Runsheng Tu

(Huanggang Normal University, China)

Abstract

People believe that the Schr?dinger 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 Schr?dinger 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 Schr?dinger 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.

Keywords: planetary model, Schr?dinger equation, quantum mechanics, classical mechanics, compatibility, meaning of wave function.

1. Introduction

Can the revolution of the earth be described by Schr?dinger equation? If it can, it will have a great impact on the existing theoretical physics. After this method is extended to all objects, there will be the Schr?dinger equation of gravitational potential energy and the Schr?dinger  equation applicable to all objects. Previously, people were bound by the uncertainty of microscopic particles and the non-localized realism, and the Schr?dinger equation of gravitational potential energy or the Schr?dinger equation describing macroscopic objects never appeared in textbooks. Since people have never tried to establish such Schr?dinger equation, it is of great significance for us to try it here. This article is an attempt, and it has been successful.

When people have to establish and apply wave mechanics or quantum mechanics, they all realize that the micro world is so different from the macro world. And recognized the notion that "the cognition, experience, rules, and theories of the macro world established by humans in theory and practice have largely failed in the micro world". So that people think that there is a huge gap between the micro world and the macro world. Specifically, the causality or determinism of Newtonian mechanics and classical electrodynamics, which are applicable to the macro world, are no longer applicable to the corresponding occasions in the micro world, but wave mechanics, natural randomness and uncertainty, which are applicable to the micro world, are not suitable for describing the movement changes of macro objects. Although there has been Ellen feaster Theorem [1] and it is the mission of condensed matter physics to explain macroscopic phenomena with microscopic theory. But this has not fundamentally bridged the gap between the micro-world and the macro-world. Because, influenced by Lenfest's theorem and condensed matter physics, the relationship between macroscopic objects and microscopic particles is the relationship between quantitative change and qualitative change. Only when the Schr?dinger equation is applied to the macro world without being limited by  mass can the gap between the macro world and the micro world be basically bridged (the difference between them is no longer dominant).

In fact, the form and application of Hamiltonian operator are not limited by the mass of the object, nor is it limited that the potential energy can only be electromagnetic interaction potential energy. Before this paper, people only used Schr?dinger equation on microscopic objects (if it is a microscopic object in a bound state, the binding force is only the electromagnetic interaction force, and the potential energy is the potential energy of the electromagnetic interaction force). We can't find the mathematical and logical basis for doing this, and we have to say that this is a habit formed by the bondage of ideas. There is no theoretical obstacle or mathematical logic obstacle in using gravitational potential energy in Hamiltonian operator. In this way, according to Schr?dinger's method, Schr?dinger  equation suitable for macro system can be established completely. The establishment of section 2 in this paper is applicable to the Schr?dinger  equation of gravitational potential energy of macro-system. In the third section of this paper, the newly established Schr?dinger equation applicable to macroscopic objects (including gravitational potential bound state system) is verified by using the known data of the earth's revolution.

Since the planetary model and Schr?dinger equation can be used to describe an object at the same time, there is no obstacle to using the planetary model in the microscopic system to which Schr?dinger  equation applies. That is to say, both microscopic and macroscopic systems can use planetary model (or classical mechanics) and wave mechanics at the same time. This gives birth to the power that can make people change their existing ideas. Have readers heard of (or seen) the Schr?dinger equation of the earth's revolution? If not, then the author's research work may lead a new trend. The successful establishment of Schr?dinger equation of planetary motion shows that we have a theoretical basis for using wave mechanics and classical mechanics at the same time.

In reference [2-7], the author calculates atoms and molecules by using wave mechanics and planetary model at the same time, which shows that the viewpoint, theory and method that "wave mechanics and classical mechanics can be used to describe the motion system at the same time" have a wide application prospect. Wave mechanics and classical mechanics are used to describe a system at the same time, which is not limited by the  mass of the system. The natural rules accumulated by the long experience in the past will not completely fail even in the micro world. This is a major conceptual revolution in the history of human development. It will also lead to a great revolution in the theory and method of basic physics.

The basic assumption in references [6-10] that "specific waves propagate along a small circle to form electrons" lays the foundation for the conclusion that "classical mechanics and quantum mechanics are compatible and can be used simultaneously on macroscopic and microscopic objects". Reference [10] derived the Schr?dinger equation that can describe the Earth's orbital motion. This article is an expanded description of the content in section 3 of reference [10]. This article introduces the significance and application of the Schr?dinger equation. It belongs to the category of macroscopic system Schr?dinger equation and its application research progress.

