## Helge Kragh

Print publication date: 2012

Print ISBN-13: 9780199654987

Published to Oxford Scholarship Online: May 2012

DOI: 10.1093/acprof:oso/9780199654987.001.0001

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# Crisis: The End of the Bohr Model

Chapter:
(p.313) 8 Crisis: The End of the Bohr Model
Source:
Niels Bohr and the Quantum Atom
Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199654987.003.0008

# Abstract and Keywords

Failed attempts to understand the anomalous Zeeman effect contributed to the feeling of crisis that by 1924 characterized parts of the physics community. The solution proposed by W. Heisenberg’s ‘core model’ only raised other problems. Another indication was the state of radiation theory and the uncertain relationship between wave theory and the light quantum, a problem that resulted in the controversial BKS (Bohr–Kramers–Slater) theory of 1924. In the wake of this theory, H. Kramers and Heisenberg constructed a formal theory of dispersion that did not rely on electron orbits but only on observable quantities, in agreement with the ‘quantum mechanics’ programme of M. Born. The development culminated with Heisenberg’s paper of August 1925, which marks the end of the Bohr model and the beginning of quantum mechanics. The chapter reconsiders the crisis and the reasons for it, in particular the role of experimental anomalies.

At the time when Bohr’s atomic model would have been celebrating its tenth anniversary, problems began to accumulate. Some of these were experimental anomalies, while others were of a logical and conceptual nature; some of them were old, others new. To the list of old problems belonged the anomalous Zeeman effect, which turned out to be intimately connected to the complex spectra and the assignment of quantum numbers to many-electron atoms. The physicists could account for the spectroscopic details of the Zeeman effect, but only by proceeding phenomenologically and abandoning any explanation of the effect based on established atomic theory. The core models introduced by Landé and Heisenberg did not fit easily with Bohr’s principles, among other reasons because they operated with half-integral quantum numbers and seemed to contradict the construction principle on which Bohr had based his theory of the periodic system. Generally speaking, the years after 1922 saw a drastic shift in the status and nature of atomic models, which became increasingly abstract, symbolic, and formal, soon leading to proposals of abandoning the model concept at all.

In the summer of 1925, while completing a major review paper on atomic and quantum theory, Van Vleck commented on the most recent developments: ‘Modern physics certainly is passing through contortions in its attempt to explain the simultaneous appearance of quantum and classical phenomena’, he said; ‘but it is not surprising that paradoxical theories are required to explain paradoxical phenomena’.1 This was an evaluation with which Bohr fully agreed.

The feeling of a crisis in atomic physics gave rise to several proposals of a radical nature, perhaps the most radical one being the idea that energy and momentum are not conserved in individual atomic processes. This idea, which featured prominently in a theory offered by Bohr, Kramers, and Slater (BKS) in 1924, was an attempt to understand radiation from atoms without making use of the concept of light quanta, which were particles non grata in Copenhagen. The BKS atom with its orchestra of ‘virtual oscillators’ was as original as it was controversial. Rejected by a majority of physicists, the theory was short-lived but nonetheless influential. Its influence was particularly strong in the dispersion theories that attracted much attention in the final phase of the old quantum theory and were greatly significant in the revolutionary change to a new (p.314) ‘quantum mechanics’, a term that Born had introduced about a year before it became a reality (but, as mentioned in Section 6.6, the term was used by Einstein as early as 1922). In the process that led to the new mechanics and at the same time made the Bohr–Sommerfeld orbital atomic theory obsolete, the centres of theoretical physics in Göttingen and Copenhagen played a dominant role. The key players in the development included the leaders of the two institutions, Born and Bohr, but no less more important were the three young physicists Kramers, Pauli, and Heisenberg.

The crisis that culminated in the spring and summer of 1925 was the result of accumulated empirical anomalies and a growing dissatisfaction with the conceptual foundation of atomic physics. It led to the abandonment of electron orbits and attempts to base atomic theory on discrete rather than continuous quantities, and it was influenced by the positivistic doctrine that a satisfactory theory of the atomic world should include only such symbols as referred to measurable quantities. The latter doctrine, known as the observability principle, featured prominently in Heisenberg’s epoch-defining paper of the late summer of 1925, but the actual role it played in the foundation of quantum mechanics was more limited. Although the emergence of the Göttingen quantum mechanics was indeed a break with the past, the break with Bohr’s old atomic theory was not complete. The basic postulates, meaning the notion of stationary states and the frequency condition ΔE = hν, survived. Perhaps the strongest link of continuity was provided by the correspondence principle, which in a mathematically sophisticated form permeated much of Heisenberg’s new theory.

# 8.1 The riddle of the Zeeman effect

‘A colleague who met me strolling rather aimlessly in the beautiful streets of Copenhagen said to me in a friendly manner, “You look very unhappy”; whereupon I answered fiercely, “How can one look happy when he is thinking about the anomalous Zeeman effect?”.’2 This is what Pauli recalled from his stay in Copenhagen, perhaps in the spring of 1923. He was not the only physicist to worry about this spectroscopic effect, which was seen as a problem for atomic theory no less serious than the helium anomaly, an indication that something was quite wrong with the Bohr–Sommerfeld theory of atomic structure. The same year, in his contribution to the Bohr issue of Die Naturwissenschaften, Born suggested that ‘the entire complex of questions connected with the anomalous Zeeman effect’ was reason enough to reconsider the existing atomic theory and possibly to reconstruct it entirely.3 The complex of questions he referred to was not only the anomalous Zeeman effect, but also the related magneto-optical Paschen–Back effect, the higher multiplet spectra discovered in 1922, and the relationship between multiplets in optical spectra and X-ray spectra. For the sake of simplicity, in this section I shall consider only the anomalous Zeeman effect.

(p.315) Physicists had been studying the complex or anomalous Zeeman effect ever since it was discovered in 1897 by the Irishman Thomas Preston. Their efforts resulted in various empirical rules, for example that the frequencies of the components were always rational fractions of those of the normal Zeeman effect. However, contrary to this effect, which Lorentz had explained immediately after its discovery, the anomalous case defied explanation in terms of both electron and quantum theory. The rules established by Preston, Runge, Sommerfeld, and other physicists were useful, but they were basically of a numerological kind. For the normal Zeeman effect, as analyzed by Debye and Sommerfeld in their papers of 1916, the magnetic contribution to the energy of a spectral term was given by mhνL, where νL = eH/4πm e c is the classical Larmor precession frequency, with m e denoting the mass of the electron. The quantum number m was interpreted as the component of the azimuthal quantum number in the direction of the magnetic field H.

The first substantial progress in accounting for the anomalous Zeeman effect was made by Landé, who in February 1921 told Bohr about his ongoing research. Presenting it as empirically rather than theoretically based, he wrote: ‘With regard to the complicated types of the Zeeman effect, I have found a few empirical rules which go considerably beyond Sommerfeld’s compilation in the Ann. d. Phys. and permit one to make predictions regarding the neon spectrum. But what these rules signify is entirely incomprehensible to me’.4 Believing at the time that Bohr might be a rival in the quest to understand the anomalous Zeeman effect, he was ‘in a hurry with this [paper] because Bohr obviously is thinking about these things; and why should a foreign country forestall us in this’. Sommerfeld informed him that Bohr was too busy with other matters and, at any rate, ‘I don’t look upon him as a foreigner’.5 In fact, Bohr was at the time working hard on his new theory of the periodic system, not to mention his new institute of theoretical physics, and he had little time to think about the regularities of complex spectra.

In a paper that appeared in June 1921 Landé expressed the energy of a spectral Zeeman term by the generalized formula

$Display mathematics$

where W 0 is the energy of the unperturbed state. The proportionality constant g was introduced empirically, the only justification for it and its assigned values being that the observed components could then be reproduced. The constant or ‘splitting factor’, as Landé called it, was assigned the value g = 1 for singlets (normal Zeeman effect), but for doublets g = 2j/(2k – 1). For triplets the dependence of g on j and k was more complicated, but in all cases it could be expressed as rational fractions similar to the one of the doublet case. Assuming as a working hypothesis that Sommerfeld’s inner quantum number j was a measure of the total angular momentum and m determined its space quantization in a magnetic field, Landé argued that for singlets and triplets m attained 2j + 1 values, namely m = 0, ±1, ±2,…, ±j. However, in the case of doublets, (p.316) where j = k and j = k – 1, he had to assign half-integral values to the magnetic or ‘equatorial’ quantum number:

$Display mathematics$

Since m can attain 2j values, the total number of magnetic states for a doublet pair is 2k + 2(k – 1) = 2(2k – 1). Not only did half-integral values of m (and then also of j) lack any theoretical basis, they also seemed to contradict Bohr’s atomic theory, which allowed only integral quantum numbers. In 1921 Landé did not propose a physical interpretation of his theory, except that he suggested that the g factors might be interpreted as ‘apparently anomalous values of e/m’ (where m stands for the electron’s mass, not the magnetic quantum number).6 Sommerfeld, who knew of Landé’s work in advance of its publication, was critical of his approach but nonetheless responded with enthusiasm: ‘Bravo, you are a magician! Your construction of the doublet Zeeman types is very beautiful’.7 Without referring to Landé by name, he told Einstein about the recent progress in understanding the anomalous Zeeman effect on the basis of the inner quantum number: ‘Light, or better, dawn really is coming to spectroscopy’.8 Perhaps so, but the dawn did not extend to the underlying quantum theory, which was still shrouded in fog.

