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We give an example of a mathematical model describing quantum mechanicalprocesses interacting with medium. As a model, we consider the process of heatscattering of a wave function defined on the phase space. We consider the casewhen the heat diffusion takes place only with respect to momenta. We state andstudy the corresponding modified Kramers equation for this process. We considerthe consequent approximations to this equation in powers of the quantityinverse to the medium resistance per unit of mass of the particle in theprocess. The approximations are constructed similarly to statistical physics,where from the usual Kramers equation for the evolution of probability densityof the Brownian motion of a particle in the phase space, one deduces anapproximate description of this process by the Fokker--Planck equation for thedensity of probability distribution in the configuration space. We prove thatthe zero (invertible) approximation to our model with respect to the largeparameter of the medium resistance, yields the usual quantum mechanicaldescription by the Schr\"odinger equation with the standard Hamilton operator.We deduce the next approximation to the model with respect to the negativepower of the medium resistance coefficient. As a result we obtain the modifiedSchr\"odinger equation taking into account dissipation of the process in theinitial model.

A Refinement of the Schr\"odinger Equation Involving Dissipation of Waves in the Phase Space