1. Laser Cooling/Trapping of atoms We will discuss this in more detail toward the end of the semester, but it is possible to slow-down (cool) atoms by passing them through a region with counter-oriented laser beams tuned to just below an optical transition [“optical molassses”; so that atoms moving toward the laser will see photons Doppler shifted onto the resonance and absorb the photon (along with its momentum), and other atoms will interact minimally with the photons]. Using this technique, along with simple evaporative cooling, you can get VERY COLD gases of atoms in an optical/magnetic trap http://www.ptb.de/en/org/4/44/443/melcol_e.jpg

2. U.C. Boulder group’s BEC This is a map of the momentum distribution in the gas cloud (measured by looking at the gas after the trap has been turned off) for various temperatures (yes, it is O(10 -7 K). The sharp white peak in the middle is the BEC showing up at ~ 200 nK

3. Bose-Einstein Condensation There is a great applet at this web sit: http://www.colorado.edu/physics/2000/bec/evap_cool.html You can slowly cool a model gas down and get BEC, or cool it down too quickly and just have all the gas leave your trap (you can even stop the gas in some state and then have it reestablish equilibrium in another smaller trap). The site shows the temperature of the gas, you can watch it cool as some molecules (the most energetic) leave the trap, and it shows the temperature at which BEC would be expected for the density in the trap. See also the movie at: http://www.colorado.edu/physics/2000/bec/images/evap2.gif

4. Another interesting applet (more for chapter 14 when we talk about cooling) Is at the related website: http://www.colorado.edu/physics/2000/bec/lascool4.html

5. Key Definitions: E=E(S,V,N) Internal energy (fundamental relation) H=H(S, P, N) = E + PV (Enthalpy) F=F(T, V, N) = E - TS (Helmholtz Free Energy) G=G(T, P, N) = E + PV –TS (Gibbs Free Energy) For hydro-static systems (volume the only external parameter). dE = TdS – PdV +  dN dH = TdS +VdP +  dN dF = -SdT –PdV +  dN dG = -SdT + VdP +  dN

6. CALM: Why “Free” Energy? Helmholtz and Gibbs free energy quantify the amount of energy available in a system that can be used to do work. In a sense, this energy is 'free‘ and available to do work. Free energy refers to the energy which is not bound in other forms, such as kinetic (or rotational) energy, i.e. it is the energy of the system available to ˜go somewhere.˜

7. Examples: Baeirlein

8. Jacobians From vol. 5 of the “Course in Theoretical Physics” by Landau and Lifshitz ; L. M. Lifshitz and L. P. Pitaevskii “Statistical Physics 3 rd Ed. Part 1 page 53) Jacobians can be extremely useful in developing non-obvious relationships among functions of several interdependent variables (as we are often wanting to do in thermodynamics). For more details and examples see extract from L&L as a pdf file on the course web site under lecture presentations.

9. Examples: Reif

10. Show that C P and C V may be related to each other through quantities that may be determined from the equation of state (i.e. by knowing V as a function of P, T, and N) (the result to this question is something you will need for question 2 of assignment number 8).

11. Examples: Reif