Quantum Mechanics as Asymptotics of Solutions of Generalized Kramers Equation

Abstract

We consider the process of diffusion scattering of a wave function given on the phase space. In this process the heat diffusion is considered only along momenta. We write down the modified Kramers equation describing this situation. In this model, the usual quantum description arises as asymptotics of this process for large values of resistance of the medium per unit of mass of particle. It is shown that in this case the process passes several stages. During the first short stage, the wave function goes to one of "stationary" values. At the second long stage, the wave function varies in the subspace of "stationary" states according to the Schrodinger equation. Further, dissipation of the process leads to decoherence, and any superposition of states goes to one of eigenstates of the Hamilton operator. At the last stage, the mixed state of heat equilibrium (the Gibbs state) arises due to the heat influence of the medium and the random transitions among the eigenstates of the Hamilton operator. Besides that, it is shown that, on the contrary, if the resistance of the medium per unit of mass of particle is small, then in the considered model, the density of distribution of probability $\rho =|\phi |^2$ satisfies the standard Liouville equation, as in classical statistical mechanics.