Harmonically Trapped Quantum Gases

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 $d+δ>2$, and a jump in the specific heat at $T_{c}$ if and only if $d+δ>4$. Specific heats for both gas types precisely coincide as functions of temperature when $d+δ=2$. The trapped system behaves like an ideal free quantum gas in $d+δ$ dimensions. For $δ=0$ we recover all known thermodynamic properties of ideal quantum gases in $d$ dimensions, while in 3D for $δ=$ 1, 2 and 3 one simulates behavior reminiscent of quantum {\it wells, wires}and{\it dots}, respectively.

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