FINAL EXAM, PHYSICS 5305, Spring, 2007, Dr. Charles W. Myles
Take Home Exam: Distributed, Monday, April 30
DUE, IN MY OFFICE OR MAILBOX, 5PM, TUES., MAY 8.
NO EXCEPTIONS!
EXAM RULE: You are allowed to use almost any resources (library books, etc.) to solve these
problems. EXCEPTION: You MAY NOT COLLABORATE WITH ANY OTHER
PERSON in solving them! If you have questions or difficulties with the problems, you may consult with me, but not with fellow students (whether or not they are in this class!), with other faculty, or with post-docs. You are bound by the TTU Code of Student Conduct not to violate this rule! Anyone caught violating this rule will, at a minimum, receive an “F” on this exam!
INSTRUCTIONS: Read all of these before doing anything else!!! Failure to follow them may lower your grade!!
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!
7. NOTE!!! The setup (THE PHYSICS) of a problem will count more heavily in the grading than the detailed mathematics of working it out.
PLEASE FOLLOW THESE SIMPLE DIRECTIONS!!!! THANK YOU!!!
NOTE!!!! YOU MUST ANSWER PROBLEM 1! WORK ANY 5 OUT OF THE OTHER 6 PROBLEMS! So, you must answer 6 problems in total. Problem 1 is worth 10 points. Each of the others is equally weighted and worth 18 points. So, 100 is the maximum points possible.
Please sign this statement and turn it in with your exam:
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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 Distribution, 2) Canonical Distribution, 3) Grand Canonical Distribution, 4) Entropy, 5) Fermi Energy, 6) Pauli Exclusion Principle, 7) Classical Statistics, 8) Quantum Statistics, 9) Partition Function, 10) Equipartition Theorem.
b. Briefly DISCUSS, using WORDS & NOT symbols, the fundamental difference between Fermions and Bosons and how this difference leads to the fundamentally different Fermi-Dirac and Bose-Einstein Statistics. (That is, what is the basic, intrinsic property that distinguishes a Fermion and a Boson?) In this discussion, be sure to include the qualitative differences one would expect between the ground states of Fermi-Dirac and Bose-Einstein statistics.
NOTE!!!! WORK ANY 5 OUT OF PROBLEMS 2 through 7!
2. Consider N identical, non-interacting magnetic atoms in a solid at thermal equilibrium at temperature T. The solid is in a static external magnetic field H. Assume that H is in the z direction. Each atom has a magnetic moment μ. Use CLASSICAL statistical mechanics to find the CLASSICAL expressions for the thermodynamic properties asked for in the following. Hints: You need to use the classical energy of a magnetic moment μ in a magnetic field H: E = -μH cosθ. θ is the angle between μ & H. The results of parts a,b,c will be different than the quantum mechanical results in Reif’s Sect. 7.8.
a. Calculate the partition function Z for this system.
b. Calculate the average (mean) z component of magnetic moment <μz>, the average (mean) z component of magnetization <Mz> and the mean energy <E> for this system.
c. Calculate the entropy S for this system.
In parts d,e,f, for notational simplicity, the mean energy <E> is replaced by the symbol E & the mean z component of magnetization <Mz> is replaced by the symbol M. For parts d,e,f, assume that the system undergoes an infinitesimal, quasi-static process in which the static external magnetic field is changed by dH. The mechanical work done by this process is dW = -MdH.
d. For this process, write, in differential form, involving the differential dE, the combined 1st & 2nd Laws of Thermodynamics for this system, assuming that the entropy S and the static magnetic field H are independent variables.
e. Use the properties of differentials and the results of part e to express T and M as appropriate partial derivatives of E.
f. Use the properties of partial derivatives and the results of part f to relate an appropriate partial derivative of M to a partial derivative of T, hence deriving one of Maxwell’s relations for this system.
