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.
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 Schrodinger equation has been considered to be a postulate of quantum physics, but it is also perceived and derived heuristically as the quantum equivalent of the classical energy relation. We indicate that the Schrodinger equation cannot be a physical postulate, and we show that the heuristic derivations wrongly consider that the particle kinetic energy is proportional to the square of the momentum operator acting on the wavefunction. The correct term is proportional to the square of the momentum. Analyzing particle interactions, we realize that particles have multiple virtual motions and that each motion is accompanied by a wave that has constant amplitude. Accordingly, we define the wavefunction as the superposition of the virtual waves of the particle. As a result, quantum mechanics becomes a local and global theory, while in the traditional formulation it is a local theory. In several interaction settings we can tell what particle motions arise and explain the outcomes in simple and tangible terms. Most importantly, the mathematical foundation of quantum mechanics becomes clear and justified, and we derive the Schrodinger, Dirac, etc. equations as the conditions the wavefunction must satisfy at each space-time point in order to fulfill the respective total energy equation.
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 big problem here. If λ=h⁄p is valid for matter waves, we must be able to obtain it for these waves using the mathematical calculations. But as you will see here, according to de Broglie's hypothesis, which considers matter wave energy is E=hf, this work is impossible; and so, we must change the matter wave energy equation to extract λ=h⁄p. The change in the equation of energy causes two important changes in the 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 we need a new atomic model for the hydrogen atom due to some problems. 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.
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.
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.
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.
Despite its successes, quantum mechanics (QM) has remained mysterious to all who have encountered it. Starting with Bohr and progressing into the present, the departure from intuitive, physical reality has widened. The connection between QM and reality is more than just a “philosophical” issue. It reveals that QM is not a correct or complete theory of the physical world and that inescapable internal inconsistencies and incongruities arise when attempts are made to treat it as physical as opposed to a purely mathematical “tool.” Some of these issues are discussed in a review by F. Laloe [Am. J. Phys. 69, 655 (2001)]. In an attempt to provide some physical insight into atomic problems and starting with the same essential physics as Bohr of e moving in the Coulombic field of the proton and the wave equation as modified by Schrodinger, a classical approach is explored that yields a remarkably accurate model and provides insight into physics on the atomic level. The proverbial view, deeply seated in the wave-particle duality notion, that there is no large-scale physical counterpart to the nature of the electron may not be correct. Physical laws and intuition may be restored when dealing with the wave equation and quantummechanical problems. Specifically, a theory of classical quantum mechanics (CQM) is derived from first principles that successfully applies physical laws on all scales. Rather than using the postulated Schrodinger boundary condition “Ψ → 0 as r → ∞,” which leads to a purely mathematical model of the electron, the constraint is based on experimental observation. Using Maxwell’s equations, the classical wave equation is solved with the constraint that the bound (n = 1)state electron cannot radiate energy. By further application of Maxwell’s equations to electromagnetic and gravitational fields at particle production, the Schwarzschild metric is derived from the classical wave equation, which modifies general relativity to include conservation of spacetime in addition to momentum and matter/energy. The result gives a natural relationship among Maxwell’s equations, special relativity, and general relativity. CQM holds over a scale of spacetime of 85 orders of magnitude — it correctly predicts the nature of the universe from the scale of the quarks to that of the cosmos. A review is given by G. Landvogt [Internat. J. Hydrogen Energy 28, 1155 (2003)].