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Mağallaẗ Al-Qirā'aẗ wa Al-Maʿrifaẗ/Mağallaẗ Al-Qirā'aẗ wa Al-Maʿrifaẗ

"Analytical and numerical study of some problems in nonlinear mechanics"

Ibtihal Mohammed ALSalem

Published Jan 1, 2024• 0 citations
Publisher

Abstract

This study deals with the topic of “Analytical and numerical study of some problems in nonlinear mechanics.” The study aimed to analyze a problem in nonlinear mechanics, which is “wave function collapse.” Which is represented by the laboratory experiment "Schrödinger's cat", the interpretations of Heisenberg and Copenhagen, and an application of the method of Lagrange's equations to explain the phenomenon of wave function collapse and the EPR paradox, and Einstein's interpretation in pure algebraic theory and his interpretation of wave collapse, in addition to the Schrödinger equation. The study concluded with a set of results, the most important of which are: that the problem Wave function collapse represents one of the measurement problems in quantum mechanics. It has been proven that it can be solved based on quantum mechanics and does not require any additional assumptions or new theories. The processes of particle creation and annihilation described based on quantum field theory play a major role in measurement processes. Decoherence theory also refers to the creation and annihilation of particles and these processes are a nonlinear consequence of quantum mechanics, in which case the wave function collapse term becomes a consequence of other statements of quantum mechanics rather than a separate assumption of quantum mechanics.

