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Journal of Physics & Optics Sciences

The Schrödinger Equation Used the Wrong Energy Relationship

Xiao En Wang

Published Nov 13, 2024• 0 citations
Publisher

Abstract

According to the energy relationship of the ground state solution of a hydrogen atom using the Schrödinger equation, the kinetic energy of an electron at twice the Bohr radius becomes zero. The electron has zero kinetic energy, which means it is dead, indicating that the energy relationship used in the Schrödinger equation is incorrect. Then, the reasons for both the chaotic energy relationship of the Schrödinger equation and the half frequency problem encountered by Bohr are analyzed. Bohr only considered Coulomb force and established an equation with centrifugal force, but he did not consider Lorentz force. So, the frequency he obtained must be multiplied by 1/2 to match the experimental value. This article also analyzes other issues related to the Schrödinger equation.

DOI
10.47363/jpsos/2024(6)278

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According to the energy relationship of the ground state solution of a hydrogen atom using the Schrödinger equation, the kinetic energy of an electron at twice the Bohr radius becomes zero. The electron has zero kinetic energy, which means it is dead, indicating that the energy relationship used in the Schrödinger equation is incorrect. Then, the reasons for both the chaotic energy relationship of the Schrödinger equation and the half frequency problem encountered by Bohr are analyzed. Bohr only considered Coulomb force and established an equation with centrifugal force, but he did not consider Lorentz force. So, the frequency he obtained must be multiplied by 1/2 to match the experimental value. This article also analyzes other issues related to the Schrödinger equation.
In the early 20th century, Planck, Einstein, Bohr, Sommerfeld, and de Broglie became the pioneers of quantum physics. Because the Schrödinger equation used the wrong energy relationship, it misled people’s understanding of quantum theory. Newtonian mechanics and Leibniz dynamics are correct. But Newton’s first law was tampered with by the British mathematician Mott during his translation, which excluded uniform circular motion from inertia; Leibniz’s kinetic energy mυ2 was modified by French engineer Coriolis to an engineering approximation of mυ2/2. Hamilton mistakenly applied the approximate kinetic energy formula of Coriolis in engineering mechanics to the dynamic analysis in theoretical physics, and established an incorrect Hamiltonian energy equation. Schrödinger ignored Sommerfeld’s research on the constant velocity of electrons in atoms, υ = αc, and thus the constant momentum and constant kinetic energy of electrons. He expressed the constant kinetic energy term of electrons as an operator that hides the velocity variable, and established the wave equation based on the incorrect Hamiltonian energy relationship. The Schrödinger equation deals with the ground state of hydrogen atoms, where electrons die when they reach twice the Bohr radius, which contradicts the stable existence of atoms. The Schrödinger ‎equation has been around for almost a century, and the quantum physics of the new millennium ‎should be a new quantum physics without the Schrödinger equation.
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.
In his doctoral thesis in 1911, Bohr reached the conclusion that at thermal equilibrium metal atom has no magnetic effect. Therefore, when establishing the hydrogen atom model in 1913, he did not consider the Lorentz force, but only the Coulomb force. The radius of the hydrogen atom he obtained is half smaller, and the orbital rotation frequency is twice larger. It must be multiplied by 1/2 to be consistent with the value of the Rydberg experiment. He proposed to calculate the photon frequency by using the average value of the zero frequency of infinite quantum number and the basic frequency, but it is not applicable to the energy levels for the ordinary quantum numbers, and later he called it the correspondence principle. In our modified Bohr model, we eliminate and solve Bohr's half frequency problem; The photon energy is equal to the difference between the two energy levels, and the photon frequency is equal to the difference between the two rotation frequencies of the electron; All excited states of hydrogen atom are elliptical orbits, and the nucleus is located in the center of the ellipse; The microscopic model of photonic structure is established, and the microscopic mechanism of atomic luminescence and quantum entanglement is given.
This chapter considers Schrodinger's model of hydrogen atom and analyzes in details the quantum wave mechanics approach to the motion of the electron in the hydrogen atom. The stationary Schrodinger equation is not easy to solve as it stands, and the whole problem looks rather complicated when expressed in terms of Cartesian coordinates. An essential difference between Bohr's and Schrodinger's models is that there are three quantum numbers, n , l , m , rather than the one number n describing a state of the electron. Moreover, in Bohr's model the atom is, at each instant, in one and only one energy state, that is., at each instant, the state of the atom is definite, but according to Schrodinger's model, the atom can exist in a superposition of states. In other words, the atom can be simultaneously in more than one energy state at a given instant. This is clearly incompatible with Bohr's model.
The validity of the de Broglie formula λ=h⁄p for matter waves has been confirmed by numerous experiments such as Davisson-Germer and G. P. Thomson experiments. But there is a major problem here. If λ=h⁄p is valid for matter waves, we must be able to obtain it for these waves using mathematical calculations. But as you will see here, according to de Broglie's hypothesis, which considers matter wave energy equal to E=hf, this work is impossible; and so, we must change the matter wave energy equation to obtain λ=h⁄p. The change in the equation of energy causes two important changes in Schrodinger’s wave equation. These two changes bring us to a new wave equation. In the fourth part of the article you will see that, for explanation of the reason for the quantization of the orbits in Bohr's atomic model, we need a new atomic model for the Hydrogen atom. The new model leads us to new results and findings. In this article, we show what the cause of the quantization of the orbits in the Bohr atomic model is.