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Substantia

Hydrogen-like quantum Hamiltonians & Einstein separability in the case of charged radical molecules

Han J.F. Geurdes

Published Sep 29, 2019• 4 citations
Publisher

Abstract

In the history of the debate about the completeness of quantum theory, Schrödinger and Einstein exchanged letters concerning the fact hat, according to Schrödinger, quantized classical mechanics in the form of the Schrödinger equation cannot be Einstein separable. In the present paper the question is raised if, next to wave-particle duality and quantum tunneling, a Schrödinger wave function can transform itself such that it no longer ”feels” the (non-relativistic instantaneous and omnipresent) Coulomb attraction of opposite charges. Looking at the separability debate between Einstein and Schrödinger this appears to be a, strange but, meaningful ques- tion. Should such transformation be possible then we can conclude that the particles described by a Schrödinger equation would be Coulomb separable. This is contrary to what Schrödinger said. Translating the mathematics to chemistry, we will look at a mesoscopic inter-molecular description of the behavior of charged radical molecules. Firstly, given a restricted experimental geometry set-up such as described in the paper. Secondly, given that the intra-molecular wave function of a charged radical molecule does not prevent the mescoscopic inter-molecular wave function to be described in the present paper. Then it is found that a transformation of mesoscopic inter-molecular wave functions is possible that entails a kind of ”immunization” for Coulomb interaction. The author acknowledges that immunization is a medical term. He has not a better term at this moment. In the appendix of the present paper, an experiment is proposed where micelle based molecules are turned into opposite charged radical molecules and are separated in the special geometry of the experiment. The generation of the opposite charged radicals can be performed with light. The method is borrowed from spin-chemistry. The separation is with ”dipole radiation”. The method is borrowed from Positronium separation. After the mathematical proof, we ask the question what kind of chemical transformation is possible to mimic the mathematical transformation of the wave function provided here in the paper. The theory given here is that the Coulomb immunity can be approximated through the geometry of the oligomerization of charged radical molecules.

