Seminar talk, 24 November 2010
Speaker: Evgeny Beniaminov
Title: Quantum mechanics as asymptotics of solutions of generalized Kramers equation
Abstract:
The talk discusses the process of diffusion (heat) scattering of waves on the phase space that oscillate at each point with frequency in proper time.
The modified diffusion Kramers equation, in which the resistance of the medium per unit of mass of particle is large, is investigated. 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, described by functions of coordinates. At the second long stage, the wave function varies in the subspace of "stationary" states according to the Schrodinger equation.
At the next stage, any superposition of waves goes to one of eigenstates of the energy operator (phenomenon of decoherence). At the last stage, due to random transitions among the eigenstates of the energy operator under the heat influence of the medium, the system goes to the mixed state of heat equilibrium (the Gibbs state).
It is also shown, that, if, on the contrary, the resistance of the medium per unit of mass of particle is small (mass is large), then in the considered model, the density of distribution of probability satisfies the standard Liouville equation, and the model does not exhibit quantum properties.
Thus, we get an example of equation whose solutions, depending on the value of parameters (the resistance of the medium and the mass of particle) model quantum or classical behavior of system.
References:
E.M. Beniaminov, Quantum mechanics as asymptotics of solutions of generalized Kramers equation, arXiv:1010.5898; Russian original: Preprint RSUH, 2010, local copy.