Derivation of thermal equations of state for quantum systems using the quasi-Gaussian entropy theory

In this article, the quasi-Gaussian entropy theory is derived for pure quan tum systems, along the same lines as previously done for semiclassical syst ems. The crucial element for the evaluation of the Helmholtz free energy an d its temperature dependence is the moment generating function of the discr ete probability distribution of the quantum mechanical energy. This complic ated moment generating function is modeled via two distributions: the discr ete distribution of the energy-level order index and the continuous distrib ution of the energy gap. For both distributions the corresponding physical- mathematical restrictions and possible systematic generation are discussed. The classical limit of the present derivation is mentioned in connection w ith the previous semiclassical derivation of the quasi-Gaussian entropy the ory. Several simple statistical states are derived, and it is shown that am ong them are the familiar Einstein model and the one-, two-, and three-dime nsional Debye models. The various statistical states are applied to copper, alpha-alumina, and graphite. One of these states, the beta-diverging negat ive binomial state, is able to provide an accurate description of the heat capacity of both isotropic crystals, like copper, and anisotropic ones, lik e graphite, comparable to the general Tarasov equation. (C) 1999 American I nstitute of Physics. [S0021-9606(99)51633-4].