Bohr's 1913 trilogy marks a turning point in the history of quantum physics for its introduction of the planetary model of the atom and its derivation of the Balmer formula for the hydrogen spectrum. It has been studied in the seminal study [1] and is discussed more recently by [2-4]. Famously, Bohr's classical paper made two utterly non-classical assumptions, according to which electrons orbited the nucleus in stationary states without losing energy while they emitted radiation only in transitions between these states.

As I will argue in this paper, Bohr's 1913 trilogy made a third crucial major assumption, which entailed key elements of the correspondence principle formulated five years later. This third assumption postulated a relation between the mechanical frequency of a stationary state and the radiation frequency in analogy with classical radiation theory. Though this relation was in direct conflict with the key assumptions of stationary states and transitions, Bohr's reasoning—as I want to show—relied heavily on it, and a major part of his initial work was devoted to clarifying how the connection of radiation and motion, the stationary states and the transitions were interrelated.

The 1913 trilogy

The main aim of Bohr's 1913 trilogy was to derive the Balmer formula from the newly-proposed atomic model. After the introduction of the stationary states and the frequency condition, the derivation follows a three-step recipe consisting of the calculation of the energy from classical mechanics, the quantization of the energy and finally the calculation of the spectral frequencies via the Planck Relation ΔE=hν. Given the simplicity of the system considered and the assumption of the stationary states and the frequency condition, the first and last step did not call for an extended argument and Bohr simply gave the relevant equations.

In contrast, the main argument is concerned with the quantization of his atomic model. To achieve this quantization Bohr established the aforementioned relation between the frequency of the emitted radiation and the mechanical frequency of the electron. Bohr introduced the relation in his first of three derivations of the Balmer formula. Considering the binding of a free electron from infinity into a stationary state inside the atom, he equated the emitted radiation frequency ν with the mean value of the orbiting frequencies of the free electron and of the electron bound within the atom:

ν=ω2.(1)

This assumption allowed Bohr to quantize the energy of the stationary states through a relation between the radiation and the motion of the electron: using (1) he could rewrite the Planck relation in terms of the quantum number τ as W=τhν=12hτω. Substituting the mechanical frequency ω for an elliptic motion, Bohr expressed the atom's energy as:

W=2πme4h2τ2.

Based on (1), Bohr had thus quantized the energy of the atom and could continue to derive the Balmer formula for the hydrogen spectrum from the frequency condition:

ν=R1τ22−1τ12,

with R being the Rydberg constant. However, this was not the final result of Bohr's paper. The basic assumption used for the quantization—the relation (1) between the frequency of the radiation and the mechanical frequency of the electron—had its roots in classical radiation theory, according to which the oscillatory properties of light, such as frequency and intensity, are determined by the frequency and amplitude of the motion of an accelerated charge.

As Bohr explicitly acknowledged, the classical radiation mechanism was replaced entirely by the fundamental assumptions of his model: radiation is produced through a transition between stationary states. Though (1) was in direct conflict with the physical assumptions of his model, Bohr's thinking nonetheless depended crucially on it. This is emphasized by the fact that Bohr knew that he could quantize the orbits on the assumption that the angular momentum of the atom is equal to nh2π.1

Exploring radiation theory

While (1) could not be interpreted as a description of a physical process in a classical sense, Bohr kept the equation and explored its conceptual significance within his new quantum theory of the atom. This was a vital concern for Bohr, first, because he needed a justification; second, because he hoped for insights on how to develop the theory by investigating which parts of classical radiation theory could still be used within the new model. The investigation of this question loomed large in Bohr's 1913 paper and continued to play an important role in his early work on the quantum theory of the atom.

For the preceding argument Bohr justified the radiation-motion relation by means of a comparison between the radiation process and the motion of the electron before and after the radiation process. This justification itself was not very convincing: it rested solely on its empirical success and the questionable mean-value argument, and Bohr decided to give a second derivation of the Balmer formula to add further plausibility to (1).

