Harmonically trapped ideal quantum gases

Abstract

We solve the problem of a Bose or Fermi gas in d-dimensions trapped by delta < 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, along with 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 wells, wires and dots, respectively. Good agreement is found between experimental critical temperatures for the trapped boson gases Rb-87(37), Li-7(3), , Rb-85(37), He-4(2), K-41(19) and the known theoretical expression which is a special 37Rb Li Rb He 19 case for d = delta = 3, but only moderate agreement for Na-27(11) and H-1(1).