In another important work on the Zeeman effect, published in May 1923, Landé developed his phenomenological theory on the basis of the core model of the atom expressed in terms of angular momentum vectors. Apart from the total angular momentum J, these vectors were the core angular momentum R and the angular momentum K of the valence electron, the three vectors being connected by the equation J = R + K. Using this model he obtained an expression for the g factor:

$Display mathematics$

Landé’s new formula agreed impressively with experiments and thus brought a much needed order to the area. However, being an empirical formula summarizing experimental data, rather than a deduction from a theory or model, it still lacked a proper theoretical explanation. And there were other problems: it postulated half-integral quantum numbers, it required that the core contributed twice to the magnetic energy of the atom, and it seemed to violate Bohr’s generally accepted construction principle. In view of the controversial half-quanta, Landé cautiously referred to the quantities appearing in his formula as ‘“apparent” angular momenta, from which the real momenta may perhaps deviate’.9 In spite of the obvious weaknesses in his work it aroused much attention in the German physics community and was widely seen as an important step towards the final understanding of the anomalous Zeeman effect. Heisenberg found Landé’s work to be ‘very beautiful’ and Pauli congratulated him with his ‘wonderful scheme of the multiplets of the Zeeman effect’.10

(p.317) The complex structure of spectral lines and their behaviour in magnetic fields was not among Bohr’s primary concerns, but of course he was well aware of the work that Sommerfeld and others had done in this area. He referred to it in his Berlin lecture of 1920, where he suggested that the problems might be understood on the basis of the correspondence principle. As he saw it at the time, the anomalous Zeeman effect did not indicate a failure of ordinary electrodynamics, but was rather to be seen as ‘connected with an effect of the magnetic field on that intimate interaction between the motion of the inner and outer electrons which is responsible for the occurrence of the doublets’.11 About two years later, in his Göttingen lectures of June 1922, Bohr dealt briefly with the new theories of the anomalous Zeeman effect proposed by Landé and Heisenberg, but apart from noting that the effect proved the classical theory to be inadequate he did not go into details. He considered Heisenberg’s recent theory to be interesting, but also to rest on assumptions that were unacceptable. His main objection to the works of Landé and Heisenberg was the half-integral quantum numbers, which he found contradicted his theory of the electron configurations of the elements. ‘My viewpoint is this’, he complained to Landé, ‘that the entire manner of quantization (half integer quantum numbers etc.) does not appear reconcilable with the basic principles of the quantum theory, especially not in the form in which these principles are used in my work on atomic structure’.12

The intense discussions Bohr had with Pauli during the latter’s stay in Copenhagen in 1922–1923 made him pay greater attention to the magneto-optical problems, which he took up in his survey of line spectra and atomic structure published in 1923 in Annalen der Physik. Referring to Landé’s first paper on the anomalous Zeeman effect, he interpreted the results by assuming that ‘the change in the motion of the atom as a whole consists in a superposed precession around an axis through the nucleus, and parallel to the field, with a frequency of precession independent of the orientation of the atom relative to the field’.13 This interpretation he claimed to rest ‘on the basis of the correspondence principle’, as usual without elaborating or clarifying the comment. Although he noted that Landé’s assigment of half-integral quantum numbers disagreed with a ‘direct analogy with the quantum theory of periodic systems’, he offered an explanation of Landé’s findings.

Bohr’s hypothesis was qualitative and quite speculative, its essence being a postulate of a non-mechanical coupling between the core’s inner orbits and the outer orbit of the valence electron. In order to explain Landé’s result that an atom placed in a weak magnetic field can take on 2(2k – 1) states, he suggested the following view:

Because of stability properties of the atom which cannot be described mechanically, the coupling of the series electron to the atomic core is subject to a constraint [Zwang] which is not analogous to the effect of an external field, but which forces the atomic core to assume two different positions in the atom, instead of the single orientation possible in a constant external field, while at the same time, as a result of the same constraint, the outer electron, instead of 2k possible orientations in an external field, can only assume 2k – 1 orientations in the atomic assemblage.14

(p.318) In a later work Bohr, translating Zwang as ‘strain’, spoke of ‘a mechanically undescribable “strain” in the interaction of the electrons which prevents a unique assignment of quantum indices on the basis of mechanical pictures’.15

By means of this hypothesis of a constraint or Zwang, blatantly ad hoc as it was, Bohr could maintain the construction principle, get the right number of anomalous Zeeman states, namely 2(2k – 1), and even account for the stability of the helium atom. The cost was that he had to accept half-integral quantum numbers. This he did reluctantly and only for the quantities j and m, which he considered to be of a more formal nature with no clear model interpretation. The numbers that really mattered, and which characterized the electron configurations of the elements, were n and k, and these he insisted had to be integers. The assumption of a half-integral azimuthal quantum number not only had consequences that ‘seem to contradict our experience about spectra’, it also ‘must be regarded as a departure from this theory [the quantum theory of periodic systems] which can hardly be substantiated’.16 In an unpublished manuscript on the applications of quantum theory, a planned sequel to his 1923 paper in Zeitschrift für Physik, Bohr returned to the problem of the anomalous Zeeman effect and the idea of a peculiar intra-atomic constraint responsible for the failure of Larmor’s theorem:

In the electronic assemblage in an atom we have to do with a coupling mechanism which does not permit a direct application of the quantum theory of mechanical periodicity systems…We are led to the view that the interplay between series electron and atomic core, at least as far as the relative orientation of the orbit of the series electron and the electronic orbits in the core is concerned, conceals a ‘constraint’ [Zwang] that cannot be described by our mechanical concepts and has the effect that the stationary states of the atom, in essential respects, cannot be compared with those of a mechanical periodicity system…According to our view, it is just this constraint that finds its expression in the regularities of the anomalous Zeeman effect and, in particular, is responsible for the failure of the Larmor theorem.17

Bohr’s discussions with Pauli resulted in a joint manuscript on the anomalous Zeeman effect, in which the effect was explained on the basis of integral quantum numbers only. Enclosing a copy of the manuscript in a letter to Landé, Bohr wrote: ‘As you will see from our note, we are always so conservative here in Copenhagen that we stick sincerely to the integral quantum numbers’.18 Bohr and Pauli submitted the manuscript to Zeitschrift für Physik in early 1923, but decided to withdraw it because they came to doubt its validity.

In another letter to Landé, Bohr wrote that his and Pauli’s manuscript was ‘a desperate attempt [Verzweiflungsversuch] to remain true to the integral quantum numbers; we hoped to find in the very paradoxes an indication of the path along which one should search for a solution of the anomalous Zeeman effect. However, in the meantime the doubts about the necessity of those paradoxes have continued to grow’.19 Latest by March, the ‘conservative’ physicists Pauli and Bohr had come to accept half-integral inner quantum numbers. Bohr, said Pauli in a letter to Landé, ‘keeps (p.319) to the integral quantization of k’, whereas he believed that ‘in the question of the quantization of j and m one has to do with novel phenomena that defy the current foundation of quantum theory’.20

# 8.2 Heisenberg’s core models

Frustrated over his failed paper with Bohr, Pauli nonetheless continued investigating the anomalous Zeeman effect, and in late April his efforts resulted in a paper of a formal rather than model-based nature. Instead of starting with the complex spectral terms of the anomalous effect, Pauli started with the simpler strong-field terms and worked his way backward by means of the Paschen–Back effect. To do so he made use of a new rule, which he introduced without attempting to justify it: ‘The sum of the energy values in all those stationary states belonging to given values of m and k, remains a linear function of the field strength during an entire transition from weak to strong fields’.21 In this way he was able to derive Landé’s g factors in the case of weak fields and without relying on specific model interpretations.

In spite of this result Pauli was far from satisfied. ‘I have for very long vexed myself with the anomalous Zeeman effect and often lost my way’, he confided to Sommerfeld. ‘For a time I was quite desperate. On Bohr’s insistence I have finally send a small paper to the Zeitschrift für Physik…I have written all of this with a tear in the corner of my eyes and am anything but delighted’.22 At the end of his paper he expressed the same kind of dissatisfaction, pointing out that his derivation was of a formal nature only. A model interpretation ‘on the basis of the currently known principles of quantum theory is hardly possible’, he concluded. One problem was the failure of Larmor’s theorem, which required g = 1; another was that ‘the appearance of half-integral values for m and j already implies a fundamental break with the framework of the quantum theory of multiply periodic systems’. This was also what he wrote to Landé: ‘The question of the normalization of the quantum numbers is presently of secondary importance to me. For I am convinced that in the anomalous Zeeman effect there is no conditionally periodic model and that something essentially new must be done’.23

Although Pauli presented his theory as phenomenological and model-independent, he was not yet ready to consider this a virtue rather than a deficiency. In his letter to Landé of 23 May he wrote that his theory of the anomalous Zeeman effect was ‘restricted to the phenomenological level and leaves out all model considerations’, yet he also said that ‘[I am] very unhappy that I have not yet succeeded in finding a satisfactory model interpretation for these so remarkably simple law-like regularities’. In fact, his theory was not entirely model-independent, as he confessed to Sommerfeld: ‘I would never have reached the given representation of the spectral terms in strong fields if I had not been guided by model representations’.24 Sommerfeld and Pauli were at the time both moving from a model-based understanding of atomic structure to one (p.320) based on empirical rules, only was Pauli moving faster. He described his vector model—at the time sometimes referred to as an Ersatz (substitution) model—in letters to Landé and Sommerfeld, and he published a detailed version of it in a paper of early 1924. As he pointed out, for weak fields his theory led to the expression

$Display mathematics$

which he could not bring into agreement with Landé’s empirical formula. In his letter to Sommerfeld of 19 July he made the proposal that the classical differential expression might have to be replaced by a difference expression:

The structure of the expressions is rather similar but one cannot make the difference disappear by changing the normalization of the indefinite additive constants involved in R, K and J.…One may also say that the two expressions are related to one another like the differential quotient $d d J 1 J$ to the difference quotient $( 1 J − 1 J − 1 )$, which seems to indicate something non-mechanical.

Much of the discussion about atomic models and the anomalous Zeeman effect in the years 1922–1925 relied on the contributions of the young Werner Heisenberg, who in his very first scientific paper from the beginning of 1922 presented an atomic model to account for the regularities that Landé had established phenomenologically. However, it was a model that differed radically from the ordinary kind of model associated with the Bohr–Sommerfeld theory. Heisenberg developed a slightly earlier semiclassical model of Sommerfeld’s into a theory which accomodated all empirical data and reproduced Landé’s formula for doublets and triplets as well as the transition from weak to strong fields as given by the Paschen–Back effect.25 To do so, he made use of half-integral quantum numbers for k and m. He justified his results by inventing a core model in which, in the case of alkali atoms, the valence electron was assumed to share half a unit of angular momentum with the core, leaving the electron with (k − ½) h/2π and the core with ½ h/2π. The doublet terms would arise from the two orientations of the angular momenta of the core and the valence electron being either parallel or antiparallel:

$Display mathematics$

The electron’s angular momentum vector was assumed to precess around the momentum vector of the core and thereby create an inner magnetic field at the site of the core, which would orient itself in, or opposite to, the direction of the total field made up of the inner field and the external field.

