  We have analytically obtained 1-particle density matrices for ideal Bose and
Fermi gases in both the 3-D box geometries and the harmonically trapped
geometries for the entire range of temperature. We have obtained quantum
cluster expansions of the grand free energies in closed forms for the same
systems in the restricted geometries. We have proposed a theorem (with a proof)
about the generic form of the quantum cluster integral. We also have considered
short ranged interactions in our analyses for the quasi 1-D cases of Bose and
Fermi gases in the box geometries. Our theoretical results are exact, and are
directly useful for understanding finite-size effects on quantum cluster
expansion of Bose and Fermi gases in the restricted geometries. Our results
would be relevant in the context of experimental study of spatial correlations
in ultra-cold systems of dilute Bose and Fermi gases of alkali atoms (i) in 3-D
magneto-optical box traps with quasi-uniform potential around the center [1],
and (ii) in 3-D harmonic traps [2, 3].
