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Progress of Theoretical Physics

Quantum Laws in Connection with Stochastic Processes

J. C. Aron

Published Apr 1, 1965• 6 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.33.726

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Zeitschrift für Physikalische Chemie•Nov 24, 2010•15 citations
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.
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 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 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.
In This essay I shall derive the laws of nature from a simple mathematical system from a postulate that reality is indeed a mathematical structure. The system can be simulated by a computer program to generate many results that agree with Quantum mechanics. Also I will show that the system can be put in regular more familiar mathematical formalism. The postulate lead to assume particles are made of random lines were one end originates in a small region representing the particle and it extends to all other points in space and some ending on other particles. The points are really nothing but random numbers, hence reality is nothing but some relation between random numbers. Moreover, the lines are responsible for the interaction by a process of crossing or not crossing or meeting.The start point and the end point of these lines define space and the length of the line is interpreted as energy, time is just a change of state. The system unifies space, time, matter, energy and interaction, all in one coherent picture, so particles and the laws of nature governing them appear naturally. The simulations generate some basic Quantum Mechanics results and the 1/r law as in quantum field Theory. There are many other results such as the hydrogen 1s level where the universal constants like c, h, e and their relation that lead to Fine Structure constant automatically fall out of the simulation. Two such simulations are carried out; one is Bohr like model and the other Schrodinger like equations solution and show the equivalency. Also, the mass of the electron appear naturally using a simulation which is an extension of the Bohr model which in turn leads to the predicting the size of the proton. The system automatically displays the non-local behavior and explains the EPR in simple terms and shows the origin of spin. Some interesting formulas connecting proton to electron mass ratio, FSC and electron g-factor are produced. While it is shown that coulomb potential is produced by line crossing, Gravity appears when lines meet at a region of Planck's length(gravity needs more investigation).
Abstract Quantum thermodynamics describes dynamic processes by means of the operators of entropy production P and time t . P and t do not commute. It exists the non-vanishing t–P commutator [ t , P ]=ik. Here the Boltzmann constant k has the physical meaning of a quantum of entropy. The t–P commutator immediately leads us to the t–P uncertainty relation ΔtΔP≥k/2. Hence the observables t and P are not sharply defined simultaneously. Similar uncertainty relations can also be expected for other pairs of conjugate variables with products of the physical meaning of an entropy. The free energy F and the reciprocal temperature (1/T) are the respective conjugate variables of an isolated system of many particles, which leads us to the F–(1/T) uncertainty relation ǀΔFǁΔ(1/T)ǀ≥k/2. It can be traced back to the t–P uncertainty relation mentioned above. In this way the Helmholtz free energy F and the temperatur T are introduced into quantum thermodynamics. The uncertainties ǀΔFǀ→0 and ǀΔTǀ→0 are negligible at low temperatures T→0, and quantum thermodynamics turns into the time-independent classical thermodynamics. Against this the uncertainties ǀΔFǀ→∞ and ǀΔTǀ→∞ grow unlimited at high temperatures T→∞, and classical thermodynamics loses its sense. In the limit of one particle the uncertainties cannot be neglected even at low temperatures. However a detailed discussion shows that the free energy f of a single particle vanishes within the whole range of temperatures T. This defines the particle entropy σ=ε/T=ak. The dimensionless entropy number a connects the particle energy ε=akT with the temperature T. The entropy number a of a single (s) independent particle can be calculated with the extended, temperature-dependent Schrödinger equation A s φ=aφ. Here A s=−(Λ 2/4π)∇2 means the dimensionless entropy operator describing the entropy number a and thus the particle entropy σ=ak. Λ is the thermal de Broglie wave length. Finally we calculate by means of quantized particle entropies σ the internal energy E, the Helmholtz free energy F, the entropy S, the chemical potential μ, and the equation of state of an ideal gas of N monatomic free particles in full agreement with classical thermodynamics. We also calculate the partition function q=V/Λ 3 of a single free particle within the volume V. Here Λ 3 is a small volume element taking into account the wave–particle dualism of a single free particle of mass m at temperature T. Extension to a system of N free particles leads us to a simple geometrical model and to the conclusion that an ideal gas of independent particles becomes instable below a critical temperature T C. T C corresponds to the critical temperature T BE of Bose–Einstein condensation.