2. Schr?dinger equation that can describe the revolution of the earth

In this article, letters with subscript e represent the physical quantities of electrons. The existing mathematical formal and explanatory systems of quantum mechanics believe that there is an insurmountable gap between the microscopic world and the macroscopic world in terms of the laws or behaviors that things follow. One specific way of expressing it is that macroscopic objects cannot use wave dynamics (including the Schr ? dinger equation), while microscopic objects do not conform to classical mechanical theories and rules. However, as long as we analyze carefully, it is not difficult to see that the first term in the Hamiltonian operator used to establish the Schr ? dinger equation is the kinetic energy operator, and the second term is the potential energy operator (which is also the potential energy function itself). In the solar system, the Earth, which is in a bound state, also has kinetic and potential energy, and conforms to the Viry's theorem. The current quantum mechanics does not limit the mass of moving objects that conform to the de Broglie wave formula. We have no reason to say that describing macroscopic objects cannot use Hamiltonian operators. The reason for using the wave function in the Schr ? dinger equation is unknown, but it is very useful in reality. Electrons and other particles, as well as macroscopic objects, are entities with static mass. We have no reason to say that only describing microscopic objects can use wave functions, while describing macroscopic objects cannot use wave functions. Because for de Broglie waves, only their wavelength is related to mass, without any upper limit on mass (i.e. macroscopic objects with large mass also have corresponding de Broglie waves).

We have no reason to believe that the 'V' in the Hamiltonian operator can only be the potential energy of electromagnetic interactions. We also have no reason to believe that the first term in the Hamiltonian operator can only apply to microscopic objects. It can be seen that we have no reason to reject the following choices: the first term in the Hamiltonian operator also applies to macroscopic objects, and the second term can be either the electromagnetic interaction potential energy function or the gravitational potential energy function. Under this premise, we can fully use the Schr ? dinger equation to describe macroscopic objects, and the Schr ? dinger equation used to describe planetary revolution is equation (4). The applicability of the Schr ? dinger equation, which was originally only applicable to microscopic objects and electromagnetic interactions, has been expanded to apply to macroscopic systems, and is applicable to bound state systems with four fundamental interactions. This article does not discuss the basic interaction systems outside of electromagnetic and gravitational interaction systems.

i?         (4)

In the equation, E is the energy eigenvalue of a classical system bound by gravity. (4) The formula is not applicable to particles with zero rest mass, but to macroscopic systems in bound states. When V=0, equation (4) still applies to unconstrained macroscopic systems [see equation (8) for details]. Simply put, the logical idea for establishing equation (4) is that since there is no reason why we cannot use Hamiltonian operators and wave functions in macroscopic systems, we may try to use the wave equation (Schr?dinger equation) to describe macroscopic systems where bound states are maintained by gravity (the description of unbound state systems is of course simpler). For the stationary Schr?dinger equation of the Earth's revolution, there is no denominator 2 in the leftmost term of equation (4). Readers can verify it themselves. If there must be a denominator of 2, it indicates that the original Schr?dinger equation is incorrect. If the denominator does not have that 2, then the table indicates that equation (8) is incorrect. This is a very serious issue that must be taken seriously.

Where r is the distance between the earth and the sun, and e is the energy of the revolution of the earth. When the potential energy V=0, equation (4) is applicable to unbound macroscopic systems. Generally speaking, the logical idea of establishing equation (4) is that since there is no reason why we can't use Hamiltonian and wave function in the macro system, we might as well try to use the wave equation (Schr?dinger  equation) to describe the macro system in which the bound state is maintained by gravity.

When Ris the distance between the earth and the sun and E is the energy of the earth's revolution state, equation (4) is the Schr?dinger equation of the earth's revolution observed on a curved surface. When R is the distance between the earth and the sun and E is the energy of the earth's revolution state, equation (4) is the Schr?dinger  equation of the earth's revolution observed on a curved surface. In this way, R is also known. Equation (4) need not be solved, but only the known constants can be substituted into Equation (4) to calculate the kinetic energy and potential energy of the earth's revolution and the total energy E of the earth's revolution. Using the three-dimensional form of Equation (4) to calculate the energy eigenvalues of the earth is unnecessary.

3. Verification of the Schr?dinger Equation of the Earth's Revolution

Among them,  ? mυ²is the potential energy of the earth's bound motion, and it is exactly twice the kinetic energy of the earth's revolution (which shows that the Schr?dinger equation of the earth's revolution guarantees the establishment of the virial theorem). This is exactly the same as the result calculated according to classical mechanics. In this way, as shown in the second equation of equation (7), the earth energy e in equation (3) is equal to . This is the classical expression of the energy of the earth's revolution, and it is the energy eigenvalue solution of equation (4). Obviously, we have proved that equation (3) holds, and we can use Schr?dinger equation to describe the revolution of the earth.

