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Engineering mechanics

New wave function and its new differential equation in quantum physics. The fuzzy logic and fuzzy sets theory of Lutfi Zadeh

Published Jan 1, 2024• 0 citations
Publisher

Abstract

It is known the quantum mechanics is different from the classical mechanics.The reason is the principle of Heisenberg's Uncertainty in quantum mechanics.The uncertainty as principle of mechanics result in wave function in it, square of which is probability.Therefore principal stochasticity of microworld which noted Bohr in famous dispute with Einstein about a God playing dice results in the quantum logic.In frame this logic the Schrodinger cat is alive and dead the same time.The probability theory with its elementary events in its topological probability space cannot extend its definitions.Fuzzy logic and fuzzy sets of Zadeh with focal elements in topological possibility space can makes it.

DOI
10.30546/2706-7734.44.7.2024.16

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It is known the quantum mechanics is different from the classical mechanics.The reason is the principle of Heisenberg's Uncertainty in quantum mechanics.The uncertainty as principle of mechanics result in wave function in it, square of which is probability.Therefore principal stochasticity of microworld which noted Bohr in famous dispute with Einstein about a God playing dice results in the quantum logic.In frame this logic the Schrodinger cat is alive and dead the same time.The probability theory with its elementary events in its topological probability space cannot extend its definitions.Fuzzy logic and fuzzy sets of Zadeh with focal elements in topological possibility space can makes it.
Recently we proposed the linguistic Copenhagen interpretation of quantum mechanics, which is called quantum language or measurement theory. This theory is valid for both quantum and classical systems. Thus we think that quantum language is one of the most powerful scientific theories, like statistics. In this paper we justify Zadeh’s fuzzy sets theory in quantum language, that is, fuzzy propositions are identified with binary measurements. This implies that the definition of “proposition” is, for the first time, acquired in the field of non-mathematics. Further, we assert that fuzzy logic is more natural than crisp logic in science. And furthermore, we discuss and solve Saussure’s linguistics, Moore’s paradox, Quine’s analytic-synthetic distinction and Lewis Carroll’s logical paradox. Therefore, from the philosophical point of view, our result gives a complete answer to Wittgenstein’s problem: “Why does logic work in our world?” and “What is a scientific proposition?” in his picture theory. That is, we simultaneously justify both Zadeh’s fuzzy sets and Wittgenstein’s picture theory in the quantum mechanical worldview.
New scientific theories have been and will always be required for the progress of science. Two examples of new theories that were advanced in the 20th century are quantum mechanics and fuzzy set theory. Both deal with the concept of uncertainty and both use mathematical tools that are essentially different from those used in classical probability theory. In both, the introduction of new concepts (theoretical terms) was necessary because phenomena and processes observed in the real world could not be explained with existing mathematical concepts. This paper examines the possibilities of (re)interpreting uncertainty in quantum mechanics with the aid of fuzzy sets and systems
Human interaction with the world is dominated by uncertainty. Probability theory is a valuable tool to face such uncertainty. According to the Bayesian definition, probabilities are personal beliefs. Experimental evidence supports the notion that human behavior is highly consistent with Bayesian probabilistic inference in both the sensory and motor and cognitive domain. All the higher-level psychophysical functions of our brain are believed to take the activities of interconnected and distributed networks of neurons in the neocortex as their physiological substrate. Neurons in the neocortex are organized in cortical columns that behave as fuzzy sets. Fuzzy sets theory has embraced uncertainty modeling when membership functions have been reinterpreted as possibility distributions. The terms of Bayes' formula are conceivable as fuzzy sets and Bayes' inference becomes a fuzzy inference. According to the QBism, quantum probabilities are also Bayesian. They are logical constructs rather than physical realities. It derives that the Born rule is nothing but a kind of Quantum Law of Total Probability. Wavefunctions and measurement operators are viewed epistemically. Both of them are similar to fuzzy sets. The new link that is established between fuzzy logic, neuroscience, and quantum mechanics through Bayesian probability could spark new ideas for the development of artificial intelligence and unconventional computing.
The fussy "roots" of quantum mechanics are traced directly to the Hamilton-Jacobi equation of classical mechanics. It is shown that the Shroedinger equation can be derived from the Hamilton-Jacobi equation. A deep underlying unity of both equations lies in the fact that a unique trajectory of a classical particle is "selected" out of many-continuum paths according to the Principle of least action. We can say that a classical particle has a membership in every path of the above set, which collapses to the winning single trajectory of a real motion. At the same time it can also be said that a quantum mechanical "particle" has different degrees of membership in a set of many-continuum paths where all of them contribute to the dynamics of the quantum mechanical particle. This allows one to provide an interpretation of the wave function as a parameter describing deterministic entity endowed by a fuzzy character. As a logical consequence of such an interpretation the complimentarity principle and the wave-particle duality concept can be abandoned in favor of a fuzzy deterministic microobject. This idea leads to a possibility of a quantum mechanical computer based on the fuzzy logic.
Till 1925 the idea of what the quantum theory would be was unknown. During 1925 three different but equivalent versions of quantum theory were proposed – Schrödinger-proposed wave mechanics; Heisenberg-developed matrix mechanics and Dirac-introduced operator theory. Considering the de Broglie’s matter waves Erwin Schrödinger, an Austrian physicist, argued that if a particle like an electron behaves as a wave then the equation of wave motion could be successfully applied to it. He postulated a function varying in both space and time in a wave-like manner (hence called wave function and denoted it as ψ). This function is generally complex and assumed to contain information about a system. Schrödinger set up a linear and time-dependent wave-like equation, called Schrödinger wave equation, to describe the wave aspect of a particle taking account of de Broglie’s relation for wavelength. Physically, |ψ(X, t)|2, where ψ(X, t) is the solution of the Schrödinger equation, is interpreted as position probability density. That is, |ψ(X, t)|2 is the probability density of observing a particle at position X at time t. ψ does not give exact outcomes of observations but helps us to know all possible events and their probabilities . Further, the probability interpretation allows us to find the average or expected result of a set of measurements on a quantum system.
Early in the development of quantum theory Bohr introduced what came to be called the Copenhagen interpretation. Specifically, the square of the absolute value of the wave function was to be used as a probability density. There followed lengthy arguments about this ranging from alternative universes to Schrodinger's cat. Einstein famously remarked "I am convinced that He (God) does not play dice." The purpose of this paper is to present a mathematical model of the measuring process that shows that the Copenhagen interpretation can actually follow from the fact that the time development of quantum systems is governed by the usual one parameter group of unitary transformations exp-iHt and that probability enters into the theory in the way it usually does in physics, namely, by having a large number of deterministic equations that can only be handled probabilistically.