On the derivation of the entropy of ideal quantum gases confined in a three-dimensional harmonic potential

In this work, the entropy functions of ideal quantum gases in a three-dimensional harmonic trap are analytically calculated using temperature as an explicit variable. Afterward, the applicability of the analytical formulas is validated by comparison with the numerical calculation. The results illustrate that the obtained functions could be applied for the whole temperature regime with a maximum relative deviation of less than 7.5% in the vicinity of the critical temperature Tc in the case of Bose gases. Meanwhile, for Fermi gases, although the analytical formula fits well at very low- and high-temperature regimes, it cannot be applied at temperature in the range [0.3-0.5] TF, where TF is the Fermi temperature. In addition, the consistency between our formulas and classical ones at significantly high temperatures is also discussed.