Homework

General Instructions

1.  HW1.  Due Thursday, April 9.  Ch. 2,3.  Multiplicity; Statistical definitions of entropy, temperature.
2.  HW2.  Due Thursday, April 16.  Ch 6.  Partition function and Equipartition.
3.  HW3  Due Thursday, April 23.  Ch 6.  Partition function and free energy.
4.  HW4  Due  Thursday, April 30.  notes + Ch 7.1.  Chemical potential and grand partition function
5.  HW5  Due Thursday, May 7.  Perfect Gases and Ideal Solutions
6.  HW6.  Due Thursday, May 14.  Degenerate Fermi Gases
7.  HW7   Due Thursday, May 21.  Semiconductor Statistics
8.  HW8   Due Thursday, May 28.  Black Body Radiation
9.  HW9  Due Thursday, June 4.  Phonons, Bose Condensates.

Back to Physics 328 Home Page.

General Instructions

• Homework assignments will be worth 10 points -- 5 points for one problem that will be graded carefully, and 5 points for the other problems that will be looked over for a solid attempt and general approach.  If grading time permits, they will be graded as 2 points per problem instead.
  
• You are encouraged to work together on your homework.  However, please read over the problems and start on them yourself before seeking help -- otherwise you won't learn the hardest part of physics, which is figuring out how to attack a problem.
• Homework must be turned in to either Prof. Olmstead or the TA by 5 pm on the day it is due to receive credit.  You may either bring your homework to class, or turn your HW in directly to Prof. Olmstead's mailbox in the department office (a folder or envelope will be placed there after class on HW due dates). 
  
• Late homework will not be accepted without prior arrangement and a VERY good reason pre-approved by Prof. Olmstead.
• There will be 9 problem sets due this quarter.
  
• Your lowest homework score will be dropped.

1.  Week 1:  Multiplicity; Statistical definitions of entropy, temperature.  Intro to Partition Function.
    DUE Thursday April 9.  5 pm.

Comments on Reading

• Read chapters 2 and 3, with emphasis on 2.1 through 2.4 and 3.1 to 3.4.  We will do a slightly more mathematical description in class, as many of you saw this in Physics 224.  Spend the time you need to learn/review this basic underpinning of statistical physics.  In class, we also will review the laws of thermodynamics and relate them to the statistical definitions of temperature and entropy.
• For week 2, read chapter 6.1-2-3.  Section 6.1 is the core of statistical mechanics.  As you read section 6.2, think about the differences between statistical probabilities and averages vs. quantum mechanical probabilities and expectation values.  For section 6.3, equipartition, we will derive the general formula in class.
  

Assignment

• 2.8  Multiplicity of sets of Einstein oscillators
• 2.42 and 3.7  Entropy and temperature of a black hole.
• 3.20  Properties of DPPH
• 3.23  Entropy of a spin system.  Note trig identities:  1 - tanh²x = 1/cosh²x; [1 + tanh x]/[1 - tanh x] = exp(2x)
• 3.34  Polymer forces.
• 6.5  Partition function of a 3-state system.
  

2.  Partition Function and Equipartition.  DUE Thursday, April 16, 5 pm.

Comments on Reading

• In week 3, we will cover most of the rest of chapter 6 -- Maxwell Boltzmann statistics; connection between partition function and Helmholtz free energy, and then apply the partition function to a multiparticle system, heading into developing the partition function for an ideal gas (which we'll finish up in week 4).  We'll also spend a little time reviewing the four energy functions -- U, H, F, G.  Since Free Energy is inherently a property of many particles, I may invert the order of sections 6.5 and 6.6 to cover composite partition functions first, and then make the connection to free energy.  We'll then use these concepts for particle-in-a-box states and after about 1.5 lectures we'll have derived the ideal gas law from first principles.
  

