NOTES AND DISCUSSIONS Comment on ‘‘Numerical analysis of Bose–Einstein condensation in a three-dimensional harmonic oscillator potential,’’ by Martin Ligare †Am. J. Phys. 66 „3..., 185–190 „1998...‡—An extension to anisotropic traps and lower dimensions

In a recent article in this journal I pointed out that the harmonic oscillator potential used to model most experiments on Bose–Einstein condensation of atomic gases simplifies the numerical analysis. In this note I extend my previous analysis to allow for anisotropic confining potentials. This extension yields a better comparison with real experiments ~which are performed in anisotropic confining potentials!, but another benefit is the added generality which allows the study of Bose–Einstein condensation in systems whose atoms’ motion is limited to one and two dimensions in severely anisotropic traps. It should be noted that for systems with a finite number of particles, such as those studied in Ref. 1 and in this Comment, there is no true critical temperature, which can be defined only in the limit of an infinite system. However, the systems under investigation exhibit rapid increases in the fraction of particles in the ground state when the temperature is lowered sufficiently and also exhibit rapid changes in the heat capacity. The effective condensation indicated by the calculations of this paper is consistent with more detailed numerical and analytical studies of condensation in harmonic and anharmonic traps. The occupation number of bosons in a distinct singleparticle state with energy e l at a temperature parametrized by b51/kT is given by the Bose–Einstein distribution,