As already mentioned, Niels Bohr published his atomic model in a three-part series of articles in the journal Philosophical Magazine in 1913 [14,15,16]. Its title would certainly be perplexing today, but at that time, this journal was the most important English forum for the latest results in physics. It is important to see that Bohr was a remarkably young scientist when he developed the model after earning his doctorate less than 2 years earlier. He worked a few months in Ernest Rutherford’s laboratory in Manchester, and his work received considerable support from the Nobel Laureate. The article itself was published with the initiative and support of Rutherford, which is expressed by a footnote attached to the title: “Communicated by Prof. E. Rutherford, F.R.S.” The date of receipt is provided only at the end of the first article, which is 5 April 1913, and the paper was published in the July 1913 issue of the journal.

Before discussing the content of the articles in more detail, it is worth examining their list of references. A total of 33 articles are cited in the first one, 17 articles in the second, of which only 4 appear in the first article as well. In the third work, there are only six references to literature apart from the ones to the first and second parts, and none of these can be found in the first article. From this trend, it is clear that Bohr considered the three articles as a single work, and he never suspected that someone would try to read the second or the third one without having read the first.

It is also striking that of the 33 references to literature in the first article, only 5 were pre-1908, i.e., at least 5 years old in 1913. Although Bohr’s work obviously drew directly on earlier publications, the author must have thought that his expected readers are already familiar with them, so there was no need to cite the sources. Such is the case, for example, with the works of Balmer and Rydberg; here, another interesting point is that in the three articles, the name Balmer was mentioned a total of ten times and Rydberg four times, but there was no formal citation of any of the articles by these scientists, which were published in the 1880s [22,23,24].

Although the first article in the series [14] essentially contains all the basic assumptions of the model, a formal introduction of these is only provided at the end of the third paper [16]. Instead of the two postulates commonly used today, Bohr originally proposed five fundamental assumptions, none of which can be derived from classical physics:

1.

That energy radiation is not emitted (or absorbed) in the continuous way assumed in the ordinary electrodynamics but only during the passing of the systems between different “stationary” states.

2.

That the dynamical equilibrium of the systems in the stationary states is governed by the ordinary laws of mechanics, while these laws do not hold for the passing of the systems between the different stationary states.

3.

That the radiation emitted during the transition of a system between two stationary states is homogeneous, and that the relation between the frequency ν and the total amount of energy emitted E is E = hν, where h is the Planck constant.

4.

That the different stationary states of a simple system consisting of an electron rotating round a positive nucleus are determined by the condition that the ratio between the total energy, emitted during the formation of the configuration, and the frequency of revolution of the electron is an entire multiple of h/2. Assuming that the orbit of the electron is circular, this assumption is equivalent with the assumption that the angular momentum of the electron round the nucleus is equal to an entire multiple of h/(2π).

5.

That the “permanent” state of any atomic system—i.e., the state in which the energy emitted is maximum—is determined by the condition that the angular momentum of every electron round the center of its orbit is equal to h/(2π).”

The first article already attracted massive interest. Its contents were presented in the autumn of 1913 at one of the physics seminars organized in Zürich, which were regularly attended by many famous native German-speaking physicists. This seminar is also worth mentioning because one of the participants, Franz Tank (1890–1981), who later authored a scientific paper on the Bohr model [41], recalled two interesting remarks about it in a letter, dated 11 May 1964 (half a century after the seminar), to Max Jammer (1915–2010), who wrote a book on the history of quantum mechanics. It is evident that contemporary physicists easily understood the essence of the theory and immediately found both the weaknesses and groundbreaking novelty in it [42].

Max von Laue (1879–1960), who was awarded the Nobel Prize in Physics a year after the notable seminar, had a devastating opinion of the model:

“Das ist Unsinn, die Maxwellschen Gleichungen gelten unter allen Umständen, ein Elektron auf Kreisbahn muss strahlen.” (This is all nonsense! The Maxwell equations are valid under all circumstances, an electron in a circular orbit must emit radiation.)

Albert Einstein, who himself became famous for questioning the previous knowledge of physics, had a markedly different view:

“Sehr merkwürdig, da muss etwas dahinter sein; ich glaube nicht, dass die Rydbergkonstante durch Zufall in absoluten Werten ausgedrückt richtig herauskommt.” (Very remarkable! There must be something behind it. I do not believe that the derivation of the absolute value of the Rydberg constant is purely fortuitous.)

Both comments reflect valid physical observations and are worth analyzing in some detail.

Max von Laue referred to the Maxwell equations. They were introduced by the Scottish mathematician and physicist James Clark Maxwell (1831–1879) in the mid-nineteenth century [43]. The theory had a huge impact on physics; it provided a unified framework of interpretation for many previously known observations and predicted the existence of several phenomena that were soon confirmed by experimental observations.