Already his first hand waving argument, however, points to some of the central aspects of his attempt to connect radiation and electronic motion: In light of the missing radiation mechanism in quantum theory, Bohr tried to recover a relation between radiation and electronic motion within quantum theory analogous to the one in classical radiation theory. He refrained from proposing a new hypothetical radiation mechanism and shied away from such assumptions until 1924. Instead he resorted to experimental, mathematical and intertheoretical arguments to construct a formal relation between radiation and matter and aimed to get insights into the properties of a radiation mechanism yet to be found.

This mixture of empirical, mathematical and intertheoretic arguments is also apparent in Bohr's second derivation of the Balmer formula. In this derivation the direction of the argument is reversed: the aim is no longer to derive the Balmer formula from the proposed atomic model but to show that the atomic model, the assumed relation between radiation and motion, and the empirical spectrum are consistent with each other. Bohr shows this by making the slightly more general claim that the energy of the quantized atom is f(τ)hω. This approach still replaces the radiation frequency by a multiple of the orbiting frequency and thus tacitly assumes a connection between radiation and motion. Determining f(τ) as a function of the quantum number τ Bohr follows the same quantization procedure as in the first derivation and gets the spectrum of the hydrogen atom with the “generalized” assumption as:

ν=πme2E22h31f(τ2)2−1f(τ1)2

In order to be equivalent to the empirical Balmer formula this function has to be f(τ)=12τ, as Bohr showed by comparing the frequency of a quantum transition and the orbiting frequencies of the stationary states in the case of high quantum numbers, in which classical electrodynamics and quantum theory predict the same numerical value for the radiation frequency.

This result implied the consistency of his major assumptions, the radiation-motion connection and the Balmer formula, which entered as an empirical constraint. For the argument the high quantum number limit plays an important role as it renders the condition for the consistency precisely and shows that the assumption of a relation between radiation and motion within quantum theory leads to the agreement of quantum theory and classical theory in this limit.

Showing the consistency of his approach was not the final conclusion of Bohr's argument. In the last part of the argument he turned to the relation between radiation and motion in general and showed that in the classical limit the radiation frequency and the mechanical frequency of his model were connected according to the equation ν=nω, with n being the difference between the quantum numbers of two states in the classical limit.2 This relation, Bohr emphasized, pointed towards an analogy between the classical and the quantum theory of radiation:

The possibility of an emission of a radiation of such a frequency may also be interpreted from analogy with ordinary electrodynamics, as an electron rotating round a nucleus in an elliptical orbit will emit a radiation which according to Fourier's theorem can be resolved into homogeneous components, the frequencies of which are nω [[5], 14].

This recapturing of the relation between orbital frequency and the radiation frequency is part of Bohr's conclusion: While classical electrodynamics in general is in contradiction with the basic assumptions of the Bohr model, the classical relation between the frequency of radiation and the mechanical frequency can be established at least in the classical limit as a formal relation. To show this is as important to Bohr as to show that his new model predicts the correct Balmer formula. As such the classical relation between motion and radiation is indispensable for Bohr's early work as a touchstone for his two derivations and it is of vital concern in the attempt to explore the break and the continuity with classical radiation theory.

The Stark effect 1914

The radiation-motion relation continued to play an important role in Bohr's works following the 1913 trilogy; after he had finished his initial papers Bohr adapted his model to account for the Stark and Zeeman effects.

Trying to explain these two effects Bohr wrote to his teacher in Manchester, Ernest Rutherford (Fig. 1), that the quantization procedure he used to deal with the two effects was “exactly analogeous [sic!]”3 to his previous paper. While he did not succeed to explain the Zeeman effect on this basis4, Bohr managed to adapt the quantization procedure of his original trilogy to the Stark effect. In his paper [7], Bohr considered a simplified model for the atom in a constant electric field. In analogy to the 1913 trilogy he simply gave the classical frequency of the electron ω and its total energy A as a function of the major axis of the orbit 2a and a constant C:

ω2=e24π2ma21∓3Ea2e

A=C−e22a∓3aeE

For the quantization of this system Bohr again used the relation between radiation frequency and orbital frequency. Bohr had reconsidered his classical limit argument and adapted it for his new problem: In the classical limit, he argued, one could rewrite the frequency condition as a differential equation. As the radiation frequency equals the mechanical frequency in this case, the Planck relation can again be rewritten in terms of a mechanical frequency ωn of the electron:

