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Theoretical and Mathematical Physics

On the nature of the quantum corrections in the case of stochastic motion of a nolinear oscillator

V. V. Sokolov

Published Oct 1, 1984• 12 citations
Publisher
DOI
10.1007/bf01038553

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A new pre-quantum level physical theory underlying the quantum mechanics has been developed and its epistemological features are mainly derived from the fluctuating classical electromagnetic zeropoint field which induces random oscillations on charged particle. In complex vector space, the average random oscillations of the charged particle are treated as complex rotations so that a particle has extended structure. It is found that the quantum nature of particles arises from the superposition of internal oscillations on the center of mass motion. Using Lagrangian mechanics in complex vector space, a generalized Newton’s equation is derived and which is shown to be acceptable for both micro and macro mechanics. The generalized Newton’s equation contains additional relativistic terms at the micro level. Further, a complex Hamilton-Jacobi equation is derived and which is used to solve quantum mechanical problems in classical approach. The presence of zeropoint field also produces modifications in the central potential and the additional terms are responsible for the effects like advance of planets perihelion and deflection of light near massive objects at macro level. Further, in the case of atoms such terms lead to the estimation of Lamb shift using stochastic electrodynamics. The consideration of extended particle structure in stochastic electrodynamics allows deriving mass correction and charge correction and consequently the estimation of anomalous magnetic moment. The fundamental deeper level theory developed here, may further enhance the efforts for research connecting micro and macro aspects of matter. Keywords: Foundations of Quantum Mechanics, Geometric Algebra, Stochastic Electrodynamics, Zeropoint Energy, Gravitation. DOI : 10.7176/APTA/88-04 Publication date : June28th 2024
Introduction The apparent generality of the title to this chapter is appropriately reduced by sufficiently precisely defining the terms used. We have in mind the natural extension to quantum mechanical settings of the ideas usually covered in a discussion of stochastic processes in classical mechanical settings. In classical mechanical settings, the fundamental process is Brownian motion, which describes the influence of external fluctuating forces on a small particle. The mathematical apparatus developed for the study of Brownian motion has been generalized and applied to irreversible thermodynamics (Onsager and Machlup, 1953), fluctuating hydrodynamics (Fox and Uhlenbeck, 1970), light scattering from fluids and mixtures (Berne and Pecora, 1976) and chemical reactions (McQuarrie and Keizer, 1981). In each case, the stochastic process involves fluctuations generated by thermal motions. We are not considering the stochastic trajectories (chaos) of classically deterministic motion in non-linear systems (Lichtenberg and Lieberman, 1983), which is an entirely separate context for stochastic processes. Our goal in this chapter is to study the effects of thermal fluctuations in quantum mechanical settings. Therefore, we exclude a discussion of interpreting quantum mechanics as a classical stochastic motion (Nelson, 1966), an idea which should have been put to rest by Grabert, Hänggi and Talkner (1979). We also exclude a discussion of quantum chaos, which is currently a very active area of research (Zaslavsky, 1981). Just as Brownian motion provides the paradigm for stochastic processes in classical mechanical settings, the Anderson–Kubo oscillator (Anderson, 1954; Kubo, 1954; Kubo, Toda and Nashitsume, 1985) is the paradigm for stochastic processes in quantum mechanical settings.
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.