According to the principles of quantum mechanics, the physical system, described by certain Hamiltonian \(\hat{H}\) , has a set of eigenstates denoted by vector \(|k\rangle \) . The vector \(|k\rangle \) is an eigenfunction of the Hamiltonian \(\hat{H}|k\rangle =\varepsilon _k|k\rangle \) where \(\varepsilon _k\) is the corresponding eigenvalue of energy or the energy level of the system. The statistical specification of the physical system implies its probabilistic description by introducing some distribution function \(w_k=w(|k\rangle )\) . The latter determines the probability to find the system in the state with vector \(|k\rangle \) . When we adhere to the mathematical positions alone, the choice of distribution functions \(w_k\) is very wide. From the physical point of view, the choice of the distribution function as the main principle of the theory should result in a set of physical consequences and conclusions fully consistent with the available experimental data and have predictive powers as well.