Eur. Phys. J. D 23, 117-124 (2003)
https://doi.org/10.1140/epjd/e2003-00005-1

Harmonically trapped ideal quantum gases

¹ Facultad de Ciencias, UNAM, Apdo. Postal 70-542, 04510 México DF, Mexico
² Instituto de Física, UNAM, Apdo. Postal 20-364, 01000 México DF, Mexico
³ Instituto de Investigaciones en Materiales, UNAM, Apdo. Postal 70-360, 04510 México DF, Mexico
⁴ Universidad Autónoma Metropolitana, Unidad Iztapalapa, 09340 México DF, Mexico

Corresponding author: ^a mdgg@hp.fciencias.unam.mx

Received: 17 July 2002
Revised: 14 October 2002
Published online: 21 January 2003

Abstract

We solve the problem of a Bose or Fermi gas in d-dimensions trapped by 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 , along with a jump in the specific heat at T_c if and only if . Specific heats for both gas types precisely coincide as functions of temperature when . The trapped system behaves like an ideal free quantum gas in dimensions. For we recover all known thermodynamic properties of ideal quantum gases in d dimensions, while in 3D for 1, 2 and 3 one simulates behavior reminiscent of quantum wells, wires and dots, respectively. Good agreement is found between experimental critical temperatures for the trapped boson gases Rb, Li, Rb, He, K and the known theoretical expression which is a special case for , but only moderate agreement for Na and H.