Thermodynamical properties of a three-dimensional free electron gas confined in a one-dimensional harmonical potential

We study the thermodynamical properties of a noninteracting electron gas confined in one dimension by a harmonicoscillator potential. The exact analytical expression for the thermodynamical potential is obtained by using a formula of contour integration. The magnetizations, magnetic susceptibilities, and the specific heats are then studied each as a function of the strength of the magnetic field in different regimes of the temperature and effective thickness. It is shown at low temperature, the magnetization, magnetic susceptibility, and the specific heat oscillate as the strength of the magnetic field increases. Especially, there exist two modes of oscillations for the specific heat in certain regimes of low temperature and effective thickness.