We give an example of a mathematical model describing quantum mechanical processes interacting with medium. As a model, we consider the process of heat scattering of a wave function defined on the phase space. We consider the case when the heat diffusion takes place only with respect to momenta. We state and study the corresponding modified Kramers equation for this process. We consider the consequent approximations to this equation in powers of the quantity inverse to the medium resistance per unit of mass of the particle in the process. The approximations are constructed similarly to statistical physics, where from the usual Kramers equation for the evolution of probability density of the Brownian motion of a particle in the phase space, one deduces an approximate description of this process by the Fokker-Planck equation for the density of probability distribution in the configuration space. We prove that the zero (invertible) approximation to our model with respect to the large parameter of the medium resistance, yields the usual quantum mechanical description by the Schrodinger equation with the standard Hamilton operator. We deduce the next approximation to the model with respect to the negative power of the medium resistance coefficient. As a result we obtain the modified Schrodinger equation taking into account dissipation of the process in the initial model, and explaining the decoherence of the wave function.
We give an example of a mathematical model describing quantum mechanical processes interacting with medium. As a model, we consider the process of heat scattering of a wave function defined on the phase space. We consider the case when the heat diffusion takes place only with respect to momenta. We state and study the corresponding modified Kramers equation for this process. We consider the consequent approximations to this equation in powers of the quantity inverse to the medium resistance per unit of mass of the particle in the process. The approximations are constructed similarly to statistical physics, where from the usual Kramers equation for the evolution of probability density of the Brownian motion of a particle in the phase space, one deduces an approximate description of this process by the Fokker-Planck equation for the density of probability distribution in the configuration space. We prove that the zero (invertible) approximation to our model with respect to the large parameter of the medium resistance, yields the usual quantum mechanical description by the Schrodinger equation with the standard Hamilton operator. We deduce the next approximation to the model with respect to the negative power of the medium resistance coefficient. As a result we obtain the modified Schrodinger equation taking into account dissipation of the process in the initial model, and explaining the decoherence of the wave function.
We give an example of a mathematical model describing quantum mechanical processes interacting with medium. As a model, we consider the process of heat scattering of a wave function defined on the phase space. We consider the case when the heat diffusion takes place only with respect to momenta. We state and study the corresponding modified Kramers equation for this process. We consider the consequent approximations to this equation in powers of the quantity inverse to the medium resistance per unit of mass of the particle in the process. The approximations are constructed similarly to statistical physics, where from the usual Kramers equation for the evolution of probability density of the Brownian motion of a particle in the phase space, one deduces an approximate description of this process by the Fokker--Planck equation for the density of probability distribution in the configuration space. We prove that the zero (invertible) approximation to our model with respect to the large parameter of the medium resistance, yields the usual quantum mechanical description by the Schrödinger equation with the standard Hamilton operator. We deduce the next approximation to the model with respect to the negative power of the medium resistance coefficient. As a result we obtain the modified Schrödinger equation taking into account dissipation of the process in the initial model.
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 ρ=∣ϕ∣2 satisfies the standard Liouville equation, as in classical statistical mechanics.
This work is an extended version of the paper arXiv:0803.2669v1[math-ph], in which the main results were announced. We consider certain classical diffusion process for a wave function on the phase space. It is shown that at the time of order 10−11 {\it sec} this process converges to a process considered by quantum mechanics and described by the Schrodinger equation. This model studies the probability distributions in the phase space corresponding to the wave functions of quantum mechanics. We estimate the parameters of the model using the Lamb--Retherford experimental data on shift in the spectrum of hydrogen atom and the assumption on the heat reason of the considered diffusion process. In the paper it is shown that the quantum mechanical description of the processes can arise as an approximate description of more exact models. For the model considered in this paper, this approximation arises when the Hamilton function changes slowly under deviations of coordinates, momenta, and time on intervals whose length is of order determined by the Planck constant and by the diffusion intensities.
Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.
In the paper, we discuss the studies of mathematical models of diffusion scattering of waves in the phase space, and relation of these models with quantum mechanics. In the previous works it is shown that in these models of classical scattering process of waves, the quantum mechanical description arises as the asymptotics after a small time. In this respect, the proposed models can be considered as examples in which the quantum descriptions arise as approximate ones for certain hypothetical reality. The deviation between the proposed models and the quantum ones can arise, for example, for processes with rapidly changing potential function. Under its action the diffusion scattering process of waves will go out from the states described by quantum mechanics. In the paper it is shown that the proposed models of diffusion scattering of waves possess the property of gauge invariance. This implies that they are described similarly in all inertial coordinate systems, i.e., they are invariant under the Galileo transformations. We propose a program of further research.
The basic equations for the statistical system of waves in a randomly fluctuating medium are presented in the same form as in quantum mechanics, assuming a Markov process for the temporal change of the medium. A simple model is chosen for the Fokker-Planck equation of the medium which gives rise to a fluctuation of the Gaussian process, and several results are given to illustrate how the methods used in quantum mechanics or quantum field theory can be applied almost without change. Thus, the physical variables are represented by linear operators, and their equations of motion are determined by an equation similar to the Heisenberg equation of motion. The system has two stationary states (corresponding to the vacuum states 〈0∣ and ∣0〉 in quantum field theory), and the rather standard methods in field theory can be used for the evaluation of the Green's functions. When the temporal changes of the medium are sufficiently large compared with those of the waves, an adiabatic approximation is possible, and the isomorphic transformation (corresponding to the unitary transformation) is employed to lead to the result that the statistical system of the waves for this model is in perfect correspondence with the waves of bosons which interact with each other only through a two-body potential.