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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Processing in (linear) systems with stochastic input

Cătălin Silviu Nuțu, Tiberiu Axinte

Published Dec 14, 2016• 1 citation
Publisher

Abstract

The paper is providing a different approach to real-world systems, such as micro and macro systems of our real life, where the man has little or no influence on the system, either not knowing the rules of the respective system or not knowing the input of the system, being thus mainly only spectator of the system’s output. In such a system, the input of the system and the laws ruling the system could be only “guessed”, based on intuition or previous knowledge of the analyzer of the respective system. But, as we will see in the paper, it exists also another, more theoretical and hence scientific way to approach the matter of the real-world systems, and this approach is mostly based on the theory related to Schrödinger’s equation and the wave function associated with it and quantum mechanics as well. The main results of the paper are regarding the utilization of the Schrödinger’s equation and related theory but also of the Quantum mechanics, in modeling real-life and real-world systems.

DOI
10.1117/12.2240955

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The paper is providing a different approach to real-world systems, such as micro and macro systems of our real life, where the man has little or no influence on the system, either not knowing the rules of the respective system or not knowing the input of the system, being thus mainly only spectator of the system’s output. In such a system, the input of the system and the laws ruling the system could be only “guessed”, based on intuition or previous knowledge of the analyzer of the respective system. But, as we will see in the paper, it exists also another, more theoretical and hence scientific way to approach the matter of the real-world systems, and this approach is mostly based on the theory related to Schrödinger’s equation and the wave function associated with it and quantum mechanics as well. The main results of the paper are regarding the utilization of the Schrödinger’s equation and related theory but also of the Quantum mechanics, in modeling real-life and real-world systems.
Quantum mechanics has been enormously successful in explaining the behavior of microsystems such as molecules, atoms, and nuclei. We have seen that quantum mechanics describes a quantum system by the wave function ψ, which is the solution of the Schro¨dinger equation. The wave function determines the probability amplitudes (whose modulus squared gives the probability) for all possible outcomes of a measurement corresponding to an observable on the system. In general, the wave function does not uniquely predict the result of a measurement; it only provides a probability distribution for all possible results of the measurement on a quantum system. Only if the system happens to be in an eigenstate corresponding to the observable being measured, is the result of a measurement of this observable completely predictable. On the other hand, the results of measurements corresponding to a conjugate or non-commuting observable, on the same system in the same state, are indeterminate. Quantum mechanical description of physical systems raises profound questions: How are the wave function and the probabilities for different outcomes of a measurement to be interpreted? Does the wave function represent a single system or an ensemble of systems? Does the physical system actually have the attribute corresponding to the measurement in question or does the measurement itself create the attribute? These questions led to different interpretations of quantum mechanics, which were vigorously debated during the years of its discovery as also in the following decades. The new idea of indeterminacy was especially difficult to reconcile with the long-held view that a physical theory should aim at predicting accurately, and without ambiguity, the result of any measurement on any system, big or small. These debates not only gave us a deeper understanding of quantum mechanics but also changed profoundly the way we view the micro-world.
To model an objective real world with absolute accurate mathematics is the symbol of modern science. However, in real life, the physical parameters could not be measured in infinite high accuracy, which makes the mathematical equations ineffective in the circumstance of turbulence and chaos. As the macro‐ and micro‐physical world could not be unified, it is believed that the laws of macro‐ and micro‐physical world are not the same. This paper proposes a new way of illustrating physical parameters, using the combination of mean ± error instead of a single numeric parameter for illustration, and establishes the corresponding algorithm and equations for means and errors, respectively. The results show that the error part of the Navier–Stokes equations in macro‐physical fluid mechanical world is just Schrodinger equation of micro‐physical quantum mechanical world. Therefore, a new method of researching complex system is proposed, and this method establishes the relationship between macro‐Newtown mechanics and micro‐quantum mechanics, so that in the macro world, the mean will follow the Navier–Stokes equations, and the error will follow the Schrodinger equations.
It is a fundamental and widely accepted assumption that a measure-ment result exists universally, and in particular, it exists for every ob-server, independently of whether the observer makes the measurement or knows the result. In this paper, we will argue that, based on an analysis of protective measurements, this assumption is rejected by the many-worlds interpretation of quantum mechanics, and worlds, if they indeed exist ac-cording to the interpretation, can only exist relative to systems which are decoherent with respect to the measurement result. In standard quantum mechanics, it is postulated that when the wave func-tion of a quantum system is measured by a macroscopic device, it no longer follows the linear Schrödinger equation, but instantaneously collapses to one of the wave functions that correspond to definite measurement results. However, this collapse postulate is ad hoc, and the theory does not tell us why and how a definite measurement result appears. This measurement problem is widely
Abstract In stark contrast to classical Newtonian mechanics, the standard interpretation of quantum mechanics associates the Schrödinger wavefunction, not in a determ- inistic way with definite predictions about a specific, individual experimental result, but only in a probabilistic manner, with |ψ(x, t )|2 acting as a probability density, in this case, for position measurements. Before analyzing this approach in more detail, we will find it profitable to briefly review some of the basic concepts of discrete and
Little is known about dark matter particles save that their most important interactions with ordinary matter are gravitational and that, if they exist, they are stable, slow moving and relatively massive. Based on these assumptions, a semiclassical approximation to the Schrödinger equation under the action of a Coulomb potential should be relevant for modelling their behaviour. We investigate the semiclassical limit of the Schrödinger equation for a particle of mass M under a Coulomb potential in the context of Nelson’s stochastic mechanics. This is done using a Freidlin-Wentzell asymptotic series expansion in the parameter ϵ=ħ/M for the Nelson diffusion. It is shown that for wave functions ψ ∼ exp((R + iS)/ϵ2) where R and S are real valued, the ϵ = 0 behaviour is governed by a constrained Hamiltonian system with Hamiltonian Hr and constraint Hi = 0 where the superscripts r and i denote the real and imaginary parts of the Bohr correspondence limit of the quantum mechanical Hamiltonian, independent of Nelson’s ideas. Nelson’s stochastic mechanics is restored in dealing with the nodal surface singularities and by computing (correct to first order in ϵ) the relevant diffusion process in terms of Jacobi fields thereby revealing Kepler’s laws in a new light. The key here is that the constrained Hamiltonian system has just two solutions corresponding to the forward and backward drifts in Nelson’s stochastic mechanics. We discuss the application of this theory to modelling dark matter particles under the influence of a large gravitating point mass.