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Journal of Applied Mathematics and Physics

The Quantum Condition That Should Have Been Assumed by Bohr When Deriving the Energy Levels of a Hydrogen Atom

Koshun Suto

Published Jan 1, 2021• 11 citations
Publisher

Abstract

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.

DOI
10.4236/jamp.2021.96084

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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.
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.
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’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.
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.
The existence of line spectra can be explained bymeans of the first “quantum”model of the atom, developed by Bohr in 1913. Although the Bohr model of the hydrogen atom was eventually replaced, it yields the correct values for the observed spectral lines, and gives a substantial insight into the structure of atoms in general. The following derivation has the objective of obtaining the energy levels of the Bohr atom. If we can obtain the energy levels, we can reproduce the hydrogen atom spectra. The derivation proceeds with three major elements: first, use force balance to relate the velocity to the radius of the electron orbit, then use a quantum assumption to get the radius, then solve for the energies.
The phase velocity of a material wave is given by the following equation: If this formula is rewritten using the Planck-Einstein relation and the de Broglie formula , it becomes: Next, the values for E and p, obtained from a relation applicable to the electron in a hydrogen atom derived by the author, are substituted into this equation. When that is done, multiple formulas relating to the phase velocity of the electron wave are derived. These formulas contain still-unknown ultra-low energy levels of the hydrogen atom and electron orbital radii corresponding to those energy levels. also appear in the formula for energy levels of the hydrogen atom derived in this paper. This serves as grounds for the existence of ultra-low energy levels in addition to the already-known energy levels of the hydrogen atom.