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NEW CONNECTIONS BETWEEN QUANTUM AND CLASSICAL EQUATIONS WITH APPLICATIONS TO THE MODELING OF ATOMIC, MOLECULAR AND ELECTRODYNAMICAL SYSTEMS

Alexandru Popa

Published Jan 1, 2011• 2 citations
OpenAlex

Abstract

In previous papers we proved the existence of exact connections between quantum and classical equations in the cases of two different quantum systems, namely the stationary atomic and molecular systems and the systems composed by electrons and very intense electromagnetic field. We proved that the geometrical elements of the wave described by the Schrodinger equation, namely the characteristic surfaces and curves, denoted, respectively, by Σ and C, are solutions of the HamiltonJacobi equations, written for the same system, in the case of multidimensional stationary systems. The Σ surfaces, which have the significance of wave surfaces, and the C curves, which are their normals, correspond to the same constants of motion as the eigenvalues of the Schrodinger equation. In three recent papers we presented a central field method for the calculation of the C curves, and of the corresponding energetic values. We have verified this method for a large number of atoms and diatomic molecules, and we found that its accuracy is comparable to the accuracy of the Hartree-Fock method applied to the same atoms and diatomic molecules. In another papers we proved that the Klein-Gordon equation for the system electron-very intense electromagnetic field is verified exactly by the function associated to the classical motion of the electron. An exact connection follows between the solutions of the Klein-Gordon and the relativistic Hamilton-Jacobi equations, for that system. This result explains the accuracy of numerous classical models from the literature, belonging to the new field studying the interactions between very intense laser beams and electron plasmas. We present now a short review of these results.

Related papers

A secondary solution of the Schrödinger and Klein-Gordon equations in modeling atomic, molecular and electrodynamic systems

Alexandru Popa

We study atomic and molecular (AM) stationary systems and electrodynamic (ED) systems, composed of an electron interacting with an electromagnetic field. We show that the Schr?dinger and Klein-Gordon equations written for these systems have a secondary solution, which is the wave function associated to a classical system. For AM systems, the wave surfaces (S surfaces) and their normals (C curves) are solutions of the Hamilton-Jacobi equation, written for the same system. The S surfaces have a periodical motion and the C curves are closed. The integral relation of the Schr?dinger equation on the C curve has a solution identical to the wave function associated to the classical motion. This solution leads to the generalized Bohr quantization relation and to the generalized de Broglie relations, which are valid in the space of the electron coordinates. An identical wave function verifies the Klein-Gordon equation, in the case of the EM systems. The above properties lead to a central field method for calculation of the energetic values and symmetry properties for AM systems, whose accuracy is comparable to the accuracy of the Hartree-Fock method. They also lead to an accurate method for modeling EM systems, which is verified by experimental data from the literature. The above properties are deduced without using any approximation.

Journal of Physics Conference Series•Mar 11, 2014•0 citations

Applications of a Property of the Schrödinger Equation to the Modeling of Conservative Discrete Systems. III

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We have demonstrated in previous papers the following property of closed, conservative and bounded systems: The energy which results from the Schrödinger equation can be rigorously calculated by line integrals of analytical functions, if the Hamilton-Jacobi equation, written for the same system, is satisfied in the space of coordinates by a periodical trajectory. In the present article, we show that this property is connected to the intrinsic wave properties of the system. This results from the equivalence between the Schrödinger equation and the wave equation, valid for conservative systems. As a consequence of the wave properties of the system, we show that the Hamilton-Jacobi equation has always periodical solutions, whose constants of motion are identical to the eigenvalues of the Schrödinger equation, written for the same system. It results that the calculation model presented in previous papers is generally valid in the case of closed, conservative and bounded systems. We present the applications of the model to the nitrogen and oxygen atoms, to the ions with the same structure, and to the He 2 , Be 2 and B 2 molecules.

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It is well known that the Schrödinger equation is equivalent to the wave equation for conservative bound quantum systems. Consequently, the motion of such a system is represented—from a mathematical point of view—by the motion of the characteristic surface of the wave equation. In this paper we present a demonstration of the periodic motion of the characteristic surface. It results that the normal curves of the characteristic surface are periodic solutions of the Hamilton–Jacobi equation written for the same system. This leads to a direct connection between the periodic solutions of the Hamilton–Jacobi equation and the wave properties of the system. The constants of motion corresponding to the above periodic solutions of the Hamilton–Jacobi equation are identical to the eigenvalues of the Schrödinger equation. These properties are proved without any approximation and they are valid for all the values of the principal quantum number.

