Thermodynamic characteristics of ideal quantum gases in harmonic potentials within exact and semiclassical approaches

We theoretically examine equilibrium properties of the harmonically trapped ideal Bose and Fermi gases in quantum degeneracy regime. analyze thermodynamic characteristics with a finite number atoms by means known semiclassical approach perform comparison exact numerical results. For gas, we demonstrate deviations energy values originating from discrete level structure show that these are observable only for small particles. observe characteristic softening phase transition features, which contrasts to predictions related approximations. provide more accurate methodology determining corrections critical temperature due