Assignment

• 6.9  Partition function of hydrogen atom.  HINT:  at large n, consider that increasing the volume of the atom does work against the surrounding gas -- take that gas to be 1 atmosphere, and note that 10⁵ Pa = 0.7 eV/(10 nm)³.
• 6.12  Temperature of interstellar space.  Compare your result to the COBE result of 2.7 K in the microwave background.
• 6.22  Partition function of higher spin value atoms.  This is relevant in my research, where we are trying to determine the spin state of impurity transition metals in an oxide or selenide matrix -- the shape of the magnetization vs. magnetic field curve at different temperatures can be used to deduce the value of j, and hence the valence of the TM impurity.
• 6.39  Atmospheric escape velocity.  Think about how this relates to whether we should be capturing the He that is trapped with natural gas deposits, or letting it vent to the atmosphere.
• Extra problem 2.A.  Equipartition and Stars.
• Extra problem 2.B.  Temperature Fluctuations in Water
• The extra problems are here in a pdf file.
  

3.   Free energy and partition function, Ideal Gas introduced.  Due Thursday, April 23.

Comments on Reading:  Schroeder assumes you already are familiar with chemical potential as he introduces the Gibbs factor and grand partition function.  We'll take time in class to show how chemical potential arises from diffusive equilibrium in the same way that temperature arises from thermal equilibrium.  You might want to read parts of chapter 5 for macroscopic applications of chemical potential.  The midterm is on Friday, April 24, and covers through the first three HW sets (The parts of Chapters 1-5 that we covered in class, plus chapter 6 and any extra material covered in lecture notes through Monday April 20).

Assignment

• 6.42.  Entropy of simple harmonic oscillator.  ALSO:  take the limits of your expression at high and low temperature (compared to hf/k), and discuss them.
• 6.48 a.  Entropy of air.  Skip part b -- we'll come back to that in another week once we have discussed chemical potential.
• 6.52 Relativistic gas partition function.  Use the same particle in a box states in wavevector as for the non-relativistic case, but the relativistic relationship between momentum (wavelength) and energy.
• Extra Problem 3.A.  Rayleigh Scattering  HERE in a pdf file.
• Extra Problem 3.B  Adiabatic Demagnetization.  HERE in a pdf file.

4.  Chemical Potential and Grand Partition Function

Comments on Reading: Schroeder treats the full electrochemical potential or gravichemical potential as what he calls the chemical potential.  Other books will treat the two separately.  It is the full electrochemical potential that is constant in equilibrium, so what Schroeder is doing is better, but you should be warned that others treat it differently.  We will take a different approach to the Gibbs Factor derivation than the text -- you should remember whichever one makes more sense to you.  We will then move on to the perfect gas for the next couple of weeks, which is a remarkably good approximation to a huge number of real-life situations.

Assignment

• Short this week due to midterm -- we'll have more Grand Partition Function practice next week.
  
• 7.2:  Hemoglobin oxygen adsorption.   Remember that each distinct energy level and number is a state to count in the Grand partition function.
  
• 7.6  Particle number averages and fluctuations.  This is similar to the derivatives of the partition function for energy.
  
• extra 4:  Tree sap in a gravichemical potential.  Here in a pdf file.  This is simpler than many students think at first glance.
  

5.   Perfect Gases and Ideal Solutions

Comments on Reading.

This week we will take a break from Chapter 7 to return to the parts of chapter 5 that weren't covered in 224, namely solutions and osmotic pressure and chemical reactions.  We will come at reactions from a different point of view than the text, using the concept of the grand partition function and how it gets modified by internal degrees of freedom.

Assignment

• 7.8  Partition functions and quantum statistics
• 7.12  Symmetry in Fermi-Dirac distribution
• Note we are doing 7.15 in class on Friday, so it is not assigned, but interesting none-the-less.
• 7.16 OR 7.17  Building up the Fermi (7.16) and Bose (7.17) distributions from scratch.  Pick either ONE of these (they are both interesting, but it gets tedious after a while).
  
