The Maxwell–Boltzmann statistics of the quantum ideal gas is studied through the canonical partition function by exactly counting discrete quantum states without the continuum approximation. Analytic expressions for energy, pressure, entropy, and heat capacity are expressed in terms of Jacobi theta functions and complete elliptic integrals. The results show typical effects of discrete energy levels in the low temperature limit while they reproduce thermodynamics of the classical ideal gas in the high temperature limit.

中文翻译：

---

典型集合中量子理想气体的麦克斯韦-波尔兹曼统计

量子理想气体的麦克斯韦-玻尔兹曼统计量是通过规范分配函数研究的，方法是精确计算离散的量子态而无需连续近似。能量，压力，熵和热容量的解析表达式以Jacobi theta函数和完整的椭圆积分表示。结果表明，在低温极限下离散能级的典型影响，同时又再现了高温极限下经典理想气体的热力学。