DOI
10.13128/substantia-633

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In the history of the debate about the completeness of quantum theory, Schrödinger and Einstein exchanged letters concerning the fact hat, according to Schrödinger, quantized classical mechanics in the form of the Schrödinger equation cannot be Einstein separable. In the present paper the question is raised if, next to wave-particle duality and quantum tunneling, a Schrödinger wave function can transform itself such that it no longer ”feels” the (non-relativistic instantaneous and omnipresent) Coulomb attraction of opposite charges. Looking at the separability debate between Einstein and Schrödinger this appears to be a, strange but, meaningful ques- tion. Should such transformation be possible then we can conclude that the particles described by a Schrödinger equation would be Coulomb separable. This is contrary to what Schrödinger said. Translating the mathematics to chemistry, we will look at a mesoscopic inter-molecular description of the behavior of charged radical molecules. Firstly, given a restricted experimental geometry set-up such as described in the paper. Secondly, given that the intra-molecular wave function of a charged radical molecule does not prevent the mescoscopic inter-molecular wave function to be described in the present paper. Then it is found that a transformation of mesoscopic inter-molecular wave functions is possible that entails a kind of ”immunization” for Coulomb interaction. The author acknowledges that immunization is a medical term. He has not a better term at this moment. In the appendix of the present paper, an experiment is proposed where micelle based molecules are turned into opposite charged radical molecules and are separated in the special geometry of the experiment. The generation of the opposite charged radicals can be performed with light. The method is borrowed from spin-chemistry. The separation is with ”dipole radiation”. The method is borrowed from Positronium separation. After the mathematical proof, we ask the question what kind of chemical transformation is possible to mimic the mathematical transformation of the wave function provided here in the paper. The theory given here is that the Coulomb immunity can be approximated through the geometry of the oligomerization of charged radical molecules.
It is an unavoidable question raised by Albert Einstein if quantum theory is complete. The wavefunctions of elements are now constructed from fitting the experimental data which has made quantum chemistry empirical for almost all elements and molecules. It is our best interest to come up with a simpler model to fit the data regardless of our wish to understand reality in a more intuitive and accurate way than classical quantum mechanics. In this manuscript, I proposed an equation that quantizes the energies orders of magnitude faster than classical quantum mechanics. I assumed that the energy density distributions along space and time in a quantum system are uniform according to the 1st law of thermodynamics. Thus, the complementary energy contributions are added to the classical solutions of the 1D particle in a box problem, making the energy density a complex distribution function over space and time. Then the concept is extended to the free rotation problem with an equation significantly different than the classical Schrödinger equation. This new equation is inspired by the energy conservation method used in solving classical mechanics problems such as the pendulum problem. The picturized energy distribution functions and associated time evolution are described in movies for comparison between example classical wave functions and the energy density functions. The radial wavefunctions for the hydrogen atom and harmonic oscillator are derived. We can see that the new equation is much simpler than Schrödinger equation in obtaining consistent quantized rotational and vibrational energy levels. Thus, it has the potential to be widely used to free up computational pressure in modern quantum chemistry calculations.
The initial purpose is to add two physical origins for the outstandingly clear mathematical description that Dirac has left in his Principles of Quantum Mechanics. The first is the “internal motion” in the wave function of the electron that is now expressed through dispersion dynamics; the second is the physical origin for mathematical quantization. Bohr’s model for the hydrogen atom was “the greatest single step in the development of the theory of atomic structure.” It leads to the Schrodinger equation which is non-relativistic, but which conveniently equates together momentum and electrostatic potential in a representation containing mixed powers. Firstly, we show how the equation is expansible to approximate relativistic form by applying solutions for the dilation of time in special relativity, and for the contraction of space. The adaptation is to invariant “harmonic events” that are digitally quantized. Secondly, the internal motion of the electron is described by a stable wave packet that implies wave-particle duality. The duality includes uncertainty that is precisely described with some variance from Heisenberg’s axiomatic limit. Harmonic orbital wave functions are self-constructive. This is the physical origin of quantization.
It is an unavoidable question raised by Albert Einstein if quantum theory is complete. The wavefunctions of elements are now constructed from fitting the experimental data which has made quantum chemistry a black box of mathematical model. It is our best interest to come up with a simpler model to fit the data regardless of our wish to understand reality in a more intuitive and accurate way than classical quantum mechanics. In this manuscript, I speculated that the energy density distributions along space and time in a quantum system are uniform according to the 1st law of thermodynamics. Thus, the complementary energy contributions are added to the classical solutions of the 1D particle in a box problem, making the energy density a complex distribution function over space and time. Then the concept is extended to the free rotation problem with a Hamiltonian slightly different than the classical Schrödinger equation. The picturized energy distribution functions and associated time evolution are described in movies for comparison between example classical wave functions and the energy density functions. The radial wavefunctions for the hydrogen atom and an ideal harmonic oscillator are then derived. Can this assumption on energy distribution and the pre-mature math in this manuscript lead to a complete quantum theory? Can the idea of energy conservation unify special relativity and quantum?
It is an unavoidable question raised by Albert Einstein if quantum theory is complete. The wavefunctions of elements are now constructed from fitting the experimental data which has made quantum chemistry a black box of mathematical model. It is our best interest to come up with a simpler model to fit the data regardless of our wish to understand reality in a more intuitive and accurate way than classical quantum mechanics. In this manuscript, I speculated that the energy density distributions along space and time in a quantum system are uniform according to the 1st law of thermodynamics. Thus, the complementary energy contributions are added to the classical solutions of the 1D particle in a box problem, making the energy density a complex distribution function over space and time. Then the concept is extended to the free rotation problem with a Hamiltonian slightly different than the classical Schrödinger equation. The picturized energy distribution functions and associated time evolution are described in movies for comparison between example classical wave functions and the energy density functions. The radial wavefunctions for the hydrogen atom and an ideal harmonic oscillator are then derived. Can this assumption on energy distribution and the pre-mature math in this manuscript lead to a complete quantum theory? Can the idea of energy conservation unify special relativity and quantum?
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.
Even after nine decades of successful run of the Quantum Mechanics (QM), different viewpoints on foundational problems of Quantum Physics are still being actively debated. That is because mathematical logic of QM often defies the physical intuition which constitutes the main spirit of Physics. De Broglie’s hypothesis of matter waves implied that the dynamic characteristics of a micro particle in motion, can be ascribed to the wave characteristics of the wavelet accompanying the particle. The Schrödinger equation models the matter-wave interactions through wavefunction ψ and effectively serves as the foundation of QM. Even though mathematical structure of the Schrödinger equation is sound and elegant, here we show a conceptual mistake in the development of this equation wherein the physical situation has not been correctly modeled in the equation. The Coulomb potential energy of the proton electron pair in Hydrogen atom is essentially the negative interaction energy between their superposed electrostatic fields which is inversely proportional to their instantaneous separation distance. Assuming the proton to be relatively fixed at the origin of an appropriate coordinate system, the potential energy of the orbiting electron will be a function of instantaneous position coordinates of the electron. This has not been properly modeled in the Schrödinger equation. The resulting errors in the solution have been quantitatively demonstrated in this paper. We have stressed the necessity of incorporating a specific correction in the potential energy term of the Schrödinger equation, after which it may facilitate the adoption of Bohmian QM.