Cold atoms Lecture 2. 4^th October, 2006 Preliminary plan for the whole fall term Back to Lecture 1. A few important points from the last class Bose-Einstein condensation: elementary approach L1: Ideal quantum gases at a finite temperature L1: Ideal quantum gases at a finite temperature L1: Ideal quantum gases at a finite temperature L1: Ideal quantum gases at a finite temperature L1: A gas with a fixed average number of atoms L1: Condensate concentration L1: Closer look at BEC L1: Closer look at BEC L1: Closer look at BEC Off-Diagonal Long Range Order Analysis on the one-particle level Coherence in BEC: ODLRO Coherence in BEC: ODLRO Coherence in BEC: ODLRO Coherence in BEC: ODLRO Coherence in BEC: ODLRO OPDM for homogeneous systems OPDM for homogeneous systems: ODLRO OPDM for homogeneous systems: ODLRO OPDM for homogeneous systems: ODLRO From OPDM towards the macroscopic wave function From OPDM towards the macroscopic wave function From OPDM towards the macroscopic wave function F.Laloë: Do we really understand Quantum mechanics, Am.J.Phys.69, 655 (2001) Thermodynamics of BEC Capsule on thermodynamics Grand canonical ensemble Thermodynamic functions of an ideal gas BEC in an ideal gas Comparison with real gases/vdW equation Homogeneous one component phase: boundary conditions (environment) and state variables Homogeneous one component phase: boundary conditions (environment) and state variables Thermodynamic potentials and all that Thermodynamic potentials and all that Thermodynamic potentials and all that Thermodynamic potentials and all that A table Digression: which environment to choose? Grand canonical ensemble Definition following Gibbs General treatment for independent particles Thermodynamic functions of an ideal gas BEC in an ideal gas Comparison with real gases/vdW equations Grand canonical ensemble - definition Grand canonical ensemble – general definition Grand canonical ensemble – finite systems Fluctuations I. – global quantities Useful identities Grand canonical statistical sum for independent bosons Grand canonical statistical sum for independent bosons Grand canonical statistical sum for independent bosons Grand canonical statistical sum for independent bosons Grand canonical statistical sum for independent bosons Grand canonical statistical sum for independent bosons Thermodynamic functions for an extended Bose gas Thermodynamic functions for an extended Bose gas Thermodynamic functions for an extended Bose gas Specific heat Isotherms in the P-V plane Compare with condensation of a real gas Compare with condensation of a real gas What will be the analogue for BEC? The end Problems Problems to Lecture 1. Problems to Lecture 2.