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A Link Merges Classical Mechanics to Quantum Theory (Part II)

Alaa S. Bayoumi

This article is a continuation of Part(I) to demonstrate more aspects for a successful merge of classical mechanics to the quantum theory. The article starts first with the deduction of the general form of the complimentary energy, next a robust firm mathematical proof is given indicates that the complimentary energy has a wave nature and it is a potential energy, then the radiation condition has been determined, a vector formula expresses the corresponding force was deduced and this force was called adhesion force. A mathematical convergence process has proved the duality of the electromagnetic wave parameters and particle parameters, which results in the existence of a “wave-particle” object. While finding the relationship between the electromagnetic wave parameters and particle parameters, a mathematical relationship was found indicating that not every moving charged mass generates a magnetic field. This relationship for the mass was linked to the effective mass tensor, The geometric representation of the constant energy surface and its relationship to the energy levels of both the orbital model (Bohr model) and the cloud model for different geometric 3D shapes were investigated as a sample of the possible geometric interpretations of the energy equation, that is by employing the triangle method, it is possible to create a geometric representation of the constant energy surfaces. The relation between the energy levels, quantum numbers, and band width is represented geometrically. As a result of proving the relationship between the motion of an electromagnetic wave and the motion of a particle associated with the wave, two new, unprecedented mathematical formulas were developed. The first formula specifies a moving charged mass and its relationship to the resulting wavelength. Thus, the radius of the charged mass is obtained in terms of its wavelength and mass density. The second formula specifies the kinetic energy of a photon. The grand potentials are the potential energy of the Newton’s Law of Universal Gravitation, Coulomb’s Law, and Inverse Square Law of Electromagnetic Waves Intensity. From the potential energy of Newton’s law of universal gravitation, the wavelength of the gravitational wave, as well as the wavelength of quantum gravity were deduced. A mathematical relationship was then deduced that determines the lengths of the functions of conic sections, and in particular the length of perimeter of ellipses. From the potential energy of Coulomb’s law of attraction, a mathematical formulation of the energy levels in general for any atom of an element was obtained. This mathematical formulation was compared with the results obtained by Bohr and Schrödinger for the hydrogen atom. As for the inverse square law of electromagnetic radiation intensity, several innovative mathematical equations have been deduced that link different parameters together. Following the contents of this article requires to be quite familiar with the contents of the first part with the same title, as well as a complete understanding of the concepts of intermediate parameters derived and used in the various equations. Otherwise, it will be extremely difficult to follow the contents of this article.

Journal of Applied Mathematics and Physics•Jan 1, 2025•1 citation

Stationary Solutions of the Klein-Gordon Equation in a Potential Field

Guang-Qing Bi, Yuekai Bi

We seek to introduce a mathematical method to derive the Klein-Gordon equation and a set of relevant laws strictly, which combines the relativistic wave functions in two inertial frames of reference. If we define the stationary state wave functions as special solutions like $Ψ(\mathbf{r},t)=ψ(\mathbf{r})e^{-iEt/\hbar}$, and define $m=E/c^2$, which is called the mass of the system, then the Klein-Gordon equation can clearly be expressed in a better form when compared with the non-relativistic limit, which not only allows us to transplant the solving approach of the Schrödinger equation into the relativistic wave equations, but also proves that the stationary solutions of the Klein-Gordon equation in a potential field have the probability significance. For comparison, we have also discussed the Dirac equation. By introducing the concept of system mass into the Klein-Gordon equation with the scalar and vector potentials, we prove that if the Schr\"{o dinger equation in a certain potential field can be solved exactly, then under the condition that the scalar and vector potentials are equal, the Klein-Gordon equation in the same potential field can also be solved exactly by using the same method.

arXiv:1008.4224•Aug 25, 2010•3 citations

The Relation between Classical Mechanics and Wave Mechanics

Yoshio Nishiyama

Introducing a dimensional unit \hbar into classical mechanics, we make classical wave functions √ρ1exp (iS1/\hbar)-√ρ2exp (iS2/\hbar) for a one-particle system in the configuration space where √ρ1exp (iS1/\hbar) corresponds to the incident path (or one half) of an orbit and √ρ2exp (iS2/\hbar) to the outgoing (or another half), and ρ's and S's are densities and actions respectively. These classical wave functions are compared with the corresponding wave functions of solutions of the Schrödinger equation for three one-particle systems, the scattering by a Coulomb potential, the linear harmonic oscillator and an electron bound by a Coulomb potential. From the essential correspondence between them it is concluded that the Schrödinger wave function describes the steady, causal but a little imaginary motion of the one-particle system as √ρexp (iS/\hbar) in the configuration space where ρ is a density and S an action, and that the states which the wave equation describes exist continuously (not discrete) and the (discrete) stationary states are the ones where √ρexp (iS/\hbar) is uniquely determined throughout an orbit. The Dirac equation shows that an electron of optics (or the electromagnetic wave equation) can be considered as the Schrödinger (or Dirac) equation for the motion of a photon, a particle, in a potential. A quantum-electrodynamical wave equation which is composed of the Dirac equation with a Coulomb potential and the electromagnetic wave equation is constructed for the steady motion of a one-electron system without recource to the so-called second-quantization method. It gives finite energy shifts. A line shift is calculated: ΔE(2S1/2 →2P1/2) \cong709 Mc/s.