4. Schr?dinger  equation of planetary model of hydrogen atom and its verification

Since the physical reality of macroscopic objects such as the earth can be described by Schr?dinger equation and planetary model at the same time, there is no reason to restrict the use of Schr?dinger equation and classical mechanical model (planetary model is one of them) to describe microscopic systems such as hydrogen atoms. We can use the planetary model and Schr?dinger  equation to describe hydrogen atoms at the same time. We still choose to observe hydrogen atoms on a curved surface (On the premise of using the planetary model, the orbital motion of electrons in hydrogen atoms is similar to the orbital motion of planets. We can observe hydrogen atoms in Riemannian space). Equation (2) is used to describe the orbital motion of electrons in hydrogen atoms, and the first term on the left of the equal sign corresponds to the kinetic energy E_k of the orbital motion of electrons.

Since the physics of macroscopic objects such as Earth can be described using both the Schr?dinger equation and the planetary model, there is no reason to restrict the simultaneous use of the Schr?dinger equation and classical mechanical models (of which the planetary model is one) to describe microscopic systems such as hydrogen atoms. We can use both the planetary model and the Schr?dinger equation to describe the hydrogen atom. We still choose to observe hydrogen atoms on a curved surface (using a planetary model, the orbital motion of electrons is similar to the orbital motion of planets. We can observe hydrogen atoms in Riemannian space). Use equation (2) to describe the orbital motion of electrons in hydrogen atoms, where the first term on the left side of the equation corresponds to the kinetic energy E_k of the electron orbital motion. The Schr?dinger equation in atoms also conforms to equation (5). No way, m_eis the mass of the electron, and mυ²/2 is the kinetic energy of the electron in the hydrogen atom (where υ is the group velocity of the electron, i.e. the group velocity of the electron's de Broglie wave). When V=0, equation (4) becomes

previously demonstrated in the macroscopic domain, this section demonstrates in the microscopic domain). This proves once again that classical mechanics and quantum mechanics can be compatible (The previous proof is that they are compatible in the macro field, and this section proves that they are compatible in the micro field).

5. Exploration on the essence of wave function ψ

Schr?dinger used the wave function (4) in those days, but he didn't know its true meaning, just regarded it as a part of mathematical tools. After him, no one clarified the essence of wave function ψ. There is still an unsolved mystery about the nature of matter wave. The first formula in the Eq. (9)accords with the classical mechanical theory.

Macroscopic objects moving in a straight line can also be described by Schr?dinger equation. For example, the Schr?dinger equation of a train moving in a straight line at the speed υ_t is

Solving this equation is very simple, just need to find the partial derivative, and use the formula (4) and the de Broglie relation: λ_t=h/p_t= h/(m_tυ_t). The solution of equation (10) is E_t=(?)m_tυ_t². This solution shows that the energy eigenvalue solution of an object in linear inertial motion is the non-relativistic kinetic energy of this object. Comparing this solution with the velocity-frequency relation υ_t=λ_tν_tof monochromatic wave, we can get

Equation (11) shows that the energy of the object's matter wave is twice its kinetic energy. This does not conform to the law of real monochromatic waves. It shows that the matter wave of an object cannot be a complete wave. This result is obviously beneficial to explore the essence of matter wave.

Since the major obstacle of "describing macroscopic objects by wave mechanics" has been removed by establishing the available Schr?dinger equation of planetary motion, we can calculate how many deuterium-like atoms are contained in the earth by using the wave mechanics method of microscopic system (deuterium atoms are calculated as simulated cells that make up macroscopic objects). The result must be very interesting.

The angular momentum of an object moving in a circle is L = r× p. Replacing P with the momentum operator , we get the operator of the orbital angular momentum of the electrons in the hydrogen atom.

Adding footnote N only emphasizes the description of N unit objects. By applying it to Apollo function ψ [see formula 3], we can get

Eliminate ψ in the above formula, and the angular momentum formula of the macroscopic object or the simulated composite particle with hydrogen atom as the unit to do the circular motion of the bound state.

L_N=Rmυ=2.658×10³⁷(m·kg·s). The mass m of the earth is 5.965×102?kg. The distance r between the sun and the earth is 149597870 kilometers (1.496 million kilometers). The average revolution linear velocity of the earth υ is: 29.783 km/s (107,220 km/h). For describing the earth's revolution with wave function, r in wave function ψ and (14) is equivalent to the radius a₀ of hydrogen atom (a₀ = 5.2918× 10^-11 meters). ?=1.054571726×10^-34 J·s. We substitute these numbers into equation (14) and we can get

It is not difficult to see that the product of the mass of N and the deuterium atom as the mass cell of macroscopic substances (including simulated composite particles with deuterium atom as the smallest unit and other compounds) is the mass of the earth. The mass of deuterium atom is 3.3688×10^-27kg. In this way, the mass of the earth calculated by wave mechanics method is

m_earth=N×3.3688×10^-27kg=2.942×10²⁴ kg.      (16)

This calculated value is of the same order of magnitude as the known Earth mass value of 5.965×102?kg . There are two main sources of error: first, the molecular structure of the earth is complex, and deuterium atoms cannot be accurately used as simulated mass cells of the earth; Second, other energies contained in the earth and the sum of binding energies in molecules are not considered.

