1. Graduate Lecture Series 29 June – 3 July, 2009 Prof Ngee-Pong Chang Lecture 3 Bose Gas & Bose-Einstein Condensate

2. Bose – Einstein Statistics 1924 1894 - 19741879 - 1955

3. Bose letter to Einstein June 4, 1924 SN Bose, Zeit f Phys v26, 178 (1924); v27, 384 (1924) The Historical Development of Quantum Theory, Jagdish Mehra and Helmut Rechenberg, p569 (2001); Writings on physics and philosophy by W Pauli, Charles Paul Enz, K. vMeyenn, R. Schlapp, p. 94 (citation of Bose papers)

4. Inclusive vs Exclusive

5. Bosons in a Box L L L Standing Wave Eigenfunction

6. At T > 0 Probability of Occupancy

7. Fermion Boson Probability of Occupancy Classical limit Large E

8. Bose-Einstein Distribution for spin-0 bosons or fugacity Positive octant E=0

9. Bose-Einstein Condensate

10. Bose-Einstein Distribution for spin-0 bosons or fugacity

11. Riemann Zeta function

12. varies monotically from z = 0 to z = 1 For small z At high temperatures

13. A t cr i t i ca l T c Critical Temperature

14. Crude estimate of T c for He 4 You can check that this gives a Tc value of 3.17 K, to be compared with the experimentally observed value of 2.18 K

15. J.C. Davies group (Cornell) http://www.iranika.ir/articles%20page/mohandesi/mechanic.superfluids1.htm

16. Bose-Einstein Condensate Term The number of zero energy bosons per unit volume

17. University of Stuttgart measurement of BEC of Chromium atoms http://www.pi5.uni-stuttgart.de/news/050303/news050303.html

18. Bose-Einstein Condensate 400 nK 200 nK 50 nK

19. Nobel Prize in Physics 2001

20. Cornell – Wieman experiment: Cooling two thousand Rubidium-87 atoms to below 170 nK using combination of laser cooling and magnetic evaporative cooling. Ketterle (MIT) experiment Cooled some hundred times more atoms (Na 23 ), and was able to demonstrate quantum mechanical interference between two BEC.

21. Magnetic Evaporative Cooling

22. For a visual applet on Magnetic Evaporative Cooling Go to http://www.colorado.edu/physics/2000/atomic_lab.html

23. For a visual applet on Laser Cooling of Atoms Go to http://www.colorado.edu/physics/2000/atomic_lab.html

24. Ketterle experiment http://online.itp.ucsb.edu/plecture/ketterle/oh/45.html

25. Nobel Prize in Physics 1997

26. Douglas Osheroff 1945 - Transition Temperature

27. Nobel Prize in Physics 1997

28. Photon Gas 2 Polarization states Photon energy N is a function of temperature At equilibrium, fugacity, z, rises to maximum value of 1

29. Pressure By comparing with the equation for U, we find that the pressure- energy relation for the photon gas is

30. Or

31. Phonon Gas ! ( k ) = ! ( k + 2 ¼n = a ) http://ocw.mit.edu/NR/rdonlyres/Electrical-Engineering-and-Computer-Science/6- 730Physics-for-Solid-State-ApplicationsSpring2003/8A2B76D2-7D99-445B- B511-EDFA06C0482B/0/lecture12c.pdf

32. Speed of sound in solid

33. Peter Debye 1884 – 1966 Verh. Deut. Phys. Ges. 15, 678-689 (1913)

35. Low Temperature Limit

36. Specific Heat of Solid at low temperature Low Temperature Limit

37. High Temperature Limit Dulong Petit Law

39. http://web.mit.edu/6.730/www/ST04/Lectures/Lecture12.pdf