The assumptions of Heisenberg’s model were arbitrary and tailored to give the results known empirically, but they did yield an explanation of the magneto-optical anomalies, at least of a sort, and for this reason the model was widely seen as a significant advance. However, the conceptual price to be paid for the instrumental (p.321) success was high, for the assumptions violated several of the most cherished theoretical principles of quantum theory. Among these were the Bohr–Sommerfeld quantum conditions, the selection rules, and the restriction of the total angular momentum to a set of discrete values. In Heisenberg’s system no such restriction could be justified. Moreover, Bohr’s construction principle was incompatible with the sharing of angular momentum between the valence electron and the core. For example, an argon atom had a closed shell or core with zero momentum, but when an electron was added, turning it into a sodium structure, the core would, according to Heisenberg, suddenly gain an angular momentum of ½ h/2π. This contradicted Bohr’s construction principle, which was based on the permanence of quantum numbers during the building up of atoms. It was no wonder that Bohr responded negatively to the new theory from Munich.

Heisenberg’s suggestive theory was not only problematical for methodological reasons, but also because of its empirical consequences. For example, in the case of alkali metals it led to values of the internal magnetic fields that were much larger than ordinarily assumed. McLennan also pointed out that according to Heisenberg electrons could be in a state characterized by an azimuthal quantum number k = ½, which seemed impossible for very heavy atoms. If the electronic orbit closest to a uranium nucleus was of this type, it would imply a shortest distance of approach equal to 4 × 10–12 cm, whereas the radius of the uranium nucleus was known to be 6.5 × 10–12 cm. ‘For reasons of this character’, argued McLennan in his address to the 1923 meeting of the British Association, ‘we are practically precluded from assigning to k, the azimuthal quantum number, a value less than 1 in defining the electronic orbits in atoms’.26 As mentioned in Section 7.2, Bohr arrived at the same conclusion, namely that an electron with quantum number k = ½ could only exist in atoms with Z 〈 68.

The various problems did not shake Heisenberg’s confidence in his core model, which was based solely on the fact that it worked. Having an opportunistic attitude, he believed that ‘success justifies the means’ (Der Erfolg heiligt die Mittel).27 Contrary to Bohr, who placed the highest value on consistency and clarity, Heisenberg was at the time willing to work with a model inconsistent with established physics if it resulted in answers that agreed with experiments. This might be an unphilosophical attitude, as Pauli later complained to Bohr,28 but from Heisenberg’s perspective this was an objection of no significance. While Bohr, Landé, and Pauli were critical, each in their own way, Sommerfeld had a greater appreciation for the work of his prodigy. To Einstein he wrote of Heisenberg’s forthcoming paper:

I have in the meantime uncovered wonderful numerical laws for line combinations in connection with the Paschen measurements and presented them in the third edition of my book. A pupil of mine (Heisenberg, 3rd semester!) has even explained these laws and those of the anomalous Zeeman effect with a model…Everything works out, but yet in the deepest sense remains unclear. I can only advance the craft of the quantum, you must make its philosophy…Set yourself to it!29

(p.322) But Einstein had no wish to engage fully in either quantum theory or quantum philosophy.

By the summer of 1923, Pauli had reached the conclusion that ‘there is no [satisfactory] model for the anomalous Zeeman effect and that we must create something fundamentally new’. He was at that time convinced that the problem of atoms with several electrons was of a physical rather than mathematical nature. In so far that one could still speak of individual electron orbits, he suggested that they ‘behave more as a system of oscillators in which the frequencies are associated not with the motion but with the transitions’.30

A few months later Heisenberg came up with something new, yet his new conception was still formulated within the framework of the core model, which Pauli distrusted. Heisenberg outlined his theory of a modification of the formal rules of quantum theory in a letter to Pauli of 9 October 1923, in which he emphasized that it was a model only in a symbolic and heuristic sense. He did not subscribe to atomic models in a physical sense, he said, but wanted to extract from them such information as could be used in constructing a new theory of a symbolic kind. This theory of the future—hopefully a near future—he envisaged as being consistently discrete. According to Bohr’s second quantum postulate of 1913, the frequency of a transition was obtained by taking the difference between the energies of two stationary states:

$Display mathematics$

Heisenberg pointed out, as Bohr had recognized from the very beginning, that this was a strange mixture of classical and quantum concepts. To remove the strangeness he suggested replacing the classical (and hence continuous) energies with quantum energies based on a difference equation of the form E = ΔF, where F = F(r, j, k, m) was a function of the quantum numbers. What he called ‘the new Göttingen theory of the anomalous Zeeman effect’ included the programmatic statement that ‘model representations have in principle only a formal sense, they are the classical analogues of the “discrete” quantum theory’.31 In late 1923 Heisenberg wrote to Sommerfeld about his new and still unpublished theory:

In the classical theory one would have ∂F/∂j = H, ∂H/∂J = ν. One now replaces the differential quotient ∂F/∂j with the corresponding difference quotient, and then everything comes out right, in particular the construction principle, the summation rule, Rydberg’s displacement law, etc. When one then contemplates what has really been done, one sees clearly that model conceptions have no real meaning. The orbits are not real, neither with respect to frequency nor energy. Obviously, only the coordinates J have a real meaning, and that only at discrete points.32

Relying on Pauli’s Ersatzmodell Heisenberg found it necessary to introduce a new quantum rule, namely, that ‘A determinate value of the coupling energy between the electron and the atomic core is not associated, as has been so far assumed, to one value of the inner quantum number j, but rather with two values’.33 In addition, he (p.323) formulated a ‘new quantum principle’ which amounted to a correspondence between the quantum-theoretical Hamiltonian of a system and the classical Hamiltonian. From Heisenberg’s symbolic core model followed Landé’s g factors and all other results of the complex spectra. To the list of negative aspects belonged the fact that ‘now one does not understand quantum theory at all’, as he phrased it in his letter to Pauli of 9 October. But this he did not necessarily see as a disadvantage:

This, however, I find rather congenial. Now the real goal must be to arrive from the symbols in a unique way at the discrete states; whether the formulae thus obtained make comprehensible sense, is doubtful to me. Born summarizes our task for the time to come in the word ‘discretization of atomic physics’. Now I wonder how much of it turns out to be correct when I continue to think about it; and I am anxious to see how you digest it.34

Heisenberg’s new theory of the anomalous Zeeman effect worked even better than his first one, and it had the methodological advantage that it was more consistent with established principles of quantum physics. In particular, it no longer violated the Aufbau principle, which helped make it acceptable in Copenhagen. In his published paper Heisenberg emphasized as a virtue how his theory provided a natural interpretation of Bohr’s principle, as he also explained privately to Bohr: ‘I can hardly imagine another interpretation of the difficulties described by you in the Kayser issue of the Annalen’.35 To his father, August Heisenberg, he wrote about Bohr’s influence: ‘I realize ever more that Bohr is the only person who, in the philosophical sense, understands something of physics’.36 While Bohr eventually approved of Heisenberg’s theory, Pauli remained intransigent. He considered it an ‘ugly’ theory, which did not even ‘yield an explanation of the half-integral quantum numbers and of the failure of Larmor’s theorem’.37 Pauli came to the conclusion that the core model was fundamentally wrong, even in the symbolic representation offered by Heisenberg. With his theory of the exclusion principle he thought he had buried the model.

In a letter to Bohr of February 1924, Pauli revealed how he looked upon the contemporary situation in the quantum theory of atoms. There were, he said, two groups of atomic physicists in Germany: one group that had began with integral quantum numbers and another that instead started with half-integral quantum numbers:

Both groups, however, have the characteristics in common that there is no a priori argument to be had from their theories that tells which quantum numbers and which atoms should be calculated with half-integral values of the quantum numbers and which should be calculated with integral values. They can decide this only a posteriori by comparison with experience. I myself have no taste for this kind of theoretical physics… I am far more radical than the ‘half-integral-number’ atomic physicists.38

Referring to the problem of interpreting the failure of Larmor’s theorem, he added: ‘Unfortunately, with respect to this main point Heisenberg’s considerations do not lead us beyond what we already know’. As to the anomalous Zeeman effect, Heisenberg had (p.324) achieved a formal description of it, but not explained it on a basis consistent with quantum theory. Pauli had no better alternative to offer, and thus his critique remained on the negative level.

There was however one important point in Heisenberg’s work with which the critical Pauli fully agreed, and this he spelled out in his letter to Bohr of 21 February: ‘According to my point of view, Heisenberg hits the truth precisely when he doubts that it is possible to speak of determinate trajectories. Doubts of this kind Kramers has never considered as reasonable. I must nevertheless insist upon this, because the point appears to me to be very important’. Pauli expressed the same kind of doubt in a letter to Eddington, in which he emphasized that ‘the quantum theory in no way calls for only a modification of the theory of light, but generally calls for a new definition of the concept of the electromagnetic field for non-static processes’.39 He believed that some of the contradictions in physics, such as the one between the wave phenomena of light and Einstein’s light quanta, were rooted in ‘the fact that we give up the laws of classical theory, but still continue to work with the concepts of that theory’. Among the concepts that had no operational meaning and therefore had to be abandoned he counted the concept of electrons moving in fixed orbits.

