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Mathematical Methods in the Applied Sciences

A new mathematical method to describe the logic of the natural physical world

Xiaotong Ding, Guanghong Ding

Published Dec 19, 2022• 1 citation
Publisher

Abstract

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.

DOI
10.1002/mma.8956

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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.
In reality, micro objects and macro objects coexist harmoniously, and each macro object depends on micro particles to exist (macro is a collection of micro objects, indicating that they have a very extensive relationship and are symbiotic). However, classical mechanics and quantum mechanics have long been regarded as binary oppositions. This is a serious disconnect between theory and phenomena, requiring the establishment of new theories and methods that are compatible between quantum and classical. Prior to this, many reasons and methods for combining quantum theory with classical theory have been found (providing numerous computational examples). The wave mechanics equations of classical mechanics laws have been written. The technical method used is still the conversion rule of the Hermitian operator in the fundamental postulates of quantum mechanics. There are no new assumptions. The key is to use the relationship between force and energy in the planetary model to obtain wave mechanics equations that include force, Newton's second law, and the law of conservation of momentum. Thus, quantum mechanics, which is compatible with classical mechanics, truly becomes a form of 'mechanics'. This kind of mechanics is generalized wave mechanics. Quantum and classical are coordinated and compatible, resulting in no clear boundary between quantum and classical. It will seriously affect the credibility of quantum decoherence and wave function collapse.
The compatibility between quantum mechanics and classical mechanics is always worth discussing. Schrödinger equation (SE) can be derived from F=ma and T+V=E. Important classical physics formulas can be derived from the SE. The SE intuitively contains the mass m and the potential energy formula (this quality m can be large enough to enter the macro range and the potential energy comes from classical attraction). These three points determine that the SE is an organic combination of wave functions and classical mechanical laws. Based on this, it can be predicted that the SE, which can be used to describe macroscopic objects, can be established. It can be observed from the experiment of electron diffraction in a magnetic field that the volatility and property of moving electrons can be presented simultaneously. This experimental phenomenon, together with the mathematical analysis mentioned above, supports the conclusion that Newton's second law still applies in the microscopic world. According to the SE, it can be proven that "the energy of a moving particle described by the law of waves is equivalent to a multiple of its kinetic energy"; There is no absolute boundary between macro and micro (or classical and quantum), and they can be compatible.
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.
There is a big mismatch between the new technology based on the laws in the micro world, and the old mechanistic viewpoint established in the macro world. This is one of the main causes of the chaotic state of recent technology. I explain historically this circumstance and show how to settle this problem by solving the so-called measurement problem of quantum mechanics. These suggest a new worldview which epistemologically unifies the macro and micro worlds, and brings the new technology into order.
People believe that the Schrödinger equation cannot be used to describe macroscopic objects like the Earth, and Newtonian mechanics cannot be used to describe microscopic systems. The old concept of the relationship between the existing laws of quantum mechanics and classical mechanics undoubtedly has a serious impact on people’s understanding of the natural world, the development of physics theories, and the application of existing physics theories. The continuous development of physics theory requires constant changes to some incorrect old concepts. The Schrödinger equation that can describe planetary motion was successfully obtained by replacing the potential energy in the Hamiltonian operator from electromagnetic interaction potential energy to gravitational interaction potential energy. If the distance between the sun and the earth is approximated as a constant, the energy eigenvalues obtained by solving the Schrödinger equation for the Earth’s revolution are completely consistent with the results obtained directly using classical mechanics. The direct significance of establishing and applying such equations is that they can simultaneously use classical mechanics and wave dynamics to describe all objects (no longer limited by the mass of the objects), simplifying the calculation process of quantum mechanics. It has been proven that classical mechanics and wave dynamics are compatible. It has been proven that classical mechanics and wave dynamics are compatible, and there is no insurmountable gap between them. This result has a huge positive impact on the theoretical updates and applications of quantum mechanics.
The aim of the paper is to seek and identify mathematical models designated to understand better the quantum scale phenomena, models which could permit to link them with the real world phenomena. This shall be made by analyzing the common features exhibited by both, normal scale and quantum scale phenomena and then, by creating mathematical models that are describing both, the quantum and the real world. One main idea of the paper, on which the involved models are built is that observed physical quantities such as position and momentum of a quantum particle, but also the laws of their transformation (of that physical quantities) from one state into another such as for example the probability wave function described by the Schrödinger’s equation, are signals transmitted out of the quantum world into the real physical world by the a certain real, not yet detected wave function, wave function, underlying the visible measurable reality and which is describing this more complex reality underlying the observable one, and unavailable not only to our limited senses but also not available, even though using the new discovered technologies. The two observed physical quantities, position and momentum, are describing a quantum system, that is to say the quantum system can be represented either as a wave function of the position of the particle or as a wave function of the momentum of the particle, this two different functions being linked one to another by means of the Fourier Transform. The paper is furthermore describing the way in which the theory of Signal Processing, a powerful and useful tool, is to be used in order to model both, quantum and large scale phenomena.