Ideal Fermi gases in harmonic oscillator potential traps

We study the thermodynamic properties of an ideal gas of fermions in a harmonic oscillator confining potential. The analogy between this problem and the de Haas–van Alphen effect is discussed and used to obtain analytical results for the chemical potential and specific heat in the case of both isotropic and anisotropic potentials. Step-like behaviour in the chemical potential, first noted in numerical studies, is obtained analytically and shown to result in an oscillatory behaviour of the specific heat when the particle number is varied. The origin of these oscillations is that part of the thermodynamic potential is responsible for the de Haas–van Alphen-type effect. At low temperatures we show analytically that there are significant deviations in the specific heat from the expected linear temperature dependence, again as a consequence of the de Haas–van Alphen part of the thermodynamic potential. Results are given for one, two, and three spatial dimensions. In the anisotropic case we show how the specific heat jumps as the ratio of oscillator frequencies varies.