Progress of Theoretical Physics•Nov 1, 1963•1 citation
In previous papers we proved the existence of exact connections between quantum and classical equations in the cases of two different quantum systems, namely the stationary atomic and molecular systems and the systems composed by electrons and very intense electromagnetic field. We proved that the geometrical elements of the wave described by the Schrodinger equation, namely the characteristic surfaces and curves, denoted, respectively, by Σ and C, are solutions of the HamiltonJacobi equations, written for the same system, in the case of multidimensional stationary systems. The Σ surfaces, which have the significance of wave surfaces, and the C curves, which are their normals, correspond to the same constants of motion as the eigenvalues of the Schrodinger equation. In three recent papers we presented a central field method for the calculation of the C curves, and of the corresponding energetic values. We have verified this method for a large number of atoms and diatomic molecules, and we found that its accuracy is comparable to the accuracy of the Hartree-Fock method applied to the same atoms and diatomic molecules. In another papers we proved that the Klein-Gordon equation for the system electron-very intense electromagnetic field is verified exactly by the function associated to the classical motion of the electron. An exact connection follows between the solutions of the Klein-Gordon and the relativistic Hamilton-Jacobi equations, for that system. This result explains the accuracy of numerous classical models from the literature, belonging to the new field studying the interactions between very intense laser beams and electron plasmas. We present now a short review of these results.
We study atomic and molecular (AM) stationary systems and electrodynamic (ED) systems, composed of an electron interacting with an electromagnetic field. We show that the Schr?dinger and Klein-Gordon equations written for these systems have a secondary solution, which is the wave function associated to a classical system. For AM systems, the wave surfaces (S surfaces) and their normals (C curves) are solutions of the Hamilton-Jacobi equation, written for the same system. The S surfaces have a periodical motion and the C curves are closed. The integral relation of the Schr?dinger equation on the C curve has a solution identical to the wave function associated to the classical motion. This solution leads to the generalized Bohr quantization relation and to the generalized de Broglie relations, which are valid in the space of the electron coordinates. An identical wave function verifies the Klein-Gordon equation, in the case of the EM systems. The above properties lead to a central field method for calculation of the energetic values and symmetry properties for AM systems, whose accuracy is comparable to the accuracy of the Hartree-Fock method. They also lead to an accurate method for modeling EM systems, which is verified by experimental data from the literature. The above properties are deduced without using any approximation.
We have demonstrated in previous papers the following property of closed, conservative and bounded systems: The energy which results from the Schrödinger equation can be rigorously calculated by line integrals of analytical functions, if the Hamilton-Jacobi equation, written for the same system, is satisfied in the space of coordinates by a periodical trajectory. In the present article, we show that this property is connected to the intrinsic wave properties of the system. This results from the equivalence between the Schrödinger equation and the wave equation, valid for conservative systems. As a consequence of the wave properties of the system, we show that the Hamilton-Jacobi equation has always periodical solutions, whose constants of motion are identical to the eigenvalues of the Schrödinger equation, written for the same system. It results that the calculation model presented in previous papers is generally valid in the case of closed, conservative and bounded systems. We present the applications of the model to the nitrogen and oxygen atoms, to the ions with the same structure, and to the He 2 , Be 2 and B 2 molecules.
It is well known that the Schrödinger equation is equivalent to the wave equation for conservative bound quantum systems. Consequently, the motion of such a system is represented—from a mathematical point of view—by the motion of the characteristic surface of the wave equation. In this paper we present a demonstration of the periodic motion of the characteristic surface. It results that the normal curves of the characteristic surface are periodic solutions of the Hamilton–Jacobi equation written for the same system. This leads to a direct connection between the periodic solutions of the Hamilton–Jacobi equation and the wave properties of the system. The constants of motion corresponding to the above periodic solutions of the Hamilton–Jacobi equation are identical to the eigenvalues of the Schrödinger equation. These properties are proved without any approximation and they are valid for all the values of the principal quantum number.
