In this article, the quasi-Gaussian entropy theory is derived for pure quantum systems, along the same lines as previously done for semiclassical systems. The crucial element for the evaluation of the Helmholtz free energy and its temperature dependence is the moment generating function of the discrete probability distribution of the quantum mechanical energy. This complicated moment generating function is modeled via two distributions: the discrete distribution of the energy-level order index and the continuous distribution 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 with the previous semiclassical derivation of the quasi-Gaussian entropy theory. Several simple statistical states are derived, and it is shown that among them are the familiar Einstein model and the one-, two-, and three-dimensional Debye models. The various statistical states are applied to copper, alpha-alumina, and graphite. One of these states, the beta-diverging negative binomial state, is able to provide an accurate description of the heat capacity of both isotropic crystals, like copper, and anisotropic ones, like graphite, comparable to the general Tarasov equation.