Heisenberg’s references to the new Göttingen theory and Born’s discretization programme in atomic physics reflected ideas that were discussed by the Göttingen physicists in the years 1923–1924, when Born gave an autumn term course on ‘Perturbation Theory Applied to Atom Mechanics’. Although these ideas were never developed into a proper theory, they may nevertheless have played a role in the development of quantum theory. In relation to his own and Heisenberg’s failed attempt to calculate the helium atom, Born remarked in his Atommechanik that calculations between two or more electrons were based on classical differential equations; on the other hand, in the interaction between atoms and radiation a difference equation, namely Bohr’s frequency condition hν = ΔE, replaced the differential equations. With respect to this asymmetry, he declared that ‘The systematical transformation of the classical mechanics into a discontinuous atomic mechanics is the goal towards which quantum theory strives’.40

Born’s interest in a discretization of physics was not new. In 1919 young Pauli had offered a critical analysis of Hermann Weyl’s new unified theory of gravitation and electromagnetism in which he objected that it was physically meaningless to speak of the field strength in the interior of an electron because the field would be unobservable even in principle.41 For this reason he suggested that the field concept needed to be modified. Inspired by Pauli’s comments, and probably also by his own work on the theory of crystal lattices, Born related them to the problems of quantum theory. In a letter to Pauli of late 1919, he wrote:

The way out of all quantum difficulties must be sought by starting from entirely fundamental points of view. One is not allowed to carry over the concept of space-time as a four-dimensional continuum from the macroscopic world of experience into the atomistic (p.325) world; the latter evidently demands another type of number-manifold to give an adequate picture. However, I have no idea of how to do that.42

The Göttingen discretization programme reflected Born’s belief that the quantum riddles could probably be solved by introducing new mathematical techniques, such as difference equations. Bohr, on the other hand, was convinced that a new mathematical formulation had to come after a physical understanding of the problems, not before it. According to Heisenberg’s recollection: ‘Bohr would always say, “Well, first we have to understand how physics works. Only when we have completely understood what it is all about can we then hope to represent it by mathematical schemes”. But Born would argue the other way, and would say, “Well, perhaps some new mathematical tool is a decisive help in understanding physics”’.43 Heisenberg was as much influenced by Born as by Bohr, and when it came to the question of the role of mathematics he tended to side with the professor in Göttingen.

While a fundamental discretization was part of the Göttingen research programme, other physicists argued that quantum phenomena did not necessitate a change from differential to difference equations. On the contrary, Einstein thought that quantum discontinuities might well be generated as solutions of continuous differential equations. Convinced that quantum theory lacked a proper logical foundation, he believed that the quantum conditions should be deduced from a set of classical differential equations, as he told Born in 1920:

I myself do not believe that the solution to the quanta has to be found by giving up the continuum…Pauli’s objection is directed not only against Weyl’s theory, but also against anyone else’s continuum theory…I believe now, as before, that one has to look for redundancy in determination by using differential equations so that the solutions themselves no longer have the character of a continuum. But how?44

At any rate, the hypothesis of a quantum physics based on a discrete rather than continuous picture of space-time, or of replacing all differential equations with difference equations, did not lead to useful results. Yet it was not forgotten and was later developed in various directions. For example, in 1930 Heisenberg proposed in a letter to Bohr that if space were discrete, structured like a three-dimensional lattice with a smallest length of h/Mc ≅ 10–15 m, where M is the mass of the proton, some of the problems of quantum mechanics might disappear. He developed a theory on this basis, but never published it.45

# 8.3 Light quanta and virtual oscillators

At the time when Heisenberg developed his second core theory of the anomalous Zeeman effect, the focus of leading quantum physicists had to some extent shifted from atomic structure to radiation theory. The unresolved question of the nature of (p.326) light and its interaction with matter was not new, of course, and, as indicated by Pauli’s letter of 1923 to Eddington, it continued to occupy the minds of the physicists. As mentioned in Section 5.2, Bohr was convinced that Einstein’s light quanta caused insuperable difficulties and could not possibly be reconciled with the wave nature of light so solidly demonstrated by interference and diffraction phenomena. However, it was only at the end of 1923 that the problem began to play a major role in Bohr’s thoughts about progress in atomic theory. In the letter to Bohr of December 1923, in which Heisenberg outlined his second core model and its application to the anomalous Zeeman effect, he suggested that his new quantum principle was connected with the interaction between atoms and radiation fields, a suggestion which undoubtedly captured Bohr’s attention with regard to the proposed theory.46 At that time Heisenberg was not yet aware that Bohr, in collaboration with Kramers and Slater, had recently arrived at some new and quite radical ideas concerning radiation and atoms.

In his Nobel lecture in Stockholm on 11 December 1922, Bohr briefly discussed Einstein’s explanation of the photoelectric effect in terms of light quanta. He admitted it to be of ‘heuristic value’, but nonetheless dismissed it because it ‘is quite irreconcilable with so-called interference phenomena [and] not able to throw light on the nature of radiation’.47 He was at the time unaware that Arthur Compton ten days earlier had presented a paper to the American Physical Society that gave strong support to Einstein’s light quanta. Compton, at the time a professor at Washington University, St. Louis, showed in experiments from 1922 that when monochromatic X-rays of wavelength λ—he used the Kα line of molybdenum—were scattered by graphite, the scattered radiation included a component of longer wavelength λ’. The shift in wavelength Δλ was found to depend on the scattering angle θ, but to be independent of the scattering material. Energy conservation (for v 〈〈c) yields hc/λ - hc/λ’ = ½mv 2, where m is the mass of the electron, and momentum conservation results in two equations involving θ. As Compton showed, a simple calculation gives for the ‘Compton effect’ the formula

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The constant quantity λC = h/mc = 0.0243 Å soon became known as the Compton wavelength. It is about 20 times smaller than the Bohr radius, the relation between the two lengths being λC/a 0 = 2πα. In a classic paper of May 1923 Compton argued that his result was incomprehensible on the basis of the classical theory of radiation but could be fully explained on the assumption of radiation quanta carrying energy and momentum given by

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(p.327)

‘This remarkable agreement between our formulas and the experiments can leave but little doubt that the scattering of X rays is a quantum phenomenon’, he concluded.48 Formulae identical to Compton’s were independently derived by Debye in Zurich, for which reason the effect was sometimes referred to as either the Compton–Debye effect or the Debye–Compton effect, but Compton’s work was earlier and the only one which included precise experiments as support of the radiation quantum hypothesis.49 Compton’s conclusions were questioned on experimental grounds by William Duane and other American physicists, and it took nearly a year until they were corroborated and generally accepted as proof of the corpuscular nature of X-rays. By the end of 1923 consensus had not yet been achieved, although a majority of physicists already agreed that the Compton–Debye interpretation was probably correct and that light quanta were real.50

Bohr did not belong to the majority, and neither did his close collaborator Kramers. According to the Dutch physicist, the explanatory power of Einstein’s hypothesis did not compensate for the problems it created. ‘The theory of light quanta may thus be compared with medicine which will cause the disease to vanish but kill the patient’, he opined.51 Bohr knew about Comptons’s work, but saw no reason to accept his conclusions with regard to the light quantum. In the late autumn of 1923 he visited the United States and Canada, giving lectures at the University of Toronto and several American universities. At Yale he was invited to give the Silliman lectures, a series of six lectures between 6 and 15 November, which attracted considerable attention.52 During his stay in the United States, he had conversations with Duane, Michelson, and other physicists concerning the Compton effect and related matters. On one occasion they discussed the possibility of testing the nature of individually scattered X-ray quanta and thereby decide whether or not energy and momentum were absolutely conserved. The possibility of such a test, which was later realized in the Compton–Simon experiment mentioned below, was suggested by the British-American physicist William Swann in a conversation he had with Bohr and Compton in November 1923.53 Swann was at the time entertaining the idea that the role of the electromagnetic field in radiation processes might be to guide light quanta.

During their discussions in the United States, Duane confirmed Bohr in his belief that the Compton effect could be explained without abandoning the wave theory. ‘I have no doubt’, he wrote to Bohr in March 1924, ‘that the shift in the wave-length that A. H. Compton has been writing about should be ascribed to this tertiary radiation [produced by photoelectrons]’. In a letter of reply Bohr informed Duane about ‘the new view as regards the connection between radiation and transition processes described in a recent paper in Philosophical Magazine by Dr Slater, Dr Kramers and myself’.54 This paper, often known as the BKS paper, had its origin in an idea of John Slater that was, however, transformed into a theory which included as a crucial element the radical hypothesis that energy and momentum are not conserved in individual atomic processes. Speculations of energy nonconservation predated Slater’s idea and coloured the way Bohr understood it.

(p.328) These speculations were in part inspired by ideas due to the Cambridge physicist Charles Darwin, Bohr’s old colleague from his time in Manchester, who in July 1919 prepared a manuscript on ‘Critique of the Foundations in Physics’. In this unpublished work, which he enclosed in a letter to Bohr, he expressed in a general way his frustration about the conceptual basis of physics. ‘It may be’, he said, ‘that it will prove necessary to make fundamental changes in our ideas of space and time or to abandon the conservation of matter and electricity or even as a last forlorn hope to endow electrons with free will’. The conflict between quantum theory and the classical wave theory rested on the assumption of strict energy conservation, and ‘I therefore claim that the possibilities to be deduced from denying the exact conservation should be thoroughly exhausted before further modifications are made’.55 Finding Darwin’s considerations greatly interesting, Bohr offered his own considerations in a letter, which was not, however, actually sent:

As regards the wave theory of light I feel inclined to take the often proposed view that…all difficulties are concentrated on the interaction between the electromagnetic forces and matter. Here I feel on the other hand inclined to take the most radical or rather mystical views imaginable. On the quantum theory conservation of energy seems to be quite out of question and the frequency of the incident light would just seem to be the key to the lock which controls the starting of the interatomic process.56

In 1922–1923 Darwin developed his vague thoughts into a dispersion theory in which energy conservation was only satisfied statistically, in the same sense as in Boltzmann’s probabilistic theory of the second law of thermodynamics. According to Darwin’s theory, when an atom was hit by a wave it would acquire a certain probability of emitting a secondary spherical wave that interfered with the incident one to produce the dispersed wave. He suggested that the emission of a spherical wave with a particular energy would be triggered by some favourable configuration of the electron orbits. As to the photoelectric effect, he found it to be ‘an impossibility in conjunction with the wave theory if energy is exactly conserved, but if only a statistical balance is required, then it becomes nothing more than one unexplained problem among others’.57

The idea of statistical energy conservation was at the time in the air and not a novelty to Bohr, who had contemplated it some years earlier (Section 5.2). Nor was it a novelty to Sommerfeld, who in the third edition of Atombau suggested that the ‘mildest cure’ for reconciling wave theory and quantum theory might be to abandon energy conservation for individual radiation processes.58 Having studied Darwin’s dispersion theory, Bohr pointed out some serious problems with it, among which he did not count its questioning of energy conservation (which Bohr considered to be ‘of a most formal nature’) and of course also not its dismissal of light quanta. What he did not like was that the theory challenged the real existence of excited stationary states, which according to Darwin might be just an epiphenomenon caused by changing electric fields.59 Bohr insisted that the stationary states were real and beyond discussion. More concretely he pointed out that it followed from Darwin’s theory that for low-intensity incident light (p.329) the probability of interference should be small, which contradicted experiments showing that even very weak light produced the same interference pattern as strong light.60

The 23-year-old Harvard physicist John Slater conceived his idea of a new radiation theory reconciling light quanta with classical wave theory while staying with Ralph Fowler in Cambridge, England, in the autumn of 1923. From Cambridge he moved on to Copenhagen, where he worked at Bohr’s institute from December 1923 until the following June. At the time he was ‘profoundly dissatisfied with Bohr’s hypothesis of instantaneous jumps from one stationary state to another’, not so much because of the jumps, but because they supposedly occurred instantaneously.61 Not having problems with light quanta he imagined that they were guided in their paths by classical wave fields generated by vibrating ‘corresponding charges’ (not orbiting electrons) in the stationary states of the atoms. Emission of radiation was not restricted to abrupt transitions from one stationary state to another, as in Bohr’s original model, but occurred continuously and with all possible transition frequencies while the atom was in a stationary state—which according to Bohr’s theory was inactive.

