The problem of constructing models for the statistical dynamics of finite-level quantum mechanical systems is considered. The maximum entropy principle formulated by E. T. Jaynes in 1957 and asserting that the entropy of any physical system increases until it attains its maximum value under constraints imposed by other physical laws is applied. In accordance with this principle, the von Neumann entropy is taken for the objective function; a dynamical equation describing the evolution of the density operator in finite-level systems is derived by using the speed gradient principle. In this case, physical constraints are the mass conservation law and the energy conservation law. The stability of the equilibrium points of the system thus obtained is investigated. By using LaSalle's theorem, it is shown that the density function tends to a Gibbs distribution, under which the entropy attains its maximum. The method is exemplified by analyzing a finite system of identical particles distributed between cells. Results of numerical simulation are presented.