This article is a continuation of Part(I) to demonstrate more aspects for a successful merge of classical mechanics to the quantum theory. The article starts first with the deduction of the general form of the complimentary energy, next a robust firm mathematical proof is given indicates that the complimentary energy has a wave nature and it is a potential energy, then the radiation condition has been determined, a vector formula expresses the corresponding force was deduced and this force was called adhesion force. A mathematical convergence process has proved the duality of the electromagnetic wave parameters and particle parameters, which results in the existence of a “wave-particle” object. While finding the relationship between the electromagnetic wave parameters and particle parameters, a mathematical relationship was found indicating that not every moving charged mass generates a magnetic field. This relationship for the mass was linked to the effective mass tensor, The geometric representation of the constant energy surface and its relationship to the energy levels of both the orbital model (Bohr model) and the cloud model for different geometric 3D shapes were investigated as a sample of the possible geometric interpretations of the energy equation, that is by employing the triangle method, it is possible to create a geometric representation of the constant energy surfaces. The relation between the energy levels, quantum numbers, and band width is represented geometrically. As a result of proving the relationship between the motion of an electromagnetic wave and the motion of a particle associated with the wave, two new, unprecedented mathematical formulas were developed. The first formula specifies a moving charged mass and its relationship to the resulting wavelength. Thus, the radius of the charged mass is obtained in terms of its wavelength and mass density. The second formula specifies the kinetic energy of a photon. The grand potentials are the potential energy of the Newton’s Law of Universal Gravitation, Coulomb’s Law, and Inverse Square Law of Electromagnetic Waves Intensity. From the potential energy of Newton’s law of universal gravitation, the wavelength of the gravitational wave, as well as the wavelength of quantum gravity were deduced. A mathematical relationship was then deduced that determines the lengths of the functions of conic sections, and in particular the length of perimeter of ellipses. From the potential energy of Coulomb’s law of attraction, a mathematical formulation of the energy levels in general for any atom of an element was obtained. This mathematical formulation was compared with the results obtained by Bohr and Schrödinger for the hydrogen atom. As for the inverse square law of electromagnetic radiation intensity, several innovative mathematical equations have been deduced that link different parameters together. Following the contents of this article requires to be quite familiar with the contents of the first part with the same title, as well as a complete understanding of the concepts of intermediate parameters derived and used in the various equations. Otherwise, it will be extremely difficult to follow the contents of this article.
We seek to introduce a mathematical method to derive the Klein-Gordon equation and a set of relevant laws strictly, which combines the relativistic wave functions in two inertial frames of reference. If we define the stationary state wave functions as special solutions like Ψ(r,t)=ψ(r)e−iEt/ℏΨ(\mathbf{r},t)=ψ(\mathbf{r})e^{-iEt/\hbar}Ψ(r,t)=ψ(r)e−iEt/ℏ, and define m=E/c2m=E/c^2m=E/c2, which is called the mass of the system, then the Klein-Gordon equation can clearly be expressed in a better form when compared with the non-relativistic limit, which not only allows us to transplant the solving approach of the Schrödinger equation into the relativistic wave equations, but also proves that the stationary solutions of the Klein-Gordon equation in a potential field have the probability significance. For comparison, we have also discussed the Dirac equation. By introducing the concept of system mass into the Klein-Gordon equation with the scalar and vector potentials, we prove that if the Schr\"{o dinger equation in a certain potential field can be solved exactly, then under the condition that the scalar and vector potentials are equal, the Klein-Gordon equation in the same potential field can also be solved exactly by using the same method.
Introducing a dimensional unit \hbar into classical mechanics, we make classical wave functions √ρ1exp (iS1/\hbar)-√ρ2exp (iS2/\hbar) for a one-particle system in the configuration space where √ρ1exp (iS1/\hbar) corresponds to the incident path (or one half) of an orbit and √ρ2exp (iS2/\hbar) to the outgoing (or another half), and ρ's and S's are densities and actions respectively. These classical wave functions are compared with the corresponding wave functions of solutions of the Schrödinger equation for three one-particle systems, the scattering by a Coulomb potential, the linear harmonic oscillator and an electron bound by a Coulomb potential. From the essential correspondence between them it is concluded that the Schrödinger wave function describes the steady, causal but a little imaginary motion of the one-particle system as √ρexp (iS/\hbar) in the configuration space where ρ is a density and S an action, and that the states which the wave equation describes exist continuously (not discrete) and the (discrete) stationary states are the ones where √ρexp (iS/\hbar) is uniquely determined throughout an orbit. The Dirac equation shows that an electron of optics (or the electromagnetic wave equation) can be considered as the Schrödinger (or Dirac) equation for the motion of a photon, a particle, in a potential. A quantum-electrodynamical wave equation which is composed of the Dirac equation with a Coulomb potential and the electromagnetic wave equation is constructed for the steady motion of a one-electron system without recource to the so-called second-quantization method. It gives finite energy shifts. A line shift is calculated: ΔE(2S1/2 →2P1/2) \cong709 Mc/s.