An important consequence of the Maxwell equations for the Bohr model is that any electric charge that changes velocity (i.e., accelerates) emits electromagnetic radiation. The negatively charged electron was discovered by J. J. Thomson in 1897 [38], and the positively charged atomic nucleus by E. Rutherford in 1911 [31]. In both cases, the discovery included the observation that the particle was much smaller than the atom itself and was part of the atom. Together, the two particles explained both the overall neutral charge and the mass of the atom, meaning that there was no need to assume the presence of anything else. Thus, there must be an attractive electrostatic interaction (the Coulomb force [44,45,46]) between the two charges, which varies inversely with the square of the distance between them. This naturally raises the question: why does this attractive force not cause the atom to collapse to a much smaller size? Rutherford had essentially addressed this question: this does not happen because the two particles move at high relative speed, resulting—analogously to the solar system (where gravity also follows an inverse-square law)—periodic motion on a closed orbit. The mass of the nucleus is at least three orders of magnitude greater than that of the electron, so the view that the motion should be described in a coordinate system fixed to the nucleus, in which the electron moves, was also natural. In an atom, unlike in the solar system, the moving particles carry electric charges. Therefore, according to the Maxwell equations, the system should continuously emit energy in the form of electromagnetic radiation. This would result in a continuous loss of energy by the electron, leading to a gradual reduction in its velocity, and ultimately to its collapse into the nucleus. For bodies much larger than electrons, this notion is not simply a theoretical prediction of the Maxwell equations but a well-established experimental fact.

The postulate of the Bohr model that most directly contradicted the physics known at the time was that no radiation is emitted and that the orbit of the electron remains stationary. From today’s perspective, Bohr was simply articulating what had already become indisputable on the basis of experimental evidence. The atom could not be regarded as an unstable structure in time, and the mass, charge, and small size of both the electron and the nucleus were directly supported by experimental evidence. In this case, one might have questioned the validity of the Coulomb law, since there was no direct evidence of the inverse squared distance dependence in the range of such small sizes, but it is unlikely that any serious physicist would have entertained this idea because the electrostatic interaction, i.e., the Coulomb law, played a crucial role in the discovery of both the electron and the nucleus.

In summary, the first two postulates formulated in Bohr’s original work essentially interpreted the clear experimental observations that contradicted the electrodynamics known at the time.

It is unlikely that Bohr’s third postulate was met with disapproval by contemporary physicists: they were closely aligned with the works of Einstein [47,48,49] and Planck [50,51,52] on electromagnetic and blackbody radiation, both of which were already widely accepted in 1913. Here, from today’s perspective, the postulate simply states that a single photon is created or absorbed during the transition between two stationary states. In the case of absorption, this is related to the fact that the duration of a photon-induced transition is so brief that the probability of another photon arriving during the process is negligible.

The fourth and fifth postulates are similar in content, which explains why they were not subsequently regarded as distinct assumptions. Using today’s usual method of formulation, this is a quantum condition; it states that an electron orbiting within an atom has an angular momentum of nh/(2π), where n is a positive integer (1, 2, 3…). The most stable state of the system is n = 1. This is the same as Eq. (2) in the chapter “Modern digression.” This postulate was key to the extraordinary success of the Bohr model, so it is worth discussing its origin and consequences at length.

Einstein’s remark already suggested that the persuasive strength of the Bohr model arises from its ability to interpret the Rydberg equation in a theoretical way and the successful derivation of the Rydberg constant as a combination of other universal constants (Eqs. 10 and 11). What is more, the value of the Rydberg constant obtained through Bohr’s derivation was practically identical to the value determined experimentally. It is important to note that the agreement between the constants in the formula must be evaluated using the values known in 1913. Bohr’s first article [14] follows the widespread scientific practice 100 years ago in that only numerical values are given without units. From today’s perspective, clarifying the use of units of measurement requires investigation. In addition, he did not use the dimension of inverse distance for the Rydberg constant, as customary today, but frequency (RH’ = 2π2me4/h3). The values he listed for the universal constants are as follows: e = 4.7 × 10−10, e/m = 5.31 × 1017, and h = 6.5 × 10−27 (centimeter–gram–second (CGS) units). These give a Rydberg constant of 3.1 × 1015, while the experimental value used by Bohr was 3.290 × 1015, with a difference of about 6%. The agreement of modern values is even more striking: using any of the first five members of the Balmer series (656.3 nm, 486.1 nm, 434.0 nm, 410.2 nm, 397.0 nm), RH = 1.097 × 107 m−1 is calculated in Eq. (1), and 1.097 × 107 m−1 can be derived from the formula in Eq. (11) as well, so there is no difference in the first four significant digits.