An+1−An=hνdAndn=hωn

Though the key assumptions for this argument only hold in the classical limit, Bohr pointed out that the equation was also satisfied by every energy and frequency expressed by quantum numbers in his original trilogy. Bohr kept it for the Stark effect and used it as a quantization condition. In order to get a quantized energy he needed to reexpress the major axis a in terms of the quantum number n. In the limiting case of a weak electric field Bohr got his quantization condition:

dnda=πemha1∓52Ea2e

Integrating this equation leads to an expression of the major axis a in terms of n, so that Bohr could reexpress the energy as a function of the quantum number n. The quantized energy levels of the Stark effect then were:

An=C−2π2e4mh2n2×1±E3h416π4e5m2n4

The first term of this energy returns the terms of Balmer formula as the electric field vanishes.

Conclusion

The relation between radiation and motion played a major role in the initial work establishing the Bohr model and its extension to the Stark and Zeeman effect. Stemming from classical radiation theory this relation was in contradiction with the major assumptions of the new model of the atom; nonetheless, Bohr's thinking relied heavily on the radiation-motion connection both in establishing his results and in clarifying the foundations of the new model.

Compared with his later statements on the correspondence principle there is a remarkable resemblance between the statements in 1913/1914 and the correspondence idea between 1918 and 1923: In both cases the relation between the orbiting frequency and the frequency of radiation and the asymptotic limit between quantum theory and classical electrodynamics were at the core of Bohr's reasoning. Also the idea of the Fourier series is mentioned, which would become ever more important in the development of the correspondence idea.5 While the core elements of the later principle are already apparent in the early phase, they are used very differently in both phases.

This shift in the usage of the basic correspondence relation was due to the development of the quantum theory of the atom as a whole. The physicists who developed Bohr's initial model mainly extended the description of the stationary states to more complicated systems with the help of celestial mechanics. In this context the relation between the radiation frequency and the orbiting frequency was replaced by Sommerfeld's quantum condition: J=∮pdq=nh. The new relation did not involve any assumptions on the radiation process or the interaction between radiation and matter, it was a statement about the action of a mechanical system and its quantization. The fruitfulness of this new quantization condition led physicists including Bohr to abandon the early quantization procedure. Through this abandonment, the frequency condition became the sole relation of radiation theory, which was needed to determine the series formulae describing atomic spectra. Whereas the loss of the classical radiation mechanism had been a vital concern for Bohr in his early works, this was no longer the case in the development of the quantum theory of the atom. The relation should, however, resurface in Bohr's thinking about the transitions in quantum theory and in this form led to the formulation of the correspondence principle.

1

[[5], 15]. Whereas the general quantum condition J=∮pdq=nh would later be developed by Sommerfeld as his starting point for the quantum theory of the atom, Bohr only mentioned the possibility of a quantized angular momentum as a side remark.

2

Bohr considered a transition N→N−n, with n≪N. The frequency of radiation is ν=π2me2E22c2h32Nn−n2N2(N−n)2. The orbiting frequency of the two stationary states are ωN=π2me2E22c2h3N3 and ωN−n=π2me2E22c2h3(N−n)3. Demanding that the ratio ν/ω had to be equal to 1, Bohr noted that ν=nω in the classical limit.

Holding that the frequencies of the Zeeman effect did not obey Ritz's combination principle and that the magnetic field did not change the energy levels of the atom, Bohr argued the magnetic field had to affect the transition process and not the stationary states. As [[2], 92] has pointed out, Bohr's assumptions were mistaken. Bohr's skepticism about the frequency condition was short-lived; after the work of Sommerfeld, Epstein and Schwarzschild, Bohr became convinced that the Zeeman effect could be explained on the basis of the assumption of stationary states and the frequency condition, which he now took as fundamental postulates of his atomic model.

5

See also [[2], 89]. Darrigol identified this as “the first germ of the analogy later extended under the name of the correspondence principle.”