Semi-classical and quantum description of an ideal Bose gas in a uniform gravitational field

TL;DR

This paper compares semi-classical and quantum calculations of an ideal Bose gas in a gravitational field, finding good agreement for large particle numbers and revealing its thermal similarity to a five-dimensional Bose gas.

Contribution

It demonstrates the consistency between semi-classical and quantum approaches for large systems and uncovers a dimensional analogy in thermal properties.

Findings

Good agreement between semi-classical and quantum calculations for large particle numbers.

The system exhibits thermal properties equivalent to an ideal five-dimensional Bose gas.

Discrepancies claimed by previous authors are resolved with proper quantum calculations.

Abstract

We consider an ideal Bose gas contained in a cylinder in three spatial dimensions, subjected to a uniform gravitational field. It has been claimed by some authors that there is discrepancy between the semi-classical and quantum calculations in the thermal properties of such a system. To check this claim, we calculate the heat capacity and isothermal compressibility of this system semi-classically as well as from the quantum spectrum of the density of states. The quantum calculation is done for a finite number of particles. We find good agreement between the two calculations when the number of particles are taken to be large. We also find that this system has the same thermal properties as an ideal five dimensional Bose gas.