimum, receive an “F” on this exam!
INSTRUCTIONS: Please read all of these before doing anything else!!!
1. PLEASE write on ONE SIDE of the paper only!! This wastes paper, but it makes my grading easier!
2. PLEASE do not write on the exam sheets, there will not be room! Use other paper!!
3. PLEASE show all of your work, writing down at least the essential steps in the solution of a problem. Partial
credit will be liberal, provided that the essential work is shown. Organized work, in a logical, easy to follow
order will receive more credit than disorganized work.
4. PLEASE put the problems in order and the pages in order within a problem before turning in this exam!
5. PLEASE clearly mark your final answers and write neatly. If I cannot read or find your answer, you can't
expect me to give it the credit it deserves and you are apt to lose credit.
6. NOTE: The words EXPLAIN and DISCUSS mean to do this briefly, using complete, grammatically
correct English sentences! They DO NOT mean to answer with only equations! Answers to such
questions which are equations only with no explanation of what they mean will receive ZERO credit.
7. NOTE!!! The setup (PHYSICS) of a problem counts more heavily in the grading than the detailed
mathematics of working it out.
NOTE: I HAVE 16 EXAMS TO GRADE!!! PLEASE HELP ME
GRADE THEM EFFICIENTLY BY FOLLOWING THESE SIMPLE
INSTRUCTIONS!!! FAILURE TO FOLLOW THEM MAY RESULT IN A
LOWER GRADE!!
THANK YOU!!!
NOTE!!!! PROBLEM 1 IS REQUIRED! WORK ANY 5 OUT OF THE OTHER 6 PROBLEMS!
So, you must answer 6 problems total. Number 1 is worth 10 points. Each of the others is equally weighted & worth
18 points. So, 100 is the maximum points possible. Please sign this statement and turn it in with your
exam:
I have neither given nor received help on this exam Signature ___________________________
1. THIS QUESTION IS REQUIRED!!
a. Briefly DISCUSS, using WORDS, with as few mathematical symbols as possible, the
PHYSICAL MEANINGS of the following terms: 1. Microcanonical Ensemble, 2. Canonical
Ensemble, 3. Grand Canonical Ensemble, 4. Entropy, 5. Fermi Energy, 6. Pauli Exclusion
Principle, 7. Classical Statistics, 8. Quantum Statistics, 9. Equipartition Theorem. 10. Partition
Function.
b. Briefly DISCUSS, using WORDS, NOT symbols, the fundamental differences between
Fermions & Bosons & how these differences lead to the fundamentally very different Fermi-Dirac
& Bose-Einstein Statistics. (That is, what are the basic, intrinsic properties that distinguish Fermions &
Bosons?) In this discussion, be sure to mention the many particle wavefunctions for both kinds of
systems & include the qualitative differences expected between the many particle ground states of
the Fermi-Dirac & Bose-Einstein systems.
c. In class, we discussed two different models to calculate the lattice vibrational contribution to the
heat capacity at constant volume, Cv, of a solid. In Ch. 7, we first discussed the Einstein Model.
Then (for a couple of lectures) we went forward to Ch. 10 to discuss the Debye model. Briefly
DISCUSS, using WORDS, NOT symbols, the major differences between the Einstein & Debye
Models. Which model gives a theoretical temperature dependence for Cv at low temperature which agrees
with experiment?
NOTE!!!! WORK ANY 5 OUT OF PROBLEMS 2 through 7!
NOTE: The chapter numbers & problem numbers in the following are all from the book by Reif!
2. Work Problem #8 in Chapter 6.
3. Work Problem #10 and #12 in Chapter 7. Treat these as two parts of one problem.
4. Work Problems #20 and #21 in Chapter 7. Treat these as two parts of one problem.
5. Work Problems #12 and #13 in Chapter 9. Treat these as two parts of one problem.
6. Work Problems #16, #17, and #18 in Chapter 9. Treat these as three parts of one problem.
7. A solid contains N identical, non-interacting atoms. Each has a nucleus with spin S = .
According to quantum mechanics, each nucleus can therefore be in any of 4 quantum states,
labeled by quantum number m which can have values m = , , -, -. The m value is a
measure of the projection of the nuclear spin along a crystal axis of the solid. The electric
charge distribution of each nucleus is not spherically symmetric, so that the energy of a
nucleus depends on it’s spin orientation with respect to the internal electric field at it’s
location. In the 4 spin states, the energies of a nucleus are given by εm = mε, where ε is a
known energy.
a. Find an expression for the nuclear contribution to the partition function Z for this
solid. Don’t forget the Gibbs correction!
b. Calculate the nuclear contribution to the mean energy <E> for this solid.
c. Calculate the nuclear contribution to the heat capacity C for this solid
d. Calculate the nuclear contribution to the entropy S for this solid.
BONUS HOMEWORK PROBLEM!!
WORTH A TOTAL OF 25 EXTRA POINTS ON YOUR HOMEWORK GRADE
(It doesn’t affect your grade on this exam)!!
8. A gas of N non-interacting hydrogen molecules (H2 molecules) is in thermal equilibrium at
absolute temperature T.
a. Assume that, for calculating vibrational properties, each H2 molecule can be treated
as a quantum mechanical simple harmonic oscillator with natural frequency ω. Find
an expression for the vibrational partition function Zvib of this gas.
b. Assume that, for calculating rotational properties, an H2 molecule can be treated as a
quantum mechanical rigid rotator. Thus, the quantized rotational energy states have
energies of the form EJ = J(J+1)(ħ)2/I where J is the rotational quantum number and
I is the moment of inertia, for which you may assume a classical “dumbbell” model.
Recall from quantum mechanics that, in addition to the quantum number J, each
rotational energy state is also characterized by a quantum number m, which can have
any of the 2J +1 values m = -J, -(J - 1), -(J - 2),…,….(J - 2), (J - 1), J. So, each
rotational energy EJ is (2J + 1)-fold degenerate. Of course, this degeneracy must be
accounted for when the partition function is calculated. Write a formal expression
(“formal expression” means leave it as a sum or an integral which can’t easily be evaluated
in closed form) for the rotational partition function Zrot of this gas. Evaluate it in the
high temperature limit. What does the phrase “high temperature limit” mean here?
c. Still in the high temperature limit, calculate the total mean energy, including
translational, vibrational, and rotational parts.
d. Calculate the specific heat at low temperatures, assuming that the temperature is still
high enough that the N H2 molecules remain in a gaseous form.