The limits of applicability of many classical (non-quantum-mechanical) theories are not sharp. These theories are sometimes applied to the problems which are, in their nature, not very well suited for that. Two of the most widely used classical approaches are the theory of diffusion stochastic process and Itô's stochastic differential equations. It includes the Brownian-motion treatment as the basic particular case. The present work shows that, for quantum-mechanical reasons, the diffusion parameter of a Brownian particle cannot be arbitrarily small since it has a nonzero minimum value. This fact leads to the version of Heisenberg's uncertainty principle for a Brownian particle which is obtained in the precise mathematical form of a limit inequality. These quantitative results can help to properly apply the theories associated with Brownian-particle modelling. The consideration also discusses a series of works of other authors.
The limits of applicability of many classical (non-quantum-mechanical) theories are not sharp. These theories are sometimes applied to the problems which are, in their nature, not very well suited for that. Two of the most widely used classical approaches are the theory of diffusion stochastic process and Itô's stochastic differential equations. It includes the Brownian-motion treatment as the basic particular case. The present work shows that, for quantum-mechanical reasons, the diffusion parameter of a Brownian particle cannot be arbitrarily small since it has a nonzero minimum value. This fact leads to the version of Heisenberg's uncertainty principle for a Brownian particle which is obtained in the precise mathematical form of a limit inequality. These quantitative results can help to properly apply the theories associated with Brownian-particle modelling. The consideration also discusses a series of works of other authors.
Abstract In stochastic mechanics, one of the interpretations of realistic quantum mechanics, the Schrödinger equation can be derived by assuming the Brownian motion (BM) of minute particles. In this study, we theoretically derive Planck’s constant (h), a fundamental constant of quantum physics, by assuming same BM. Einstein investigated the BM of a resonator caused by photons in a black body; meanwhile, in this study, BM is considered to be caused by the particle and wave natures of light. As a result, as the functional form of h, hN=cε04πe2β1 h N = 4 π e 2 c ε 0 1 β (e: elementary charge, ε0 ε 0 : permittivity, c: speed of light) was derived, where the undetermined coefficient 2β β 2 represents the probability that the interaction between electrons and electromagnetic waves becomes particle-like absorption in one cycle of oscillation. The theoretical value of h was calculated to be cε04πe2π3 4 π e 2 c ε 0 π 3 by obtaining β from the transition conditions of the BM model and uncertainty relation. This theoretical value is within 0.3% of the most recent value of h. Besides, Nelson’s assumed diffusion coefficient in the stochastic mechanics, which is essential in obtaining Schrödinger’s equation, was proved to be obtained from the BM model and the uncertainty relation. This finding and the conclusion that the Planck’s constant can be mathematically derived, provide support for Nelson’s stochastic mechanics.
The theory of stochastic motion is formulated from a new point of view. It is shown that the fundamental equations of the theory reduce to Schrödinger's equation for specific values of certain parameters. A generalized Fokker-Planck-Kolmogorov equation is obtained; with other values of the parameters, certain approximations reduce this to the Smoluchowski equation for Brownian movement. In particular, the potential function in the Schrödinger equation differs in the two cases. The usual uncertainty relations appear in a natural way in the theory, but in a broader context. A single theory thus covers both similarities and differences between quantum-mechanical and Brownian motion. Furthermore, possibilities for broadening nonrelativistic quantum mechanics are brought out and, as an example, the possible corrections due to non-Markoffian terms are briefly studied.
The Schrödinger free particle equation in 1+1 dimension describes second-order effects in ensembles of lattice random walks, in addition to its role in quantum mechanics, and its solutions represent the continuous limit of a property of ensembles of Brownian particles. In the present paper, the classical Schrödinger and Dirac equations have been derived from the Brownian motions of a particle, and it has been shown that the classical Schrödinger equation can be transformed into the usual Schrödinger quantum equation on applying the Heisenberg uncertainty principle between position and momentum, while the Dirac quantum equation follows from its classical counterpart on applying the Heisenberg uncertainty principle between energy and time, without applying any analytical continuation.
It is shown that the motion of a Brownian particle in the Smoluchowski approximation may be described by a Schrödinger-like equation defining a complex probability amplitude whose norm is the same as the stochastic probability density. Furthermore, the quantum dynamical operators have a physical meaning which arises in a natural way, from the stochastic nature of the process. These operators satisfy the usual commutation relations and thus the uncertainty principle. Here the constant ℏ is replaced by a parameter depending on the characteristics of the system. In particular, the potential-energy operator for a Brownian particle subject to no external forces can be interpreted as a Rayleigh dissipative function.
In the context of quantum mechanics, Schrödinger's equation has many possible interpretations, none of which appear to be entirely satisfactory. One reason for this is that there is no known microscopic model which yields both Schrödinger's equation and an accompanying ‘sensible’ interpretation. In contrast to this the diffusion equation is well known as a phenomenological equation obtainable from the microscopic model of Brownian motion. In this case the microscopic model provides an unambiguous interpretation of the partial differential equation. In this article we review work which shows that Schrödinger's equation by itself is also easily obtained from a lattice random walk version of the Brownian motion model. In this system, Schrödinger's equation arises by projection and the interpretation of wave functions is as direct and unambiguous as the interpretation of density in the diffusion case. However the direct interpretation is compatible with quantum mechanics only in a statistical sense in the context of ensembles of particles. This shows that measurement theory is the aspect of quantum theory which harbours the problems of interpretation, and we suggest the study of this and related systems to clearly illuminate the boundary between dynamics and measurement.
The dynamics and kinematics of reciprocal diffusions were examined in a previous paper [J. Math. Phys. 34, 1846 (1993)], where it was shown that reciprocal diffusions admit a chain of conservation laws, which close after the first two laws for two disjoint subclasses of reciprocal diffusions, the Markov and quantum diffusions. For the case of quantum diffusions, the conservation laws are equivalent to Schrödinger’s equation. The Markov diffusions were employed by Schrödinger [Sitzungsber. Preuss. Akad. Wiss. Phys. Math Kl. 144 (1931); Ann. Inst. H. Poincaré 2, 269 (1932)], Nelson [Dynamical Theories of Brownian Motion (Princeton University, Princeton, NJ, 1967); Quantum Fluctuations (Princeton University, Princeton, NJ, 1985)], and other researchers to develop stochastic formulations of quantum mechanics, called stochastic mechanics. We propose here an alternative version of stochastic mechanics based on quantum diffusions. A procedure is presented for constructing the quantum diffusion associated to a given wave function. It is shown that quantum diffusions satisfy the uncertainty principle, and have a locality property, whereby given two dynamically uncoupled but statistically correlated particles, the marginal statistics of each particle depend only on the local fields to which the particle is subjected. However, like Wigner’s joint probability distribution for the position and momentum of a particle, the finite joint probability densities of quantum diffusions may take negative values.