We have analytically obtained 1-particle density matrices for ideal Bose and Fermi gases in 3-D box geometries for the entire range of temperature. We also have obtained quantum cluster expansions of the grand free energies in closed forms for the same systems in the restricted geometries. 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 results are exact and are directly useful for a postgraduate course on statistical mechanics or that on many-body physics.

中文翻译：

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受限几何中量子气体簇扩展的有限尺寸效应

我们已经通过分析获得了在整个温度范围内 3-D 盒几何形状的理想 Bose 和 Fermi 气体的 1 粒子密度矩阵。对于受限几何中的相同系统，我们还获得了封闭形式的大自由能的量子簇扩展。在对盒几何中玻色和费米气体的准一维情况的分析中，我们还考虑了短程相互作用。我们的结果是准确的，对统计力学或多体物理学的研究生课程直接有用。