• 5.75  Comparison of class and book derivation  In class on Friday or Monday, we will derive the chemical potential of solution from the free energy of an ideal mixture.  The book takes the approach of adding solute molecules to a container of solvent.  This problem asks you to think about the range of validity of the two approaches and show they give the same result.
• 5.76  Osmotic pressure and desalination
  
• extra 5.  Partial pressures  SORRY this never got linked to.  Moved to Next Week's Assignment.
  

6.  Chemical Reactions and Fermi Gases

Comments on Reading

Degenerate Fermi Gases are present in metals, semiconductors and stars.  We will come back to the particle in a box states, and see what happens when will fill them with Fermions at a density much larger than the quantum density.  Electrons in a metal are at low enough energies that they are non-relativistic.  When a white dwarf collapses to a neutron star, however, the electrons go relativistic.  When the neutron star collapses to a black hole, it is the neutron Fermi gas that goes relativistic.  The book covers stars only in a HW problem -- we'll spend some time on them in class.  We will then move on to systems at finite temperature, and look at heat capacity and Pauli paramagnetism, and then semiconductors.  Again, the book contains the bare minimum equations, and puts all the applications into HW problems.  We'll spend an extra couple of lectures on semiconductors next week, covering material that isn't in the book.

Assignment

• Extra 6.A  Partial Pressures (relevant to last week, but forgot to link to it)
• Extra 6.B  Polymer stability. Using the chemical equilibrium equations to look at a real system.
  
• Extra 6.C  Thermal ionization of hydrogen.  This extends the example in the text.
  
• 7.28  Chemical Potential of a 2D electron gas
• 7.22 Relativistic Fermi Gas.  You will need this result for 6.D.
  
• Extra 6.D  Neutron Star
• LINK for the non-text problems.
  

7.   Semiconductor Statistics.  Due Thursday, May 21

We will return to the text this week for Bose gases, including photons (black body radiation), phonons (including heat capacity of a solid) and Bose condensates (superfluid He and ultracold gases).

Assignment

• Since semiconductors are not covered well in the text, this weeks problems are all from outside the text.
• Extra 7.A.  Semi-insulating GaAs. 
  
• Extra 7.B.  Extrinsic vs. Intrinsic contributions to the conductivity.
  
• Extra 7.C.  Degenerate semiconductors.
• Optional 7.D.  Fields in a pn-junction.  This problem is more E&M than Stat-Mech, and is included for the interested student.
  
• Link for the non-text problems
  

8.  Black Body Radiation.  Due Thursday, May 28

Comments on Reading

There is no class Monday, May 25 due to the Memorial Day Holiday.  The rest of the week we'll look at phonons and Bose Condensates.  The book does a reasonable job on both topics, though we'll approach condensation slightly differently in class.

Assignment

• 7.41  Einstein A and B coefficients.
• 7.43  Solar radiation. 
  
• 7.46  Free energy of the photon gas.
• 7.56  Venus as a greenhouse
  

9.  Bose Excitations and Condensates.  Due Thursday, June 4.

Comments on Reading

The final week of the quarter we will spend Monday on superfluids (see notes) and magnons (covered in problem 7.64).  Wednesday we will cover the Ising model (8.2), and then both tie up loose ends and review on Friday in preparation for the final exam on Monday.  The problems on magnons and the Ising model are optional, since you don't really have enough time to learn it in time for the HW to be due.

Assignment

• 7.60  Heat capacity of copper [note -- This is a classic problem.  There was a Prof. at Berkeley who was famous for always asking in the oral qualifying exam -- "What's the specific heat of a penny?"]
• 7.63  2D Solid
• 7.66  Bose condensation of Rubidium
• Extra 8.A.  2-D Bose Condensate Considerations.    Updated on Monday June 1 with a hint on the DOS for the oscillator.
  
• Extra 8.B.  Fountain Effect
• link for extra.