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arXiv:quant-ph/9609023

Quantum Mechanics as a Classical Theory XIV: Connection with Stochastic Processes

L. S. F. Olavo

Published Sep 27, 1996• 0 citations
arXivHTMLPDF

Abstract

In this paper we are interested in unraveling the mathematical connections between the stochastic derivation of Schrödinger equation and ours. It will be shown that these connections are given by means of the time-energy dispersion relation and will allow us to interpret this relation on more sounded grounds. We also discuss the underlying epistemology.

DOI
10.48550/arxiv.quant-ph/9609023

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In this paper we are interested in unraveling the mathematical connections between the stochastic derivation of Schrödinger equation and ours. It will be shown that these connections are given by means of the time-energy dispersion relation and will allow us to interpret this relation on more sounded grounds. We also discuss the underlying epistemology.
In this paper we explore the mathematical and epistemological connections between the stochastic derivation of the Schr\"odinger equation and the one proposed by ourselves in previous papers. It will be shown that these connections are accomplished by means of the fluctuation-dissipation theorem, to which we may unambiguously relate the symbols and physical references of both approaches. As a by-product of our investigation, it will be possible to interpret the time-energy dispersion relation on sounder grounds. It will also be possible to discuss the superposition principle and to interpret it on a quite simple basis. The origin of the stochasticity and its relation to stability will be also addressed and the bridge to an axiomatic formulation of stochastic electrodynamics will be constructed.
Introduction. There exists a very comprehensive theory concerning the time development of probabilistic systems. This is the Theory of Stochastic processes (1). Quantum mechanics as described by the Schrödinger equation rests very heavily on probabilistic interpretation, and yet, after forty years of remarkably rapid advances and successes, it is not clear how it is related to stochastic theory. The formal similarity of certain aspects of, and equations used in these two theories are well known. The diffusion equation corresponds to the Schrödinger equation, a correspondence only marred by a square root of minus unity appearing with the time derivative in the latter equation. The Kolmogoroff–Chapman relation for probabilities in Brownian motion has an analogue (2) in quantum mechanics, not for probabilities though, but for the complex probability amplitudes. The uncertainty relation also has a stochastic analogue (3). Some of these analogous characteristics are actually symptomatic of the gulf which separates these two theories. There are solutions to the diffusion equation which have a very clear physical probabilistic meaning. This is not the case with the corresponding and formally similar solutions to the Schrödinger equation and is partly due to the √ − 1 mentioned earlier. The probabilistic interpretation of the solutions to the diffusion equation is direct and immediate. On the other hand, the probabilistic interpretation of the solutions, ψ of the Schrödinger equation involve the introduction of the probability density, p = ψ*ψ. One great difference between these two theories lies in their each having distinct principles of linear superposition. In Stochastic Theory probabilities are superposed and in quantum mechanics amplitudes or wave functions are superposed.
It is shown that, with suitable conventions, Schrödinger's equation can be deduced from the lows of stochastic diffusion: uncertainty relations, concrete meaning of mass-energy equivalence, minimal constants of time and length and vanishing of negative energy also appear resulting from the basic conventions. An extension to mechanics with spin is worked out, and a model of Dirac's fluied is elaborated. Stochastic assumptions also lead to the energy states and quantum numbers of the hydrogen atom. Suggestions are made on the physical interpretation of the model, on its relation to the wave-particle dualism, and on the meaning of quantized states.
We derive the equations of quantum mechanics and quantum thermodynamics from the assumption that a quantum system can be described by an underlying classical system of particles. Each component ϕjϕ_jϕj​ of the wave vector is understood as a stochastic complex variable whose real and imaginary parts are proportional to the coordinate and momentum associated to a degree of freedom of the underlying classical system. From the classical stochastic equations of motion, we derive a general equation for the covariance matrix of the wave vector which turns out to be of the Lindblad type. When the noise changes only the phase of ϕjϕ_jϕj​, the Schrödinger and the quantum Liouville equation are obtained. The component ψjψ_jψj​ of the wave vector obeying the Schrödinger equation is related to stochastic wave vector by ∣ψj∣2=⟨∣ϕj∣2⟩|ψ_j|^2=\langle|ϕ_j|^2\rangle∣ψj​∣2=⟨∣ϕj​∣2⟩.
We present a modified version of Nelson's seminal paper on the derivation of the time-dependent Schrödinger equation which draws on the equation of motion of a particle that moves under the influence of a classical force field and additional stochastic forces. The emphasis of our elaboration is focused on the implication of allowing stochastic forces to occur, viz. that the energy E of the particle is no longer conserved on its trajectory in a conservative force field. We correlate this departure Δ E from its classical energy with the energy/time uncertainty relation where Δ t is the average time for Δ E to persist. The stability of atoms, the zero-point energy of oscillators, the tunneling effect and the diffraction at slits are shown to be directly connected with the occurrence of such energy fluctuations. We discuss and rederive Nelson's theory entirely from this point of view and generalize his approach to systems of N particles which interact via pair forces. Achieving reversibility in a description of particle motion that is akin to Brownian motion, represents a salient point of the derivation. We demonstrate that certain objections raised against Nelson's theory are without substance. We also try to put the particular worldview of this version of stochastic quantum mechanics into perspective with regard to the established Copenhagen interpretation.