The Rydberg equation was found essentially by systematic trial and error on the basis of experimental data. However, the probability of finding the formula in Eq. (11), which connects the Rydberg constant to other universal constants, by systematic trial and error is very small, as it contains the fourth power of one of the constants. It should also be noted that in 1890, at the time of the publication of the Rydberg equation [32, 32], the elementary charge, the mass of the electron, or the Planck constant were still unknown. Einstein’s remark is therefore entirely understandable: there must be a correlation behind the convergence of the theoretical and the experimental value, and an achievement so significant that the direct use of the Maxwell equations in the atomic size range can be safely abandoned.

The interpretation of the physical background of the fourth postulate, which determines the angular momentum of the electron, will be discussed later, but all the justifications in today’s textbooks refer to scientific phenomena that were only established significantly later in time. One might also suspect that Bohr used a kind of reverse engineering, i.e., he found a relationship between the electron velocity and the radius of the orbit proceeding from Eq. (3), used one of them as a parameter to perform the derivation, and then, knowing this and the experimental value of the Rydberg constant, realized that initial assumption 2 will give a result that matches the observations well. Today, when it is customary to work with physical properties in the equations and not values, this certainly seems to be a feasible route, but for Bohr, the equations involved numerical values, which would have made such a reverse-engineering process significantly more difficult.

Although there is no trace of this in the wording of the postulates, Bohr’s first article contains an explicit reference to how this assumption was formulated. The original sentence reads:

“Let us now assume that, during the binding of the electron, a homogeneous radiation is emitted of a frequency ν, equal to half the frequency of revolution of the electron in its final orbit.”

Maintaining the usage of the notation introduced earlier: the circumference of the circular path is 2πrn, and the electron speed is vn, from which the frequency of the angular motion (ν) is:

Bohr’s assumption yields that the total kinetic energy of the electron must be an integer (n) multiple of this hν/2 unit. That is:

$$\frac_^}=nh\frac_}_}}$$

(13)

Equation (2) follows from Eq. (13) by rearrangement. Bohr did not explain why the frequency of the radiation emitted is half the frequency of the angular motion of the electron and not equal to it. It is reasonable to assume that he might have considered this self-evident given that half of the period elapses between the two extreme positions of the motion. However, it is also possible that, knowing the course of the whole derivation, he assumed the division by 2 because this was how the match with the experimental data became convincing. In contrast to the full backward derivation path, the need for a division by 2 is easily recognized even if one works with numerical values only. So, the origin of the postulate is essentially that Bohr hypothesized a relationship between electromagnetic radiation and the frequency of the angular motion of the electron. This is not a derivation in the strict sense—nor is one expected of a postulate; its only test is how well the conclusions drawn from it match physical reality.

It can also be observed in the original articles that Bohr cited many papers on spectroscopy, but in his analysis, he relied only on the Balmer and the Paschen series [34], and for the latter, he cited the original data series as well. He refers to the additional spectral lines of hydrogen as a kind of prediction, despite the fact that the existence of the Lyman series had already been published by that time [33]—in a journal from which Bohr cited other papers. Another interesting feature of the spectroscopic references is that two “element names”, ionium [53] and nebulium [54], which are not known in modern chemistry, are present in the titles. Of course, this does not imply that the corresponding spectral lines used are not real.

At the end of this section, it is worth highlighting two interesting details concerning the Bohr model. One is the value of the Bohr radius calculated from Eq. (6), 5.292 × 10−11 m. This means that the diameter of a hydrogen atom is roughly one ten-thousandth of a micrometer. On the basis of experimental observations, this was already considered the approximate size range of atoms in 1913. However, the numerical value itself was not suitable for such an accurate comparison as was possible with the Rydberg constant because it was (and still is) impossible to attribute a well-defined experimental size to an atom, especially in the gaseous state. Equation 6, along with the definition of a0, can also be derived naturally in the wave-mechanical model of the atom, which is why a0 is often used as the unit of distance in that context.

The other remark is that—taking the Bohr model literally—the hydrogen atom should not be pictured as a sphere, but as a very thin disc (Fig. 2). The motion of the electron always remains in the plane defined by the velocity vector of the electron and the position vector drawn from the nucleus to the electron. In multi-electron atoms, one may imagine different electrons orbiting the nucleus in different planes, resulting in an overall spherical distribution, but for the hydrogen atom containing a single electron, for which the Bohr model was remarkably successful, there is no escaping its inherently planar structure within the framework of classical physics.