Exam III (Final Exam) PHYSICS 4302, Spring, 2013, Dr. Charles W. Myles

Exam III (Final Exam) PHYSICS 4302, Spring, 2013, Dr. Charles W. Myles Take Home Exam: Distributed, Monday, May 6 DUE, NOON, TUESDAY, May 14!! 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!! It wastes paper, but it makes my grading easier! 2. PLEASE don’t write on the exam sheets, there is no room! If you don’t have paper, I’ll give you some. 3. PLEASE show ALL work, writing down at least the essential steps in the problem solution. Partial credit will be liberal, provided that essential work is shown. Organized work, in a logical, easy to follow order will receive more credit than disorganized work. Problems for which just answers are shown, without the work being shown, will receive ZERO credit! 4. The setup (PHYSICS) of a problem counts more than the mathematics of working it out. 5. PLEASE write neatly. Before handing in the solutions, PLEASE: a) put problem solutions in numerical order, b) number the pages & put them in order, & c) clearly mark your answers. If I can’t read or find your answer, you can't expect me to give it the credit it deserves. 6. NOTE!! The words “DISCUSS” & “EXPLAIN” below mean to write English sentences in the answer. They DON’T mean to answer using only symbols. Answers to such questions containing only symbols without explanation of what they mean will get ZERO CREDIT!!! It would also be nice if undergraduate physics majors would try to write complete, grammatically correct English sentences! NOTE: I HAVE 14 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!! NOTE!!!! QUESTION 1 IS REQUIRED! ANSWER ANY 4 OUT OF THE OTHERS! So, you must answer 5 questions total. Each of them is equally weighted & worth 20 points. So, 100 is the maximum points possible. Please sign the following statement and turn it in with your exam: I have neither given nor received help on this exam Signature ______e 1. NOTE!!!! THE FOLLOWING QUESTIONS ARE REQUIRED!!! a. Briefly DISCUSS, using WORDS, with as few mathematical symbols as possible, the PHYSICAL MEANINGS of the following: 1) Microcanonical Ensemble, 2) Canonical Ensemble, 3) Grand Canonical Ensemble, 4) Entropy, 5) Fermi Energy, 6) Pauli Exclusion Principle, 7) Maxwell-Boltzmann Statistics, 8) Fermi-Dirac Statistics, 9) Bose-Einstein Statistics. 10) Bose-Einstein Condensation. 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. NOTE!!!! ANSWER ANY 4 OUT OF QUESTIONS 2 through 6! NOTE! All chapter numbers & problem numbers in what follows refer to the book by Reif 2. Work Problems #14 & 15 of Chapter 7. Treat these as two parts of one problem.

3. Work Problem #19 of Chapter 7. Use Classical (Maxwell-Boltzmann) Statistical Mechanics to solve this problem.

4. Work Problem #5 of Chapter 9. Use Quantum Statistical Mechanics to solve this problem.

5. Work Problems #12 and #13 of Chapter 9. Treat these as two parts of one problem.

NOTE!!!! ANSWER ANY 4 OUT OF QUESTIONS 2 through 6! 6. A gas of N non-interacting hydrogen (H2) molecules is in thermal equilibrium at 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 2 energies of the form EJ = [J(J+1)(ħ) ]/(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? That is, “high temperature” in comparison with what? 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 H2 molecules remain in a gaseous form.

NOTE!!! There are BONUS HOMEWORK PROBLEMS on the next page!!!! BONUS HOMEWORK PROBLEMS!! WORK ON PROBLEMS 7 & 8 IS WORTH UP TO AN EXTRA 50 POINTS (25 points each) ON YOUR HOMEWORK GRADE (Work on them doesn’t affect your grade on this exam)!! 7. 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 2 2 2 2 2 2 box: εr = (ħ π )[(nx) + (ny) + (nz) ])/[2mL ]. nx, ny, nz, are integers and m is the particle mass. Start with the canonical ensemble partition function Z for this gas. In Ch. 9 of Reif’s book, it is shown that the natural logarithm of Z has the form (μ is the chemical potential): ln(ZBE) = - μβN - ∑r ln{1 – exp[β(μ - εr)} for a Bose-Einstein gas. ln(ZFD) = - μβN + ∑r ln{1 + exp[β(μ - εr)} for a Fermi-Dirac gas. a. Use the 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. (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 total particle number N. That is discussed in detail in Reif’s book, Sect. 6.9 & again in Sect. 9.6. (Note: This equation of state should convince you that the classical “Ideal Gas Law” is not valid for quantum mechanical gases!)

NOTE: In parts b and c, I want NUMBERS for Pf, & Tf, not just formal results with mathematical symbols! b. Consider an adiabatic, quasi-static expansion of this gas from an initial volume Vi = V to a final 5 2 volume Vf = 10V. If the initial pressure is Pi = 1 atm = 10 N/m , 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. In this calculation, neglect the interaction of the gas with the container walls. c. Now, specialize to the case of a Fermi-Dirac gas. For Fermions, it is shown in Ch. 9 of Reif’s 2 book that the mean energy at low temperatures T depends on T as E = E0 + AT , where E0 and A are constants. In the process described in part a, if the initial temperature was Ti = 10K and the 3 initial volume was Vi = 1.0 m , calculate the final temperature Tf . To obtain a NUMBER for Tf, let the constant A = 3,000 Joules/K. (Note: The constant E0 should not be needed!)

8. A classical monatomic, NON-IDEAL gas with N particles, in thermal equilibrium at temperature T, is confined to a volume V. In Ch. 10, it is shown that the natural logarithm of the partition function Z 3 2 for this system has the form: ln(Z) = ( /2)N ln(A/β) + ln(ZU) – Nln(N) + N, with A = (2πm/h ), β = -1 (kBT) . 2 nd ln(ZU) has the APPROXIMATE form: ln(ZU) = N ln(V) - (N /V)B2(T). B2(T) is called the “ 2 Virial Coefficient”. It is a complicated integral which depends on the form of the interaction potential between two particles. Ch. 10 also discusses that this Z gives an APPROXIMATE equation of state 2 for this gas of the form P = kBT[n + B2(T)n ]. Here, n = (N/V) is the particle number density. There are other important properties of a gas besides it’s equation of state. For example: a. Derive expression for the mean energy Ē 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. (You may have to look at Ch. 8 to find out how to calculate this!) Note: For each part, assume that B2(T) is a known function of temperature & express your answers in 2 2 general in terms of it and it’s temperature derivatives (dB2/dT), (d B2/dT ), etc.