BOSE-EINSTEIN CONDENSATION IN A HARMONIC POTENTIAL

We examine several features of Bose-Einstein condensation (BEG) in an external harmonic potential well. In the thermodynamic limit, there is a phase transition to a spatial Bose-Einstein condensed state for dim ension D greater than or equal to 2. The thermodynamic limit requires maintaining constant average density by weakening the potential while increasing the particle number N to infinity, while of course in real experiments the potential is fixed and N stays finite. For such finite ideal harmonic systems we show that a BEC still occurs, although with out a true phase transition, below a cel tain ''pseudo-critical'' temp erature, even for D=1. We study the momentum-space condensate fraction and find that it vanishes as 1/root N in any number of dimensions in the thermodynamic limit. In D less than or equal to 2 the lack of a mo mentum condensation is in accord with the Hohenberg theorem, but must be reconciled with the existence of a spatial BEC in D=2. For finite s ystems we dei ive the N-dependenee of the spatial and momentum condens ate fractions and the transition temperatures, features that may be ex perimentally testable. We show that the N-dependence of the 2D ideal-g as transition temperature for a finite system cannot pel sist in the i nteracting case because it violates a theorem due to Chester, Penrose, and Onsager.