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The aim of the investigation is to consistently incorporate quantum corrections in the transport model for applications to nanoscale semiconductor devices. This paper is comprised of two parts. Part I derives a set of two semiclassical equations in which the dynamics of the dispersion of the single-particle wave function is accounted for in addition to that of the expectation value of position. The model is founded on an approximate description of the wave function that eliminates the need of the Ehrenfest approximation. This leads to a set of two Newton-like single-particle equations for position and dispersion. In Part II, it will be shown that the Lagrangian form of the single-particle equations naturally lends itself to the incorporation of such extended dynamics into the statistical framework. The theory is suitable for different levels of applications: description of the single-particle ballistic dynamics, solution of the generalized Boltzmann equation by the Monte Carlo method or other methods, and solution of the continuity equations in the position-dispersion space.