The article aims to develop the stochastic interpretation of quantum mechanics by E. Nelson based on balancing the intra-systemic contradiction (i.e., antisymmetry) between “order” and “chaos”. For the set task, it is proposed to combine two mutually opposite system-forming principles: “the principle of least action” and “the principle of maximum entropy” into one, the “principle of averaged efficiency extremum”. In a detailed consideration of the averaged states of a chaotically wandering particle, the time-independent (stationary) and time-dependent stochastic Schrödinger-Euler-Poisson equations are obtained as conditions for finding the extremals of the globally averaged efficiency functional of the stochastic system under study. The resulting stochastic equations coincide with the corresponding Schrödinger equations up to coefficients. In this case, the ratio of the reduced Planck constant to the particle mass is expressed through the averaged characteristics of a three-dimensional random process in which the considered wandering particle participates. The obtained stochastic equations are suitable for describing the quantum states of stochastic systems of any scale.
The article aims to develop the stochastic interpretation of quantum mechanics by E. Nelson based on balancing the intra-systemic contradiction (i.e., antisymmetry) between “order” and “chaos”. For the set task, it is proposed to combine two mutually opposite system-forming principles: “the principle of least action” and “the principle of maximum entropy” into one, the “principle of averaged efficiency extremum”. In a detailed consideration of the averaged states of a chaotically wandering particle, the time-independent (stationary) and time-dependent stochastic Schrödinger-Euler-Poisson equations are obtained as conditions for finding the extremals of the globally averaged efficiency functional of the stochastic system under study. The resulting stochastic equations coincide with the corresponding Schrödinger equations up to coefficients. In this case, the ratio of the reduced Planck constant to the particle mass is expressed through the averaged characteristics of a three-dimensional random process in which the considered wandering particle participates. The obtained stochastic equations are suitable for describing the quantum states of stochastic systems of any scale.
In this article, the following results are obtained: the process of a randomly wandering particle having a size and a continuous trajectory of motion is considered; (b) based on the study of this probabilistic process, a derivation of the Schrodinger equation is proposed; the Planck constant h is expressed in terms of the characteristics of the stationary stochastic process under consideration; the probability distribution density of the n-order derivative for a n-times differentiable stationary stochastic process is determined; it is shown that the obtained generalized Schrodinger equation is applicable not only to describe the possible states of microcosm objects, but also for discrete states of a number of macrocosm stochastic systems
espanolEl objetivo del articulo es obtener una ecuacion estocastica que describa el estado promedio de una particula errante caoticamente, independientemente de su tamano. Como resultado de un analisis se obtuvo la integral de la accion promediada de una particula errante caoticamente en la representacion de coordenadas. La integral resultante resulto ser funcional de la funcion de onda ψ(x,y,z,t) (92). La ecuacion estocastica de Euler-Poisson (102) fue encontrada por el calculo de las variaciones, cuyas soluciones son los extremos del funcional (92). En el caso estatico ψ(x,y,z,t) = ψ(x,y,z), la ecuacion se reduce a la ecuacion de Schrodinger generalizada independiente del tiempo (113). Una caracteristica distintiva de las ecuaciones estocasticas es el hecho de que son adecuadas para describir la dinamica y la estatica de estados promediados de particulas caoticamente errantes de cualquier escala (es decir, son adecuadas para describir objetos estocasticos del micro y macrocosmos) EnglishThe aim of the article is to obtain a stochastic equation that describes the averaged state of a chaotically wandering particle, regardless of its size. As a result of an analysis it was obtained the integral of the averaged action of a chaotically wandering particle in the coordinate representation. The resulting integral turned out to be a functional of the wave function ψ(x,y,z,t). The stochastic Euler-Poisson equation was found by the calculus of variations, the solutions of which are the extremals of the functional. In the static case ψ(x,y,z,t) = ψ(x,y,z), the equation is reduced to the generalized time-independent Schrodinger equation. A distinctive feature of stochastic equations and is the fact that they are suitable for describing the dynamics and statics of averaged states of chaotically wandering particles of any scale (i.e., they are suitable for describing stochastic objects of the microcosm and macrocosm).
Non‐relativistic quantum systems are analyzed theoretically or by numerical approaches using the Schrödinger equation. Compared to the options available to treat classical mechanical systems this is limited, both in methods and in scope. However, based on Nelson's stochastic mechanics, the mathematical structure of quantum mechanics has in some aspects been developed into a form analogous to classical analytical mechanics. We show here that finding the Nash equilibrium for a stochastic optimal control problem, which is the quantum equivalent to Hamilton's principle of least action, allows to derive two things: i) the Schrödinger equation as the Hamilton‐Jacobi‐Bellman equation of this optimal control problem and ii) a set of quantum dynamical equations which are the generalization of Hamilton's equations of motion to the quantum world. We derive their general form for the non‐stationary and the stationary case. For the harmonic oscillator, the stationary equations lead to the coherent states, and we establish a numerical procedure to solve for the ground state properties without using the Schrödinger equation. image
In (1), it is suggested that one may obtain the Schrodinger equation by using Hamilton’s equations as a starting point, i.e dp/dt = - dH/dq and dq/dt = dH/dp. (1) defines new variables phi(x,t) = ap+ibq and phi*(x,t) and writes: iC dphi/dt partial = {phi, H} and iC dphi*/dt partial ={ phi*, H}, where {} are the Poisson brackets. (1) then changes phi(x,t) into a stochastic variable which still follows these Hamilton-like relations iC dphi/dt = Integral dx H1(x1,x) phi(x) + b(x,t) where H= Integral dx1dx H1 phi*(x1)phi(x) and b(x,t is a random function. In other words, Hamilton’s theory with an added stochastic b(x,t) is applied to a stochastic variable in order to obtain the Schrodinger equation. The catch, however, is that (1) introduces a number of conditions and assumptions to simplify the problem and actually obtains iC dphi/dt = Integral dx H1(x1,x)phi(x) with b(x,t) gone. We suggest that this implies that one may apply the Hamilton-like form directly to a stochastic variable without b(x,t) present. To strengthen this argument, one may consider a bound state in which b(x,t) may be said to average to 0. Basically, the idea of (1) seems to be that quantum mechanics follows from introducing stochastic physics linked to usual Newtonian trajectory physics, in this case Hamilton’s theory. We argue that this is very similar to the classical action approach we used in (2) to obtain exp(-iEt+ipx) from the classical free particle action applied to stochastic motion and so we compare the two approaches.
The stochastic theory of non-relativistic quantum mechanics presented here relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the classical trajectories of the particles are identical to the sample functions of a diffusion Markov process, whose conditional probability density is proposed as a substitute for the wave function. The Schroedinger equation and the so-called Nelson equations are used to determine the diffusion tensor and the drift vectors characteristic of such a process. It is then possible to write down the forward and backward Kolmogorov equations that are used to determine the conditional probability density, as well as the Fokker-Planck equation that is used to determine the normal probability density. This method is applied to several simple cases and the results obtained are compared with those of the orthodox theory. Among the most important differences let us mention that any system evolving freely from any original state returns spontaneously to its ground state, that the definitions of the particles velocities and momenta are impossible because the sample functions of a diffusion Markov process nowhere possess a derivative with respect to time and that the so-called collapse of the wave function is a mere mirage explained by the updating choice between new and older initial conditions in the resolution of partial differential equations. We finally attempt to extend the proposed theory to the domain of the relativistic quantum mechanics. It is promising but is clearly unfinished because it has not been possible up to now to solve the Nelson and Kolmogorov equations, except in the very simple case of a free particle.