Harmonically Trapped Quantum Gases

TL;DR

This paper analyzes the thermodynamics of quantum gases trapped by multiple harmonic potentials in various dimensions, revealing conditions for Bose-Einstein condensation and how trapping modifies their thermodynamic behavior.

Contribution

It provides a generalized framework for understanding the thermodynamics of harmonically trapped quantum gases in arbitrary dimensions and potential configurations.

Findings

Bose-Einstein condensation occurs if and only if d + δ > 2.

Specific heat exhibits a jump at T_c if and only if d + δ > 4.

Trapped systems mimic free gases in d + δ dimensions.

Abstract

We solve the problem of a Bose or Fermi gas in $d$-dimensions trapped by $% \delta \leq d$ mutually perpendicular harmonic oscillator potentials. From the grand potential we derive their thermodynamic functions (internal energy, specific heat, etc.) as well as a generalized density of states. The Bose gas exhibits Bose-Einstein condensation at a nonzero critical temperature $T_{c}$ if and only if $d+\delta >2$, and a jump in the specific heat at $T_{c}$ if and only if $d+\delta >4$. Specific heats for both gas types precisely coincide as functions of temperature when $d+\delta =2$. The trapped system behaves like an ideal free quantum gas in $d+\delta $ dimensions. For $\delta =0$ we recover all known thermodynamic properties of ideal quantum gases in $d$ dimensions, while in 3D for $\delta =$ 1, 2 and 3 one simulates behavior reminiscent of quantum {\it wells, wires}and{\it dots}, respectively.