Cold atoms Lecture 6. 29^th November, 2006 Preliminary plan/reality in the fall term The class before last: Interacting atoms L4: Scattering length, pseudopotential The class before last: Mean-field treatment of interacting atoms L4: Many-body Hamiltonian and the Hartree approximation L4: Gross-Pitaevskii equation at zero temperature Gross-Pitaevskii equation – homogeneous gas Previous class: Field theoretic reformulation (second quantization) L5: Field operator for spin-less bosons L5: Operators Previous class: Bogolyubov method L5: Basic idea L5: Hamiltonian of the homogeneous gas L5: Approximate Hamiltonian L5: Bogolyubov transformation L5: Bogolyubov transformation – result L5: More about the sound part of the dispersion law Particles and quasi-particles Trying to understand the Bogolyubov method L5: Action of the field operators in the Fock space Average values of the field operators in the Fock states L5: Hamiltonian conserves the particle number L5: Hamiltonian conserves the particle number L5: Hamiltonian conserves the particle number First try: Coherent ground state Reformulation of the Bogolyubov requirements New vacuum and the shifted field operators General case: the approximate vacuum General case: the Bogolyubov transformation Second try: Quasi-averages and broken symmetry Zero temperature limit of the grand canonical ensemble Degenerate ground state Degenerate ground state Rovnovážná struktura molekul AB[3 ]The end On the way to the mean-field Hamiltonian On the way to the mean-field Hamiltonian On the way to the mean-field Hamiltonian On the way to the mean-field Hamiltonian Variational approach to the condensate ground state Variational estimate of the condensate properties Variational estimate of the condensate properties Variational estimate of the condensate properties Variational estimate of the condensate properties Variational estimate of the condensate properties Variational estimate of the condensate properties Variational estimate of the condensate properties