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Journal of Applied Mathematics and Physics

The Incompleteness of Quantum Mechanics Demonstrated by Considerations of Relativistic Kinetic Energy

Koshun Suto

Published Jan 1, 2020• 2 citations
Publisher

Abstract

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.

DOI
10.4236/jamp.2020.82016

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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.
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.
The Bohr radius was derived using the non-relativistic de Broglie approximation, not the relativistic de Broglie equation. In this work, we will derive a relativistic Bohr radius from both the relativistic de Broglie approximation and relativistic kinetic energy. The Bohr model of the hydrogen atom has since been replaced by modern quantum mechanics and the Schrodinger equation. It is well known that the Schrodinger equation confirms the first Bohr radius as the most probable radius, but the Schrodinger equation itself is non-relativistic. The Dirac equation can likely be used, but there seem to be very few publications on the relativistically corrected Bohr radius. This is something we will discuss.
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. 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’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.
In this chapter we are going to make a very detailed study of the one-electron atom using relativistic quantum mechanics. The one-electron atom is the simplest bound system that occurs in nature, and plays a central role in both classical and quantum theory. There are two reasons for this. Firstly, it is a model that can be solved exactly at many levels of theory — the classical Bohr atom, the non-relativistic quantum mechanical one-electron atom, and the relativistic one-electron atom with spin zero and spin 1/2. Secondly, most of our understanding of multielectron atoms, molecules and solids is based on this model. Of course, a good description of the hydrogen atom is provided by non-relativistic Schrödinger theory. However, once the relativistic quantum theory had been discovered it was of great importance to validate the theory by describing hydrogen at least as well as Schrödinger theory. As we shall see, the Dirac theory was exceedingly successful in this, and furthermore, at several points in the theory the non-relativistic limit can be taken and the effect of relativity in one-electron atoms can be seen explicitly. This connection with the simpler theory will be made here wherever insight may be gained from doing so. One of the key advances brought about by the advent of quantum theory was that it defined eigenfunctions as well as the eigenvalues predicted by the Bohr theory.