However, there was no real contradiction between Slater’s mechanism and the Bohr theory, for Slater’s guiding field was not an ordinary electromagnetic field but a kind of ghost field that carried neither energy nor momentum. Still, in a purely formal sense it could be described as the electromagnetic field containing all the possible Bohr transition frequencies. These frequencies were imagined to be those of a set of virtual harmonic oscillators, not the frequencies belonging to the actual motion of electrons. Thus while an atom in a stationary state emits a virtual field, it is still stationary in Bohr’s sense since it does not emit a real electromagnetic wave. When, according to Slater, a real light quantum is emitted, the atom changes discontinuously to a new stationary state and begins emitting virtual waves with frequencies corresponding to this state. In the case of a collection of many atoms, each one contributes with its own virtual field to the fields produced by all the others. The atoms ‘communicate’ with one another. Bohr convinced himself that Slater’s picture was ‘far more harmonious from the point of view of the correspondence principle’ than the traditional one, where spontaneous radiation was connected only with transitions between stationary states.62

In early December 1923 Slater described his hypothesis in a letter to Kramers: ‘It seems possible to suppose that there is an electromagnetic field, produced not by the actual motion of the electrons, but with motions with the frequency of possible emission lines…and amplitudes determined by the correspondence principle, the function of this field being to determine the motion of the quanta’.63 The amplitudes of the wave field determined the probability of a quantum being emitted with a particular frequency, but Slater did not think of the probabilistic feature as very important. ‘It should be noted’, he wrote in one of his manuscripts, ‘that the only place where chance and discontinuity comes into the theory is in emission; once a quantum is emitted, the rest of the process is prescribed exactly as in the classical (p.330) theory’. He hoped that ‘when the dynamics of the inside of atoms are better known, chance may be eliminated there also’.64

Slater’s idea of light quanta guided by a system of virtual ghost-like waves had some similarity to Louis de Broglie’s contemporary ideas associated with his new theory of matter waves. It has been suggested that Slater was inspired by de Broglie’s theory, parts of which were probably known to him, but according to Slater this was not the case. ‘De Broglie’s work was not known in Copenhagen during the time, the first half of 1924, when I was there’, he recalled.65 Although de Broglie’s theory of matter waves was probably unknown to Bohr and his associates at the time, his work on atomic structure and X-ray spectroscopy were well known (if not appreciated, see Section 7.4). In the late development of the Bohr–Sommerfeld theory, de Broglie’s theory of matter waves played almost no role at all. Bohr did refer to the famous doctoral thesis of the French physicist, but only in July 1925.

When Slater arrived in Copenhagen, his ideas were immediately taken up, discussed and substantially revised—almost beyond recognition—by Bohr and Kramers. The note he submitted to Nature in late February 1924 was in important ways influenced by the views of the two Copenhagen physicists, one result being that the light quanta no longer appeared in a significant way. In fact, Slater mentioned that Kramers’ objections to his original idea of a radiation field guiding discrete quanta had forced him to abandon the idea. At the time he did not seem to have considered it a defeat or great loss. As he explained in a note from the summer of 1925, shortly after the Bohr–Kramers–Slater theory had been refuted by experiments, ‘I became persuaded that the simplicity of the mechanism obtained by rejecting a corpuscular theory more than made up for the loss involved in discarding conservation of energy and rational causation’.66 On the other hand, he later recalled differently: ‘To my consternation I found that they [Bohr and Kramers] completely refused to admit the real existence of the photons…This conflict, in which I acquiesced to their point of view but by no means was convinced by any arguments they tried to bring up, led to a great coolness between me and Bohr, which was never completely removed’.67

The mechanism that Slater described in early 1924 was in terms of ‘virtual fields’ and ‘virtual oscillators’, terms that were coined by Bohr and Kramers, but, as far as the concept of virtual oscillators are concerned, can be found in an earlier paper by Ladenburg and Reiche. According to Slater:

Any atom may, in fact, be supposed to communicate with other atoms all the time it is in a stationary state, by means of a virtual field of radiation, originating from the oscillators having the frequencies of possible quantum transitions, and the function of which is to provide for statistical conservation conservation of energy and momentum by determining the probabilities of quantum transitions…The part of the field originating from the given atom itself is supposed to induce a probability that that atom loses energy spontaneously.68

Slater’s ideas were quickly transformed into the more elaborate and significantly different Bohr–Kramers–Slater (BKS) theory, which was, however, the product of (p.331) Bohr and Kramers rather than Slater.69 In spite of its break with the radiation mechanism of Bohr’s atomic model, Bohr found Slater’s idea of emission of (virtual) radiation during a stationary state appealing. He had recently introduced a ‘coupling principle’, which included the notion of ‘latent’ forces that controlled transition probabilities in atoms. Slater’s virtual fields were somewhat similar to Bohr’s latent forces, a similarity Bohr alluded to in a letter of early 1925, when the BKS theory was still alive and well:

It was just the completion which your suggestion of radiative activity of higher quantum states apparently lent to the general views on the quantum theory with which I had been struggling for years which made me welcome your suggestion so heartily. Especially I felt it was far more harmonious from the point of view of the correspondence principle to connect the spontaneous radiation with the stationary states themselves and not with the transitions.70

The intimate connection between the BKS theory and the correspondence principle also appears in a letter Bohr wrote to Michelson in February 1924. Albert Michelson, America’s first Nobel laureate in physics, had established his entire career on the wave theory of light, and Bohr assured him that ‘it appears to be possible for a believer in the essential reality of the quantum theory to take a view which may harmonize with the essential reality of the wave theory conception’. The reason for Bohr’s assertion that quanta and waves could be harmonized was this:

In fact on the basis of the correspondence principle it seems possible to connect the discontinuous processes occurring in atoms with the continuous character of the radiation field in a somewhat more adequate way than hitherto perceived…I hope soon to send you a paper about these problems written in cooperation with Drs. Kramers and Slater.71

He said much the same in the published paper, in which he expressed his belief that the new theory would allow them ‘to arrive at a consistent description of optical phenomena by connecting the discontinuous effects occurring in atoms with the continuous radiation field in a somewhat different manner from what is usually done’.72

The BKS paper on ‘The Quantum Theory of Radiation’ appeared in the May issue of Philosophical Magazine and at the same time a German translation was published in Zeitschrift für Physik. It was a paper very much in Bohr’s style—verbose, discursive, repetitive, and qualitative—and was presented as a further development of Bohr’s research programme based on the correspondence principle. ‘The present paper’, it said in the introduction, ‘may in various respects be considered as a supplement to the first part of a recent treatise by Bohr, dealing with the principles of quantum theory’.73 The lengthy paper included only a single equation, which was Bohr’s frequency condition hν = E 1E 2. Among the issues discussed in the BKS paper was the time of duration of the stationary states and the corresponding sharpness of the states, an issue which Bohr had previously dealt with in his memoir on the application of quantum theory to atomic structure. Bohr now argued that there is a ‘limit of definition of the motion and of the energy in the stationary states’ which manifests itself in a finite line (p.332) width. The postulate of the stationary states, he said, ‘imposes an a priori limit to the accuracy with which the motion in these states can be described by means of classical electrodynamics’.74 In other words, he suggested an inverse relationship between the uncertainty in energy ΔE and the lifetime Δt, a relationship of roughly the same form as the one that later appeared in Heisenberg’s famous energy–time uncertainty relation based on quantum mechanics:

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The new Copenhagen theory was basically an attempt to reconcile the continuous electromagnetic field with the physical picture of the discontinuous quantum transitions in atoms (but not, as in Slater’s original idea, with the light quanta). To harmonize continuity with discontinuity, or at least to correlate the two concepts, strict causality had to be abandoned, and from this it followed that energy and momentum conservation changed from being absolutely valid to having only statistical validity. As mentioned above, Bohr had discussed this possibility in a conversation with Compton and Swann in November 1923, before he knew about Slater’s ideas.

