Improved models for ideal Fermi gas and ideal Bose gas using quantum phase space

The current models of ideal Fermi gas and ideal Bose gas are often considered as compatible with quantum theory. In this work, however, it is shown that improvements should be introduced into these models to rigorously take into account the quantum nature of phase space which is related to the uncertainty principle. The construction of the improved models which are considered is based on the use of the concepts of phase space representation of quantum mechanics and quantum phase space that were introduced and developed recently. Hamiltonian operator of a gas particle and its eigenstates are firstly determined and the use of quantum statistical mechanics leads to the deduction of the thermodynamics properties of the ideal gas itself. The explicit expressions of thermodynamic quantities such as the thermodynamic particles number, the internal energy, the grand canonical potential and the pressure as well as the state equations, corresponding to the improved models, are established for both bosons and fermions. The corrections introduced by these expressions are expected to be particularly significant at low temperature and for confined volume. They permit in particular to highlight the existence of quantum shape and size effects. It is also shown that the thermodynamics functions and variables are directly related to the momenta quantum statistical variances. Well-known relations corresponding to classical ideal gas model can be retrieved from the improved models as asymptotic limits at high temperature and for large volume.