DOI
10.21608/mrk.2024.336745

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This study deals with the topic of “Analytical and numerical study of some problems in nonlinear mechanics.” The study aimed to analyze a problem in nonlinear mechanics, which is “wave function collapse.” Which is represented by the laboratory experiment "Schrödinger's cat", the interpretations of Heisenberg and Copenhagen, and an application of the method of Lagrange's equations to explain the phenomenon of wave function collapse and the EPR paradox, and Einstein's interpretation in pure algebraic theory and his interpretation of wave collapse, in addition to the Schrödinger equation. The study concluded with a set of results, the most important of which are: that the problem Wave function collapse represents one of the measurement problems in quantum mechanics. It has been proven that it can be solved based on quantum mechanics and does not require any additional assumptions or new theories. The processes of particle creation and annihilation described based on quantum field theory play a major role in measurement processes. Decoherence theory also refers to the creation and annihilation of particles and these processes are a nonlinear consequence of quantum mechanics, in which case the wave function collapse term becomes a consequence of other statements of quantum mechanics rather than a separate assumption of quantum mechanics.
Abstract The problem of the wave function collapse (a problem of measurement in quantum mechanics) is considered. It is shown that it can be solved based on quantum mechanics and does not require any additional assumptions or new theories. The particle creation and annihilation processes, which are described based on quantum field theory, play a key role in the measurement processes. Superposition principle is not valid for the system of equations of quantum field theory for particles and fields, because this system is a non-linear. As a result of the creation (annihilation) of a particle, an additional uncertainty arises, which “smears” the interference pattern. The imposition of such a large number of uncertainties in the repetitive measurements leads to the classical behavior of particles. The decoherence theory also implies the creation and annihilation of particles, and this processes are the consequence of non-linearity of quantum mechanics. In this case, the term “collapse of the wave function” becomes a consequence of the other statements of quantum mechanics instead of a separate postulate of quantum mechanics.
The problem of measurement in quantum mechanics is that the quantum particle in the course of evolution, as described by the linear Schrodinger equation, exists in all of its possible states, but in measuring, the particle is always detected in only one of its states. This property is called the "collapse of the wave function" and was formulated by Von Neumann as one of the postulates of quantum mechanics. However, it remains unclear at what point in time and under what laws this transition occurs. This article demonstrates that the collapse of the wave function may be due to the creation or annihilation of particles (quasi-particles). The processes of the creation or annihilation of particles play a key role in the measurements and are described on the basis of quantum field theory. The system of equations of quantum field theory of particles and fields is non-linear; as a result, the principle of superposition does not hold for the theory. The collapse of the wave function is a consequence of this non-linearity and occurs at the moment of creation (annihilation) of a particle. This result demonstrates that the wave function collapse can occur in both microscopic and macroscopic systems. Understanding the mechanisms of the collapse of the wave function can lead to the creation of microscopic devices involved in the calculations based on quantum computing.
The basic idea of quantum mechanics is that the property of any system can be in a state of superposition of various possibilities (or eigen states).This state of superposition is also known as wave function and it evolves linearly with time in a deterministic way in accordance with the Schrodinger equation.However, when a measurement is carried out on the system to determine the value of that property (say position), the system instantaneously transforms to one of the eigen states and thus we get only a single value as an outcome of the measurement.Quantum measurement problem seeks to find the cause and exact mechanism governing this transformation.In an attempt to solve the above problem, in this paper, we will first define what the wave function represents in real-world and will identify the root cause behind the stochastic nature of events.Then, we will develop a model to explain the mechanism of collapse of the quantum mechanical wave function in response to a measurement.In the process of development of model, we will explain Schrodinger cat paradox and will show how Born's rule for probability becomes a natural consequence of the measurement process.
We use a two states quantum spin system S, and thus considering the particular case of three anticommuting elements and the measurement of . We evidence that, during the wave collapse, we have a transition of standard commutation relation of the spin to new commutation relations and this occurs during the interaction of the S system with the macroscopic measurement system M. The reason to accept such view point is that it causes the destruction of the interferential factors and of the fermion creation and annihilation operators of the S system without recourse to further elaborations based on the use of Hamiltonians or other methods. By this formulation we propose a new method in attempting to solve the problem of wave function collapse. The concept of Observable , in use in standard quantum mechanics, is resolved in an abstract entity to which is connected a linear hermitean operator that signs mathematically the operation that we must perform on the wave function in order to obtain the potential and possible values of the observable. It does not commute with a number of other operators characterizing the system and the non commuting rules have a fundamental role in quantum mechanics .They have a logic that must be analyzed in each phase of the non measuring and the measuring processes. When we consider the dynamics of wave function collapse we must account that the observed observable becomes a number ,with proper unity of measurements ,during the measurement, thus the linear hermitean operator to which is connected before the measurement, disappears and in its place it appears a new operator that maintain the non commutativity with the other operators to which the old and disappeared operator was connected.
This thesis is an attempt to reconstruct the conceptual foundations of quantum mechanics. First, we argue that the wave function in quantum mechanics is a description of random discontinuous motion of particles, and the modulus square of the wave function gives the probability density of the particles being in certain locations in space. Next, we show that the linear non-relativistic evolution of the wave function of an isolated system obeys the free Schrödinger equation due to the requirements of spacetime translation invariance and relativistic invariance. Thirdly, we argue that the random discontinuous motion of particles may lead to a stochastic, nonlinear collapse evolution of the wave function. A discrete model of energy-conserved wavefunction collapse is proposed and shown to be consistent with existing experiments and our macroscopic experience. In addition, we also give a critical analysis of the de Broglie-Bohm theory, the many-worlds interpretation and dynamical collapse theories, and briefly analyze the problem of the incompatibility between quantum mechanics and special relativity.
In his famous thought experiment, Schrodinger(1935) imagined a cat that measures the value of a quantum mechanical observable with its life. Since Schrodinger’s time,no any interpretations or modifications of quantum mechanics have been proposed which gives clear unambiguous answers to the questions posed by Schrodinger’s cat of how long superpositions last and when (or whether) they collapse?In this book appropriate modification of quantum mechanics is proposed.We claim that canonical interpretation of the wave function ψ=с1ψ1+с2ψ2 is correct only when the wave function supports ψ1 and ψ2 essentially overlap. When these ones have separated supports we claim that canonical interpretation of the wave function ψ is no longer valid for a such cat state.Possible solution of the Schrodinger’s cat paradox is considered. We pointed out that the collapsed state of the cat always shows definite and predictable outcomes even if the cat also consists of a superposition:|cat>=с1|alive cat>+с2|death cat>.Using new quantum mechanical formalism the EPRB-paradox is considered.We find that the EPRB-paradox can be resolved by nonprincipal and convenient relaxing of the Einstein’s locality principle.