Following Slater, the basic physical assumption of the BKS theory was to associate each atom with an unspecified number of virtual oscillators that produced a virtual radiation field through which the atoms entered in a mutual communication. It remained unclear what the enigmatic virtual oscillators were, except that they were abstract and unphysical quantities that could not be directly observed. While the virtual field was not endowed with measurable properties, it transmitted probabilities for transitions in other atoms. In this way, the field established a communication between distant atoms, but one which merely changed the probability that a transition would occur. Bohr and Kramers (and, nominally, Slater) expressed the basis of their theory as follows:

We will assume that a given atom in a certain stationary state will communicate continually with other atoms through a time-spatial mechanism which is virtually equivalent with the field of radiation which on the classical theory would originate from the virtual harmonic oscillators corresponding with the various possible transitions to other stationary states…As regards the occurrence of transitions, which is the essential feature of the quantum theory, we abandon on the other hand any attempt at a causal connexion between the transitions in distant atoms, and especially a direct application of the principles of conversation of energy and momentum so characteristic for the classical theories.75

According to the BKS view, the experimentally established validity of the conservation laws was nothing but the result of a statistical average over a great number of individual events. For the individual events the laws would not be exactly satisfied. Considering the Compton effect, Bohr and his coauthors argued that Compton’s light-quantum interpretation was not the only one and that the effect could easily be explained on the basis of the BKS theory: ‘In contrast to this [Compton’s] picture, the scattering of the (p.333) radiation by the electrons is, in our view, considered as a continuous phenomenon to which each of the illuminated electrons contributes through the emission of coherent secondary wavelets’.76 In the BKS explanation of the Compton effect, the shift in wavelength was attributed to a Doppler shift. However, this explanation indicated what the three authors called ‘a feature strikingly unfamiliar to the classical conceptions’, namely an incompatibility between the kinematical and the wave-theoretical description. Kramers and Heisenberg later called attention to ‘the curious fact that the centre of these spherical waves moves relative to the excited atom’.77

While hoping that the new Copenhagen theory would turn out to be correct, Kramers and Bohr tended to emphasize its unfinished nature. According to Kramers, writing at a time when experiment had not yet settled the matter: ‘The new conceptions of the postulates, which require the independence of the atom processes, is in no way a completed theory; it is only an attempt to throw a little light in the great darkness of our ignorance about the course of the atomic processes, and it should, for the time being, be conceived as essentially a working-programme for the theorists’. This is how he wrote in a chapter added to the German version of his and Holst’s book on the Bohr atom, in which he gave a pedagogical and fairly detailed account of the BKS theory, or what he consistently called ‘Bohr’s new view’. As to the theory’s element of acausality, he preferred to consider it ‘rather a matter of taste’, although there is little doubt that his own taste (and Bohr’s as well) was acausal. To Kramers, the principle of causality was a fact of experience rather than a logical necessity, and ‘one could easily imagine that it breaks down for atomic processes’. Similarly, he thought that one should keep an open mind with respect to a violation of the law of energy conservation. Interestingly, Kramers suggested that large-scale energy nonconservation processes might occur in the depths of space. There are indications, he said, that ‘in the hot stars,…the principle of energy conservation cannot be used just like that, but that in these bodies there occurs, so to speak, a spontaneous creation of energy which contributes to maintain the enormous radiation of energy of the stars into space’.78

The BKS theory aroused immediate and great attention, if little enthusiasm. The chemist Fritz Haber, who visited Bohr in 1924, was exposed to a one-and-a-half-hour lecture on the BKS radiation theory of which he understood little. And what he thought to have understood was a misunderstanding based on Bohr’s rejection of the light quantum. Bohr, he reported to Einstein, ‘strives with all fibres back to the classical world’.79 As to any enthusiasm outside Copenhagen, it was largely limited to Born and Schrödinger. In the United States, Swann also found the theory attractive. In a letter to Bohr of May 1924 Schrödinger declared himself ‘extremely sympathetic’ to the main parts of the theory and in particular to its renunciation of strict causality and energy conservation, which he, as a student of the Viennese physicist Franz Exner, had long been fond of. ‘Your new point of view means a far-reaching return to classical theory, as far as radiation is concerned’, he wrote, wondering about the difference between ‘virtual’ radiation and the ‘real’ radiation caused by transitions between stationary orbits.80

(p.334) Schrödinger misunderstood Bohr’s intentions, which he interpreted in accordance with his own preferences. Contrary to what Haber and Schrödinger thought, Bohr had no wish to ‘return to classical theory’. In a paper of September 1924 published in Die Naturwissenschaften, Schrödinger publicly endorsed the BKS theory, which he interpreted in accordance with his own holistic approach to physics: ‘A certain stability of the world order sub specie aeternitatis can only exist through the interrelationship of each individual system with the rest of the entire world’.81 Moreover, Schrödinger spoke of the ‘Exner–Bohr conception of the energy principle’, which he related to speculations of the statistical nature of thermodynamics in a manner foreign to Bohr’s thinking.

Most other physicists, including Sommerfeld, Heisenberg, Compton, Ehrenfest, Franck, and Pauli, were either sceptical or directly opposed to the BKS theory.82 In response to a letter from Bohr, including a draft version of the BKS theory, Pauli declared that he felt mystified: ‘I laughed a little…about your warm recommendations of the words “communicate” and “virtual”…On the basis of my knowledge of these two words…I have tried to guess what your paper is all about. But I have not succeeded’.83 Of particular interest is the response of Einstein, who had earlier expressed his worries over causality in relation to radiation theory. ‘Can the quantum absorption and emission of light ever be understood in the sense of the complete causality requirement, or would a statistical residue remain?’ he asked Born in 1920. Four years later, after having learned about the BKS theory, he spoke out forcefully against it:

Bohr’s opinion about radiation is of great interest. But I should not want to be forced into abandoning strict causality without defending it more strongly than I have so far. I find the idea quite intolerable that an electron exposed to radiation should choose of its own free will, not only its moment to jump off, but also its direction. In that case, I would rather be a cobbler, or even an employee in a gaming-house, than a physicist.84

As we learn from a letter to Ehrenfest, Einstein gave a colloquium on the BKS theory in May 1924, in which he gave his reasons for rejecting the theory. ‘This idea is an old acquaintance of mine, whom, however, I do not regard as a respectable fellow [einen reellen Kerl’]’, he told Ehrenfest.85 Born was more inclined to consider the BKS theory a respectable fellow. Seriously interested in the theory, he thought that the valuable part of it was the idea of emission of radiation while an atom stays in a stationary state. In a reply to a letter from Born on the subject, Bohr stressed that ‘the assumption of a coupling between the changes of state in distant atoms by means of [virtual] radiation precludes the possibility of a simple description of the occurrences in terms of visualizable pictures’.86

Bohr and Kramers realized that their theory, if it were to be seriously considered in the physics community, had to make testable predictions. Applying it to the Compton effect they concluded that the direction of a recoil electron after scattering by a monochromatic X-ray would not be uniquely determined, as required by the conservation laws, but display a wide statistical distribution. As early as June 1924, Walther (p.335) Bothe and Hans Geiger in Berlin proposed an experiment to test the theory by simultaneously measuring the scattered X-rays and the recoil electrons. This was one of the first experiments using electronic coincidence devices, and it was not until April 1925 that they had their final result ready; it was ‘incompatible with Bohr’s interpretation of the Compton effect’.87 Using a cloud chamber to determine the direction of electrons recoiling from X-ray scattering, Compton and Alfred Simon reached the same conclusion a little later (Figure 8.1). In what they described as ‘a crucial test’ they provided ‘a direct and conclusive proof that…energy and momentum are conserved during the interaction between radiation and individual electrons’.88

Einstein was pleased, if by no means surprised. In a short manuscript written during a visit to Rio de Janeiro, he referred to the still unpublished experiment of Bothe and Geiger. ‘At the time of my departure from Europe’, he ended the manuscript, ‘the experiment was not yet completed. However, according to the results attained so far, such statistical dependence appears to exist. If this is confirmed, then there is a new important argument for the reality of light quanta’.89

Bohr had for a year defended the BKS theory and taken it very seriously; as seriously as his coauthor Kramers and much more so than his other coauthor Slater. The

Fig. 8.1. The Compton–Simon experiment of 1925 was designed to test the effect of X-ray quanta scattered by electrons. On the radiation quantum hypothesis, if a recoil electron is ejected an angle θ, the scattered X-ray quantum proceeds in a definite direction ϕ. On the BKS theory, the scattered X-rays are spherical waves and there should be no correlation between the directions of recoil electrons and the directions in which the effects of scattered X-rays are observed. Compton and Simon used a cloud chamber in which each recoil electron produced a visible track. From a series of stereoscopic photographs they concluded that there was a strong correlation, hence that the BKS theory was wrong.

Source: Compton and Simon 1925, p. 291.

(p.336) experimental verdict was a disappointment, yet it was inescapable and generally accepted as a crucial refutation of the theory.90 In a mood of resignation he wrote to Fowler that ‘there is nothing else to do than to give our revolutionary efforts as honourable a funeral as possible’.91 While Bohr accepted that the BKS theory was wrong, he did not yet accept the light quantum or (as it would be coined the following year) the photon.92 Having received the news from Geiger, he maintained that, ‘in spite of the existence of a coupling [between quantum processes], conclusions about a possible corpuscular nature of radiation lack a sufficient basis’.93 Kramers was equally convinced that the refutation of the BKS theory did not provide a final answer to the problem of the nature of light. In a letter to the American physicist Harold Urey, who had stayed in Copenhagen 1923–1924, he wrote: ‘Still the wave description of optical phenomena must be preserved in some way, and we think that Slater’s original hypothesis contains a good deal of truth’.94

As to Slater, his response to the failure of the BKS theory was different from those of Bohr and Kramers: he merely returned to his original view of a virtual field guiding the radiation quanta.95 After having returned to Harvard, but before the final Geiger–Bothe experiment, Slater developed his ideas into a theory of quantum optics that appeared in the spring of 1925. The theory made use of some of the key ideas of the BKS paper, including that energy and momentum were not precisely conserved for individual atomic processes. It followed from Slater’s theory (as it did from the BKS theory) that, as a result of the finite lifetime of the stationary states, spectral lines must have a minimum width, an inference that he later saw as an anticipation of the energy-time uncertainty relation.96 By adopting the BKS idea of frequencies emitted for all possible transitions, and not only for the ones turning up spectroscopically, he thought he had solved the old conundrum of the ‘free will’ of electrons. For then ‘the atom is under no necessity of knowing what transitions it is going to make ahead of time’.97

Bohr publicly abandoned the BKS theory in an addendum to a paper of July 1925 in which he had adopted a BKS-inspired approach to investigate collision and capture processes. While arguing in the paper that some kinds of atomic collision violated strict conservation of energy and momentum, in the addendum he admitted this to be a mistake. ‘If this way out is now closed’, he suggested, ‘we are probably forced to recognize in the capture phenomenon a new feature of the supramechanical stability of the stationary states that cannot be described in space-time pictures’.98

According to Pauli, the theory of Bohr, Kramers, and Slater was a ‘reactionary Kopenhagener Putsch’, and he considered it ‘a magnificent stroke of luck’ that the revolt had been crushed so quickly by the experiments in Berlin and St. Louis. As he stressed in a letter to Kramers, it was the kinematic concepts that needed modification, not the concept of energy:

