We give an example of a mathematical model describing quantum mechanicalprocesses interacting with medium. As a model, we consider the process of heat scatteringof a wave function defined on the phase space. We consider the case when the heat diffusiontakes place only with respect to momenta. We state and study the corresponding modifiedKramers equation for this process. We consider the consequent approximations to this equationin powers of the quantity inverse to the medium resistance per unit of mass of the particle inthe process. The approximations are constructed similarly to statistical physics, where fromthe usual Kramers equation for the evolution of probability density of the Brownian motion ofa particle in the phase space, one deduces an approximate description of this process by theFokker–Planck equation for the density of probability distribution in the configuration space. Weprove that the zero (invertible) approximation to our model with respect to the large parameterof the medium resistance, yields the usual quantum mechanical description by the Schr¨odingerequation with the standard Hamilton operator. We deduce the next approximation to the modelwith respect to the negative power of the medium resistance coefficient. As a result we obtainthe modified Schr¨odinger equation taking into account dissipation of the process in the initialmodel, and explaining the decoherence of the wave function.
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