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Wrong Application of Schrödinger Equation In Hydrogen Atom

Hung-Kuk Oh, Ho-gyun Jeong

Published Jun 1, 2014• 0 citations
OpenAlex

Abstract

The conventional atomic model did not contain nuclear electrons because of the uncertainty principle. It made us not explain life energies. The improved principle does not take out the nuclear electrons. Under the minimum energy level of the conventional model, it did not consider the electrons motion. The electron is absorbed to the proton and then it is rebounded at the mass wall of the proton. Finally it repeats a round trip, which is called as Kronig-Penny quantum mechanical. Model most of atom bondings and molecular are composed of this round trip.

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The conventional atomic model did not contain nuclear electrons because of the uncertainty principle. It made us not explain life energies. The improved principle does not take out the nuclear electrons. Under the minimum energy level of the conventional model, it did not consider the electrons motion. The electron is absorbed to the proton and then it is rebounded at the mass wall of the proton. Finally it repeats a round trip, which is called as Kronig-Penny quantum mechanical. Model most of atom bondings and molecular are composed of this round trip.
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.
The Schrödinger equation is the cornerstone of quantum mechanics and is used to determine the energy levels of the hydrogen atom. However, in the teaching of Quantum Chemistry, many students tend to regard the solutions of the Schrödinger equation as purely mathematical results without fully understanding the physical origin of quantum energy levels. In this paper, an approximate model of the hydrogen atom is first developed based on the Heisenberg uncertainty principle and the Coulomb potential. This model is then used to derive the Bohr radius and the ground-state energy by minimizing the total energy function. Subsequently, the time-independent Schrödinger equation is formulated and solved by the method of separation of variables in spherical coordinates to obtain the exact energy spectrum of the hydrogen atom. The results demonstrate that both the approximate model and the exact solution predict the same ground-state energy, E1 = −13.6 eV, while the Schrödinger equation further generalizes this result to the quantized energy spectrum, En = −13.6/n2. These findings indicate that the solutions of the Schrödinger equation are not "miraculous" mathematical outcomes but rather the inevitable consequences of the fundamental principles of quantum mechanics. This approach provides a clearer physical interpretation of quantum energy levels and offers significant pedagogical value for the teaching of Quantum Chemistry.
It is shown that a solution of the classical Kepler problem with a binomial potential explains the stationary orbits of the hydrogen atom, its discrete spectrum, and trajectories of scattered electrons with energies from several electron-volts to relativistic values. The Schrodinger equation with the binomial potential takes the form of a classical wave equation. Hence, the binomial potential is a more appropriate model for understanding atomic phenomena than the model based on the Coulomb potential.
The atom model of Quantum Mechanics (QM) was conceived from an unsolved paradox. Indeed, Schrödinger’s equation has been deducted by considering a free electron, but it is applied for the atom, where the electron is inside a potential. In order to eliminate the nonsense, quantum theorists proposed a ridiculous postulate: they claim it makes sense to use the equation because it gives results in agreement to experimental data. The unsolved paradox evidences that Schrödinger’s equation cannot be applied to the physical conditions considered in the QM atom model, and that his equation actually requires some special conditions not considered in the theory (for instance, the electron helical trajectory, rejected by Heisenberg). The banishment of the aether has introduced several paradoxes in the development of Theoretical Physics. And because the theorists have neglected other paradox (from the mathematical probability the spectacular successes of Bohr’s hydrogen atom cannot be accidental), these two unsolved paradoxes introduced dramatic consequences in the development of Nuclear Physics.
The Schrödinger equation describes the basic physical models in a very contradictory way. And its canonization, which gives a catastrophic divergence of the energy of electronic levels with an increase in the mass of an atom, in principle, stalled the development of science. Nevertheless, it allowed, albeit roughly, to systematize and describe many phenomena of the microworld, primarily the hydrogen atom. And approximate descriptions of many “quantum” effects were obtained, but with the use of numerous adjustable parameters. So the bottom line from the century of development of quantum theory (QT) is, in fact, the development of some quantum mechanical methods.
This paper discusses Bohr-Somerfield quantization theory and tries to explain how all earlier propositions about energy at atomic level failed to correspond to observed facts and further how this model solved the mathematical discrepancies leading to a near perfect model of energy exchange at the atomic level. The author has attempted here to show the historical connections, logical justifications as well as mathematical expressions for the energy of electron in hydrogen atom and prove how Bohr’s celebrated theory of atomic structure is an application of Planck’s theory of quanta to the Rutherford nuclear atom in an attempt, extraordinary fruitful, to define the nature of the orbits in which the electrons might revolve round the nucleus and to explain the origin of spectral lines of the elements. It further shows how Sommerfeld extended the model so as to include elliptical orbits. The introduction of the elliptic orbits has no new energy for the electron of the hydrogen atom. Bohr had also solved the dilemma arose due to the Rutherford’s atomic model and he admitting the failure of the classical theory, applied with remarkable success of the quantum theory to the Rutherford nuclear atom with revolving electron, which leads us to understand the significance of the Bohr’s atom model.  Keywords: Electromagnetic theory, Model, Electrical attraction, Energy, Atom.