  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 take into account the
quantum nature of phase space related to the uncertainty principle. The
construction of the improved models 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
grand canonical potential, the thermodynamic particles number, the internal
energy 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 classic al
ideal gas model can be retrieved from the improved models as asymptotic limits
at high temperature and for large volume.
