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Atoms

The Relativistic Bohr Radius and Its Agreement with the Dirac Most-Probable Radius

Espen Gaarder Haug

Published Jul 17, 2026• 0 citations
Publisher

Abstract

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 The Bohr model of the hydrogen atom has since been replaced by modern quantum mechanics and the Schrodinger equation.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.

DOI
10.3390/atoms14070059

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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 The Bohr model of the hydrogen atom has since been replaced by modern quantum mechanics and the Schrodinger equation.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.
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.
This research examines the probability percentage of the existence of the hydrogen atom's electron in the space surrounding the nucleus of the mentioned atom. From the calculations of the semi-classical Bohr atomic model, a specific radius for the hydrogen atom's electron was obtained (with good approximation a_0=5.3×〖10〗^(-10) m); However, after the wave property of particles was introduced and this property was extended to the electron, Schrödinger formulated the wave equation of matter and its probabilistic physical interpretation was proposed by Born. In this paper, by applying the Schrödinger equation to the hydrogen atom, we aim to determine the probability of finding the electron of the hydrogen atom at the Bohr radius, as well as at distances smaller and greater than it.
The Bohr model is historically important in the understanding of quantum physics. It gives many important results for the hydrogen atom including atomic size and ionization energy. It is sometimes referred to as a semi-classical calculation, because it includes the physics of angular momentum quantization without the full quantum mechanical description of an electron wave. It is tremendously useful for gaining physical intuition of the quantum nature of the atom. In the Bohr model of the hydrogen atom, an electron has a circular orbit with speed v and radius r about the proton. The Bohr quantization condition is that an integer number of electron (de Broglie) wavelengths must fit to the atomic circumference.
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.