3. Work Problem #7 in Chapter 6 of the book by Reif.
4. The partition function z for a single one-dimensional, quantum mechanical, simple harmonic oscillator of frequency ω in thermal equilibrium at temperature T is given in Eq. (7∙6∙10) of Reif’s book. Consider a system of N such oscillators. They are identical and non-interacting.
a. Calculate the partition function Z for this system. Don’t forget the Gibbs correction!
b. Calculate the mean energy of this system. DISCUSS the physical meaning of the two terms in the mean energy.
c. Calculate the entropy of this system.
d. Calculate the heat capacity of this system.
e. Evaluate the heat capacity of part d in the limit of high temperatures (Ћω >> kBT). DISCUSS the physical meaning of this limit.
f. Evaluate the heat capacity of part e in the limit of low temperatures (Ћω << kBT). DISCUSS the physical meaning of this limit.
NOTE!!!! WORK ANY 5 OUT OF PROBLEMS 2 through 7!
5. Consider a quantum mechanical, ideal gas of N identical, structureless particles in thermal equilibrium at temperature T and confined to volume V. Let the container be a cubic box of side L so that V = L3. The quantized energies of the single particle quantum states are of the usual form for a particle in a box: ε = (Ћ2π2)[(nx)2 + (ny)2 + (nz)2])/[2mL2]. Here, nx,ny,nz, are integers and m is the particle mass.
a. Start with the logarithm of the partition function Z for this gas, which is in Reif’s book. [ln(Z) for a Bose-Einstein gas is in Eq. (9∙6∙9), ln(Z) for a Fermi-Dirac gas is in Eq. (9∙7∙4)]. Use the general relations between ln(Z), the mean energy E, and the mean pressure P to show that the equation of state of this gas is PV = (⅔)E, independent of whether the gas is composed Fermions or Bosons. (However, it won’t hold if the Bosons are photons, which have a different equation of state.) You might also need to use the general relation between the chemical potential μ and the total particle number N. (Note: This result should convince you that the classical “Ideal Gas Law” is not valid for quantum mechanical gases!)
b. Consider an adiabatic, quasi-static expansion of this gas from an initial volume Vi = V to a final volume Vf = 10V. If the initial pressure is Pi = 1 atm = 105 N/m2, calculate the final pressure Pf of the gas in this process. This result should ALSO be independent of whether the gas is composed Fermions or Bosons (again, the Bosons can’t be Photons!). In this calculation, neglect the interaction of the gas with the container walls. NOTE: I want a NUMBER for Pf, not just a formal result with mathematical symbols!
c. Now, specialize to the case of a Fermi-Dirac gas. For Fermions, it is shown in Ch. 9 of Reif’s book [see Eq. (9∙17∙18)] that the mean energy at very low temperatures T depends on T as E = E0 + AT2, where E0 and A are constants. In the process described in part b, if the initial temperature was Ti = 10K and the initial volume was Vi = 1.0 m3, calculate the final temperature Tf . To obtain a NUMBER for Vf, let the constant A = 3,000 Joules/K.
6. A classical monatomic, NON-IDEAL gas with N particles confined to volume V is in thermal equilibrium at temperature T. In Ch. 10 of Reif’s book, which we didn’t have time for in class, it is shown that the an approximate equation of state for this gas is P = kBT[n + B2(T)n2]. Here, n = (N/V) is the number density and B2(T) is called the “2nd Virial Coefficient”. In Ch. 10, it is also shown [see Eq. (10∙4∙4)] that B2(T) can be expressed in terms of an integral, called I(β) in Ch. 10, involving the temperature parameter β and the pair potential of interaction between the particles. Of course, there are other important properties of a gas besides it’s equation of state. For example:
a. Derive an expression for the mean energy E of this gas.
b. Derive an expression for the heat capacity at constant volume, Cv of this gas.
c. Derive an expression for the entropy S of this gas.
d. Derive an expression for the chemical μ potential of this gas.
Note: For each of parts a,b,c,d you may assume that B2(T) and I(β) are known functions and you may express your answers in general in terms of either of B2(T) or I(β) and various derivatives of them.
7.
a. Work Problem #7 in Chapter 9 of the book by Reif.
b. Work Problem #22 in Chapter 9 of the book by Reif.