It is true that, in cases where interference phenomena are present, we cannot define definite ‘trajectories’ for the electrons in an atom; and just as little as it would be justified to doubt the existence of electrons for this reason, would it be justified to doubt the existence of light (p.337) quanta because of interference phenomena. For probably every unprejudiced physicist it can now be regarded as proved that light quanta are just as much (and just as little) physically real as electrons. But to neither of them should the classical kinematic concepts in general be applied.99

What Pauli called a reactionary Copenhagen revolt was a failed revolution, but the failure was not without fruitful consequences. In spite of its short lifetime, the BKS theory was important in paving the way for a greater understanding that methods and concepts of classical physics could not be carried over to a future quantum mechanics. In particular, the theory appeared prominently in Kramers’ theory of dispersion of 1924 and its further development into the Kramers–Heisenberg dispersion theory of 1925, the final step before Heisenberg’s formulation of quantum or matrix mechanics. In 1929, looking back on the development of quantum theory, Heisenberg singled out the BKS theory as the high point in the crisis that led to the new quantum mechanics. In his view, the theory ‘contributed more than any other work at that time to a clarification of the situation in quantum theory’.100 However, the historical significance of the BKS theory should be understood in the proper and restrictive sense that one element in it—the idea of virtual oscillators—played a fruitful role.101 The important dispersion theory of Kramers was formulated in the language of the BKS theory, but it did not actually rely on it (see the next section). Likewise, it was the virtual oscillators, and not the BKS theory as a whole, that played an essential role in the reasoning that led Heisenberg to quantum mechanics.

# 8.4 Dispersion theories

The dispersion of light in gases and other refractive media was successfully explained by classical electron theory by assuming the atom to contain a number of elastically bound electrons with characteristic frequencies ν0.102 When hit by an electromagnetic wave of frequency ν the electrons would vibrate, the collective result being a characteristic dependence of the refraction index n (or dielectric constant ε, related to the refraction index as n$ε$) on the frequencies ν and ν0. In terms of the polarizability α, which for weak fields is the ratio P/E between the dipole moment and the electric field, the classical dispersion formula could be written in the form

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The term f i was interpreted as either the ‘strength’ of the oscillating charge or the number of dispersion electrons in an atom. With the advent of Bohr’s atomic model, the picture of elastically bound electrons lost its validity and had to be replaced by one with electrons moving on stationary orbits around a nucleus. But how, then, could (p.338) dispersion be explained? This was an old question, going back to the very beginning of Bohr’s theory. At the meeting of the British Association in September 1913 a sceptical Lorentz had asked the young Dane how his model of the atom could possibly account for the well-established classical dispersion formula (Section 3.1). Bohr could offer no answer.

The first theories of dispersion relating to Bohr’s model, as proposed by Debye and Sommerfeld in 1915 and by Davisson in 1916, were based on particular atomic and molecular models and assumed that the perturbations of the electrons in their stationary orbits could be calculated by classical electrodynamics. Understandably, Bohr found theories of this kind to be untenable because they conflicted with his fundamental distinction between orbital frequencies and the radiation frequencies caused by quantum jumps (see Section 3.9). In his withdrawn 1916 paper to Philosophical Magazine, he severely criticized the Debye–Sommerfeld theory, pointing out that it involved ‘considerable formal difficulties due to the contrast between the assumption of stationary states and ordinary electrodynamics’. We must assume, he said, that

[T]he dispersion in sodium and potassium vapour depends essentially on the same mechanism as the transition between different stationary states, and that it cannot be calculated by application of ordinary electrodynamics from the configuration and motions of the electrons in these states…If the above view is correct and the dispersion depends on the mechanism of transition between different stationary states, we must, on the other hand, assume that this mechanism shows a close analogy to an ordinary electrodynamic vibrator.103

The ‘close analogy’ would soon become known as the correspondence principle.

(p.339) In his 1923 paper on the fundamental postulates of quantum theory, Bohr stated that ‘the phenomena of dispersion must thus be so conceived that the reaction of the atom on being subjected to radiation is closely connected with the unknown mechanism which is answerable for the emission of the radiation on the transition between stationary states’. He praised the work of Ladenburg, who ‘has tried in a very interesting and promising manner, to set up a direct connection between the quantities which are important for a quantitative description of the phenomena of dispersion according to the classical theory and the coefficients of probability appearing in the deduction of the law of temperature radiation by Einstein’.106 Together with Fritz Reiche, his colleague at the University of Breslau, Ladenburg offered a more detailed version of his dispersion theory in the 1923 issue of Naturwissenschaften marking the tenth anniversary of Bohr’s atomic theory.107 In this paper, Ladenburg and Reiche made use of the correspondence principle to argue that although the detailed mechanism behind dispersion was unknown, the classical theory could be carried over to the domain of quantum physics.

A major improvement of Ladenburg’s dispersion theory occurred with two notes published by Kramers in 1924. They were both to some extent based on the BKS theory, but Kramers’ results did not rely crucially on this theory. ‘You perhaps noticed his [Kramers’s] letter to Nature on dispersion’, Slater wrote to Van Vleck in late July 1924; ‘the formulas & that he had before I came, although he didn’t see the exact application’.108 What Kramers used in his published derivation was only the idea of the atom as an assemblage of virtual oscillators, but he did not rely on either statistical energy conservation or the lack of coupling between atomic processes. Moreover, he emphasized in his second note that the virtual oscillators should be understood ‘only as a terminology suitable to characterise certain main features of the connexion between the description of optical phenomena recognized and the theoretical interpretation of spectra’.109 Likewise, Van Vleck thought that the virtual oscillators were ‘in some ways very artificial’, yet he found the idea valuable because it seemed to be the only way to avoid the ‘almost insuperable difficulty that it is the spectroscopic rather than the orbital frequencies…which figure in dispersion’.110

A recognized master in the application of the correspondence principle, Kramers argued that Bohr’s principle required an extension of Ladenburg’s result, which was only valid for transitions to the ground state. For this reason it had nothing to say about transitions in the limit of large quantum numbers. To satisfy the requirements of the correspondence principle Kramers needed a dispersion formula consisting of two parts: one corresponding to processes where an atom decays spontaneously from a higher to a lower state, and another in which the atom returns to the excited state by a transition induced by the incident light. While the first part corresponded to absorption frequencies νa, the second related to emission frequencies νe. The resulting formula can be written as

(p.340)
$Display mathematics$

Kramers referred to the second part, which formally corresponds to emission oscillators of negative mass, or negative value of e 2/m, as ‘negative dispersion’. For atoms in the ground state this term vanishes and the formula reduces to Ladenburg’s result. The appearance of a negative dispersion term lacked experimental evidence, but Kramers argued that it was needed to establish connection by means of the correspondence principle to the classical dispersion formula. It could, he suggested, be understood as an instance of the stimulated emission predicted by Einstein in his radiation theory of 1916. The negative term was criticized by the American physicist Gregory Breit, at the University of Minnesota, whose critical note gave rise to Kramers’ second letter to Nature.111 Experimental support for the negative dispersion was first reported in 1928 by Ladenburg and his collaborators.

Two other features of Kramers’ dispersion theory need to be mentioned. First, he emphasized as an advantage of his formula that ‘it contains only such quantities as allow of a direct physical interpretation on the basis of the fundamental postulates of the quantum theory of spectra and atomic constitution, and exhibits no further reminiscence of the mathematical theory of multiple periodic systems’.112 Second, to come from the classical frequency of motion to the radiation frequency in the limit of large quantum numbers, he replaced the differential quotients of the classical formula by difference quotients. This would soon turn out to be an important step. Kramers’s dispersion programme culminated in a joint paper with Heisenberg, but even before this paper his work was recognized as an important advance. Ladenburg told him ‘how much it pleases me that you have managed to give a correspondence derivation of the relation between dispersion and transition probabilities’. He thought that, ‘In this way a solid basis has now been created’.113

Kramers’ work was also appreciated by Van Vleck, who regarded it ‘a distinct advance in the problem of reconciling dispersion with quantum phenomena’.114 Van Vleck was himself a main contributor to the theory of dispersion, in which field he competed with the European physicists and obtained results of no less significance than they did. Thus, he arrived independently at Kramers’ dispersion formula, including the negative dispersion term, and he did so without relying of the BKS theory. Having learned from Slater about Kramers’ work, he wrote to his colleague and rival in Copenhagen: ‘The concept and introduction of the virtual-oscillator formula is entirely yours, and I always refer to the “Kramers dispersion formula”, but I had developed the perturbation theory method for absorption etc. prior to learning of any of your work’.115

Although Bohr was strongly interested in dispersion theory and closely followed Kramers’ work on the subject, he did not participate actively in it. Nonetheless, the theory was sometimes—rather unfairly to Kramers, it would seem—considered the (p.341) result of a collaboration between the two Copenhagen physicists. According to Heisenberg, writing to Landé, ‘The beautiful thing about the new Bohr and Kramers dispersion theory is precisely that one now knows (or suspects)…how the quantum mechanics will look’.116

An important paper by Born published in August 1924 took its starting point in the ‘considerable progress’ made with the BKS theory and Kramers’ use of it to explain the dispersion of light. Born’s paper, entitled ‘Über Quantenmechanik’, merits attention not only because it introduced ‘quantum mechanics’ as the name for the quantum theory of the future, but also because it was a serious attempt to formulate the general structure of this still unborn theory. Although Born made use of virtual oscillators as ‘the real primary thing’, he stressed that his theory was independent of the ‘still disputed conceptual framework of that theory [BKS], such as the statistical interpretation of energy and momentum transfer’.117 The abstract theory included Kramers’s dispersion theory, which it connected to Heisenberg’s theory of multiplets and the anomalous Zeeman effect. From a formal point of view its most important result was that it offered a ‘formal passage from classical mechanics to a “quantum mechanics”…in the sense of a transition from differential to difference equations’.118 According to Born, if a transition from a stationary state of quantum number n to another state of quantum number n – τ was characterized by some classical function F(n), the function could be translated into its quantum-theoretical analogue by a replacement of the kind

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What has been called ‘Born’s correspondence rule’ was known earlier to Kramers, who referred to it in his second note on dispersion theory, but Born arrived at it independently.119 He emphasized, again in agreement with Kramers, that whereas the classical dispersion formula as derived from perturbation theory contained action variables and other reminiscences of mechanical orbits, in its quantum-theoretical translation only observable quantities appeared. However, the appeal to observable quantities was of a purely formal nature: Born’s theory did not lead to calculations that could be compared to spectroscopic or other measurements.