The mass of the earth is calculated by wave mechanics, which proves that wave mechanics is effective in dealing with planetary motion. The results of this analysis can at least remind us why we can use the wave function ψ when describing macroscopic and microscopic objects. It also supports the theory of wave element material structure proposed in references [3-6].

6. Summary

The revolution of the earth is absolutely in a plane, and in a certain period of time, R is a certain value. If we assume that the ball moves randomly in three-dimensional space (it is an uncertain motion without orbit), and solve the equation similar to Eq. (1), there will definitely be extraneous root. If the state of the electrons in the hydrogen atom is certain, the calculation by using the three-dimensional Schr?dinger equation will also lead to unrealistic root growth. The research results of this paper increase this possibility.

A conclusion of Section 4 is that the classical mechanical method and the quantum mechanical method are compatible and can be used at the same time, whether describing macroscopic objects or microscopic objects. For convenience, we call this conclusion conclusion 1. The use of classical mechanical methods means that the described object is deterministic, realistic and localized, and conforms to determinism. It can be seen that conclusion 1 cannot absolutely deny that micro-objects are also deterministic, localized and causal (only in a narrow range or under certain conditions can people show uncertainty, non-localization, unreality and indecision). This is an important inference according to conclusion 1.

Conclusion 1 and its inference show that the gap between macro-system and micro-system can be eliminated or reduced. Wave mechanics and classical mechanics can be used to describe objects from micro to macro at the same time, which can simplify the calculation process. Determinism, localization, realism and determinism cannot be completely denied in the microscopic system. This obviously has great influence on the interpretation system of quantum mechanics. The obstacles to establishing localized real quantum mechanics are also much smaller.

At that time, Schr?dinger did not explain the reason why he used the wave function of formula (4) in the Schr?dinger equation of hydrogen atom [that is, he did not specify the meaning of formula (4), but only used it as a tool]. We can be sure that the revolution of the earth is definitely not a wave like Eq. (4). This paper proves that "the correct result can be obtained by using equation (4) when describing the revolution of the earth". This result further strengthens the concept that "wave function ψ is a tool in wave mechanics". Unless both microscopic and macroscopic objects are made of waves. If there are no particles but waves in the constituent elements of matter, we can start to establish the theory of wave element material structure.

It is recognized that "human beings have not yet loved the combination of relativity (or gravity theory) and quantum mechanics." This paper proves that we can use Newton's mechanics and quantum mechanics at the same time (that is, Schr?dinger equation, the basic equation of Newton's gravitational interaction potential energy and wave mechanics) to describe the motion and microscopic system of celestial bodies. This is the compatibility and combination of Newton's gravity theory and quantum mechanics to a certain extent (although gravity is not quantized, it is combined in another way).

References

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[2] P. Ehrenfest. Bemerkung über die angen?lte Gültig keit der klassischen Mechanik innerhalb Quantenmeehanik. Z. Phys. 45, 455-457 (1927).

[3] R. Tu. Progress and Review of Applied Research on New Theory of Electronic Composition and Structure. Infinic Energy, 2024. 167: 35-51.

[4] Tu Runsheng. Principle and application of experimental method for measuring the interaction energy of electrons in atoms. International academic research report, 2016. 2(8): 187-200.

[5] Runsheng Tu. (2019) If the Wave Function Collapse absolutely in the Interaction, how can the Weird Nature of Particles are Born in the Interaction: A Discussion on Quantum Entanglement Experiments, Indian Journal of Science and Technology. 12(8): 1-10.

[6] Runsheng Tu. Quantum Mechanics’ Reurn to Local Realism, Cambridge Scholars Press. 2018

[7] Runsheng Tu. Some Successful Applications for Local-Realism Quantum Mechanics: Nature of Covalent-Bond Revealed and Quantitative Analysis of Mechanical Equilibrium for Several Molecules. Journal of modern physics, 5(6). 2014.

[8] Runsheng Tu. A Wave-Based Model of Electron Spin: Bridging Classical and Quantum Perspectives on Magnetic Moment. Physical Science International Journal,

[9] R. Tu. (2024). Progress and Review of Applied Research on New Theory of Electronic Composition and Structure. Infinic energy,  167: 35-51.

[10] Runsheng Tu. (2024). A Review of Research Achievements and Their Applications on the Essence of Electron Spin, Advances in Theoretical & Computational Physics. 7(4): 01-19.

All the work in this manuscript was completed by the corresponding author alone, and there is no conflict of interest. There is no conflict of interest with the reviewers. This manuscript has not been submitted to any other publishing institution. Only submitted to scientific report journals.