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Applied Physics Research

The Relationship Enfolded in Bohr’s Quantum Condition and a Previously Unknown Formula for Kinetic Energy

Koshun Suto

Published Jan 29, 2019• 17 citations
Publisher

Abstract

Bohr’s quantum condition is an indispensable assumption for classical quantum theory. However, strictly speaking, Bohr's quantum condition does not hold when deriving the energy of an electron forming a hydrogen atom from the perspective of the theory of relativity. In this paper, it is thought that the relationship enfolded in Bohr's quantum condition, i.e., is suitable as a new quantum condition to replace Bohr’s quantum condition. Also, in quantum mechanics, the energy of an electron is derived based on the theory of relativity, as exemplified in the theory of Sommerfeld. However, this paper points out that the previous energy formula based on the theory of relativity is mistaken. It also proposes a previously unknown formula for the kinetic energy of an electron.

DOI
10.5539/apr.v11n1p19

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Bohr’s quantum condition is an indispensable assumption for classical quantum theory. However, strictly speaking, Bohr's quantum condition does not hold when deriving the energy of an electron forming a hydrogen atom from the perspective of the theory of relativity. In this paper, it is thought that the relationship enfolded in Bohr's quantum condition, i.e., is suitable as a new quantum condition to replace Bohr’s quantum condition. Also, in quantum mechanics, the energy of an electron is derived based on the theory of relativity, as exemplified in the theory of Sommerfeld. However, this paper points out that the previous energy formula based on the theory of relativity is mistaken. It also proposes a previously unknown formula for the kinetic energy of an electron.
Bohr assumed a quantum condition when deriving the energy levels of a hydrogen atom. However, this quantum condition was not a logically derived formula. The support of the de Broglie’s condition was necessary to give physical meaning to Bohr’s quantum condition. However, contemporary physics denies the existence of the waves which de Broglie thought actually existed. Since the theory of relativity are not taken into account in Bohr’s theory, the derived energy levels do not have the correct values. Thus, the author has previously proposed , a quantum condition superior to Bohr’s quantum condition. However, although this quantum condition was mathematically valid, it could not provide physical meaning. This paper has improved that point. When the ratio of the momentum vector of an electron moving inside a hydrogen atom and the momentum vector of the photon possessed by the electron satisfy a certain condition, the two are orthogonal. The condition for these two vectors to be orthogonal is the quantum condition derived in this paper. It is thought that the state where these two vectors are orthogonal is stable, and this state becomes the steady state of the hydrogen atom.
Bohr assumed a quantum condition when deriving the energy levels of a hydrogen atom. This famous quantum condition was not derived logically, but it beautifully explained the energy levels of the hydrogen atom. Therefore, Bohr’s quantum condition was accepted by physicists. However, the energy levels predicted by the eventually completed quantum mechanics do not match perfectly with the predictions of Bohr. For this reason, it cannot be said that Bohr’s quantum condition is a perfectly correct assumption. Since the mass of an electron which moves inside a hydrogen atom varies, Bohr’s quantum condition must be revised. However, the newly derived relativistic quantum condition is too complex to be assumed at the beginning. The velocity of an electron in a hydrogen atom is known as the Bohr velocity. This velocity can be derived from the formula for energy levels derived by Bohr. The velocity v of an electron including the principal quantum number n is given by αc/n. This paper elucidates the fact that this formula is built into Bohr’s quantum condition. It is also concluded in this paper that it is precisely this velocity formula that is the quantum condition that should have been assumed in the first place by Bohr. From Bohr’s quantum condition, it is impossible to derive the relativistic energy levels of a hydrogen atom, but they can be derived from the new quantum condition. This paper proposes raising the status of the previously-known Bohr velocity formula.
It is thought that quantum mechanics is the physical science describing the behavior of the electron in the micro world, e.g., inside a hydrogen atom. However, the author has previously derived the energy-momentum relationship which holds inside a hydrogen atom. This paper uses that relationship to investigate the relationships between physical quantities which hold in a hydrogen atom. In this paper, formulas are derived which hold in the micro world and make more accurate predictions than the classical quantum theory. This paper concludes that quantum mechanics is not the only theory enabling investigation of the micro world.
The energy levels of a hydrogen atom, derived by Bohr, are known to be approximations. This is because the classical quantum theory of Bohr does not take the theory of relativity into account. In this paper, the kinetic energy and momentum of an electron in a hydrogen atom are treated relativistically. A clearer argument is developed while also referring to papers published in the past. The energy levels of a hydrogen atom predicted by this paper almost match the theoretical values of Bohr. It is difficult to experimentally distinguish the two. However, this paper predicts the existence of an n = 0 energy level that cannot be predicted even with Dirac’s relativistic quantum mechanics. The only quantum number treated in this paper is n. This point falls far short of a finished quantum mechanics. However, even in discussion at the level of this paper, it can be concluded that quantum mechanics is an incomplete theory.
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.
Bohr assumed a quantum condition when deriving the energy levels of a hydrogen atom. However, this quantum condition was not a logically derived formula. The support of the de Broglie condition was necessary to give physical meaning to Bohr’s quantum condition. However, this paper shows that Bohr’s quantum condition can also be derived from formulas of classical mechanics. Even if an attempt is made to rewrite Bohr’s quantum condition as a relativistic quantum condition, it is not possible to derive the relativistic energy levels of a hydrogen atom. However, if the true quantum condition built into Bohr’s quantum condition is used, then the relativistic energy levels can also be derived, in addition to the nonrelativistic energy levels of a hydrogen atom. In this way we can conclude that the quantum condition vn/c = a/n advocated by this paper is superior as a quantum condition to that of Bohr.