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Exact spatial density of ideal Bose atoms in a two-dimensional isotropic harmonic trap

We have proposed an exact analytical solution to the problem of Bose–Einstein condensation (BEC) of harmonically trapped, two-dimensional (2D), and ideal atoms. It is found that the number of atoms in vapor is characterized by an analytical function, which involves a series expansion of q-digamma functions in mathematics. We employ the analytical solution to calculate the spatial density n(r;T,N) of ideal Bose atoms in a 2D isotropic harmonic trap. The first main finding in this paper is that when Bose atoms are in the normal state, the density profile exhibits a 2D dark spot in the center and a bright ring outside. The second main finding is that when Bose atoms are in the BEC state, the density profile exhibits a 2D sharp peak with an extremely small radius. The third main finding is that the central density is a monotonically increasing function of the number of atoms N, but is a monotonically decreasing function of temperature T. The fourth main finding is that the number density of atoms of condensate exhibits a 2D Gaussian distribution in the center while the number density of atoms of vapor exhibits a 2D bright annulus with zero density at the origin.