1. What is an ideal gas? pV=NRT or non-interacting free particles
2. Review Z formula.
3. Stirling: n! approx n^n e^(-n) sqrt(2 pi n ).
4. Easy route: Z gives F. Use Stirling. Needed to make F extensive.
5. Internal energy, heat capacity, entropy, equations of state.
6. Entropy of mixing.
7. Gibbs’ paradox.
8. Non-ideal gas.

• Which series do you know?
• Why can we replace sum by integral?
• Design ideal gas thermometer.

Two days of ideal gas and three days of two level system is not the best. Next time do three days ideal gas. Did not get to entropy of mixing and so on. Design ideal gas thermometer is a good activity, but was poorly executed, needs development. Notes are not optimal, once you ahve the partition function you have the Helmholtz free energy, and you do not need other approaches.

1. Importance of two level systems, energy cost versus desired property.
2. N impurities, each in two states, no interaction
3. Possible energy eigenstates
4. Degeneracies
5. Partition function (Complete derivation, as well as Z1^N.)
6. Helmholtz free energy. Where is V? What happened to N! Show that N! would be wrong for S.
7. Entropy, internal energy, heat capacity, Schottky anomaly
8. Interpretation in one particle states: U = N(e1P1+e2P2)

• What do you expect for the energy at high T. 50/50 or E2?
• Calculate 1/N dU/dbeta and compare with fluctuations.
• Conductivity of a doped semiconductor as a function of T.

Starting out with a motivation of two level systems, ask the question what do you expect for the energy of a two level system at high temperature? There were good arguments for E2, only one group argued 50/50 based on Boltzmann probabilties. Also, what is a high temperature? kT » E1 or » E2-E1. It took a while to have them discover it is E2-E1. It is important that the total free energy is Fimp+Fback, with the latter having strong V dependence and giving pressure. Notation binomial (N choose m) is good to review quickly, they know, but are not certain. Review quickly why that is a good distribution

1. N! counts traps, not particles. n1! Is in formula!
2. Heat capacity peak, general principle.
3. Population inversion and negative temperatures.
4. Lasers.
5. NMR

• Low and high temperature expansion of energy
• Semiconductors, ne, nh product

We did spend a lot of time on the low and high temperature limits. First I asked what they would expect. That did not give simple answers. They tended to go into the math right away. next we did the math, and even there we needed reminders, like (1+2x)/(1+x) cannot ignore the x in the denominator.