In a programmatic passage in his Atommechanik, Born summarized what he called the fundamental ideas of a future quantum mechanics, a theory that would be based solely on observable properties such as the frequency and intensity of light emitted by an atomic system. As to the orbital frequencies and distances of electrons, as they appeared in the existing theory, ‘these quantities are, as a matter of principle, not accessible to observation’. For this reason, ‘our procedure is just a formal computational scheme which, for certain cases, allows us to replace the still unknown quantum laws by computations on a classical basis’. Concerning the true laws of quantum physics, he wrote:

(p.342) Of these true laws we would have to require that they only contain relations between observable quantities, that is, energy, light frequencies, intensities, and phases. As long as these laws are still unknown, we must always face the possibility that our provisional quantum rules will fail; one of our main tasks will be to delimit the validity of these rules by comparison with experience.120

Kramers’ two notes of 1924 were rather sketchy and left out many details. It was only when he joined forces with Heisenberg, who spent the period from September 1924 to April 1925 at Bohr’s institute, that his theory reached completion. The result of their collaboration, a detailed article published only in March 1925, was mostly written by Kramers and was to a large extent an elaboration of his earlier ideas. Based on the correspondence principle, the two physicists gave a full and systematic explanation of dispersion, extending Kramers’ theory to cover also the kind of incoherent or anomalous dispersion predicted by the Austrian physicist Adolf Smekal. According to Smekal, who based his arguments on the light-quantum hypothesis, the scattered radiation should include terms with a frequency smaller and larger than the frequency ν of the incident radiation: if Δν is the frequency difference between two stationary states, the incoherent radiation would have the frequencies |ν ± Δν|. In agreement with the Copenhagen attitude to light quanta, Kramers and Heisenberg reproduced the Smekal effect by means of the correspondence principle and without making use of Einstein’s hypothesis of the corpuscular nature of light.121

As in Kramers’s earlier theory, the Kramers–Heisenberg dispersion theory was detached from any intuitive model of the atom and visualizable concepts such as the orbits and velocities of revolving electrons. Its formulae contained ‘only the frequencies and amplitudes which are characteristic for the transitions, while all those symbols which refer to the mathematical theory of periodic systems will have disappeared’. Again, some of its results were achieved by following the discretization procedure, namely ‘by interpreting the differential quotients…as differences between two quantities, in analogy to Bohr’s procedure in the case of frequencies’.122 Kramers and Heisenberg acknowledged the close connection of their theory to the recent BKS theory, which was however more important to the senior author than the junior author. While Kramers tended to see the work as a triumph of the BKS approach, to Heisenberg the connection was of little significance. What mattered to him was the formal structure of the theory, the formulae themselves and the fact that they agreed with the observability principle by depending only on observable quantities.

In his Faraday Lecture of 1930, Bohr described Heisenberg’s symbolic quantum mechanics as ‘a most ingenious completion of the trend of ideas characterised by Kramers’ adaption of Lorentz’ classical theory of the optical dispersion phenomena to the quantum theory of spectra’. He likewise singled out Kramers’ general dispersion theory in a memorial address of 1952. This work, he said, ‘actually proved a stepping stone for Heisenberg, who shortly afterwards in a most ingenious way accomplished a rational formulation of quantum mechanics in which all direct reference to ordinary (p.343) pictures of mechanical motion was finally abandoned’.123 Heisenberg agreed that Kramers’ work had a significant impact on the process that led to quantum mechanics: ‘One felt that one had now come a step further in getting into the spirit of the new mechanics. Everybody knew that there must be some new kind of mechanics behind it. Nobody had a clear idea about it, but still one felt that the dispersion formula was a good step in the right direction’.124

While recognized as an important theoretical advance, from an empirical point of view there was little to recommend the Kramers–Heisenberg theory. In fact, there was no direct experimental evidence for its results—as little as there was for Kramers’s earlier version or for Born’s generalized theory. The explanation of the Smekal effect related to experiment, but this phenomenon was only demonstrated in 1928, when the Indian physicist Chandrasekhara Raman, in collaboration with his student Kariamanikam Krishnan, discovered the Raman effect or what at the time was sometimes referred to as the Smekal–Raman effect.125 The lack of empirical confirmation did not count heavily in how physicists received the Kramers–Heisenberg theory. Experts in atomic and quantum theory found the work important and stimulating for theoretical and conceptual reasons, not because it was able to account for known experimental phenomena relating to dispersion and scattering or because it predicted new phenomena. To Heisenberg, it was a fruitful ‘failure’. Its importance was methodological, or so he later recalled: ‘This attempt was a failure. It got me into an impenetrable thicket of complicated mathematical formulae, from which I found no way out. But the attempt confirmed me in the attitude that one should not ask about electron orbits inside the atom at all.’126

If there were no electron orbits, then there was no Bohr model of the atom in the classical sense either. Indeed, by the summer of 1924 the visualizable Bohr or Bohr–Sommerfeld model of the atom was fading and was no longer considered a candidate for the real structure of atoms. The constitution of the atom in terms of a tiny positive nucleus surrounded at great distances by a system of electrons was left untouched, and so was the postulate of stationary states (not orbits), but few leading physicists believed in the planetary analogy—that the electrons actually moved in definite orbits whose geometry was characterized by quantum numbers. Objections to the orbital model had been around for some time, raised in particular by Pauli. By Christmas time 1923 he and Heisenberg agreed that ‘model conceptions have no real meaning’ and that ‘the orbits are not real’, as Heisenberg expressed it in his letter to Sommerfeld of 8 December.127 One year later, Pauli praised his former professor for having left out atomic models in his exposition of complex spectra in the fourth edition of Atombau:

The model concepts are now in a severe, fundamental crisis, which, I believe, will finally end with a further radical sharpening of the contrast between classical and quantum theory…[T]he conception of definite and unambiguously determined electron orbits in the atom can hardly be sustained. One now has the strong impression with all models, that we speak there a language that is not sufficiently adequate for the simplicity and beauty of (p.344) the quantum world. For this reason I found it so beautiful that your presentation of the complex structure is completely free of all model-prejudices.128

The shift in attitude to orbital models may be illustrated by the 6 July 1923 issue of Die Naturwissenschaften celebrating the tenth anniversary of the Bohr atom. The issue included articles by Coster and Kramers on Bohr’s ‘second atomic theory’ accompanied by two-colour figures of the orbital structure of atoms from hydrogen to radium, taken from the plates Bohr used in his lectures.129 These pictorial models still had a certain credibility, but not for long. In the same issue Born explicitly criticized the planetary atom analogy routinely used in more popular expositions of the theory. ‘The similarity of atoms to planetary systems has only limited validity’, he pointed out. ‘It becomes increasingly probable that not only new assumptions will be needed in the ordinary sense of physical hypotheses, but that the entire system of concepts in physics will have to be restructured in its foundations’.130

Even more than Born and Heisenberg, Pauli campaigned against the kind of orbital pictorial models that he seems to have associated with Kramers rather than Bohr. It was as if such models emotionally offended him. In his letter to Bohr of 12 December 1924, in which he enclosed his manuscript on the exclusion principle, Pauli proudly mentioned that he had avoided the term ‘orbit’ in his work altogether. ‘I believe that the energy and [angular?] momentum values of the stationary states are something much more real than “orbits”’, he said. In a footnote he poked fun at ‘our good friend Kramers and his colourful picture books’, a reference to the Kramers–Holst book and its pictorial atomic models. In his letter to Kramers of 27 July 1925, quoted above, Pauli once again stressed his lack of belief in orbits, adding that in a recent report on superconductivity from the Leiden laboratory, ‘I have again noticed with horror traces of your spirit in the form of figures!’131 (Figure 8.2). As Pauli saw it, in so far as one could speak of atomic models at all, it had to be a mathematical and not a pictorial model.

# 8.5 From anomalies to crisis

Among many physicists in the period 1923–1925 there was a mounting feeling that the existing quantum theory of the atom had come to a dead end and that further search for substantial progress within the framework of the Bohr–Sommerfeld theory was futile. The old framework had to be replaced by a new one. What is often referred to as a deep-seated attitude of crisis included the expectation that a way out of the morass of anomalies and inconsistencies necessitated a radically new foundation of atomic physics, probably one where there was no place for visualizable models and where mathematical symbols referring to observables would replace mechanical terms. The general feeling of crisis was not new to German physicists, but it acquired a new and more specific dimension with the experimental and conceptual problems that plagued quantum theory.132

(p.345)

Fig. 8.2. The electronic and lattice structure of indium according to Bohr’s atomic theory and as used by Kamerlingh Onnes in a paper of 1924 on superconductivity. Kamerlingh Onnes was in contact with Kramers in Copenhagen, who provided him with this and other pictures of metal atoms.

Source: Kamerlingh Onnes 2004.

As demonstrated by statements from Pauli, Born, Heisenberg, and Bohr, by the beginning of 1925 the quantum crisis was real. With the recognition that the simple Bohr–Sommerfeld orbital theory was wrong, earlier successes became puzzles. This is what Heisenberg referred to in a letter to Pauli of 21 June 1925 in which he reported what he felt was his painfully slow progress in ‘fabricating quantum mechanics’. Heisenberg expressed surprise that Pauli in an earlier letter had wondered about the failure of mechanics, because: ‘If anything like mechanics were true, one would never understand the existence of atoms. Evidently, there exists another, a “quantum mechanics”, and one must only wonder about the fact that the hydrogen atom agrees, as far as the energy constant is concerned, with something computed classically’.133 Shortly after the quantum revolution, Bohr considered the same issue, the surprising empirical success of the semi-classical model of the hydrogen atom. ‘The atomic theory has passed through a serious crisis in these last years’, he wrote to Oseen in Sweden. Looking back on the development, he reflected:

At the present stage of the development of the quantum theory we can hardly say whether it was good or bad luck that the properties of the Kepler motion could be brought into such (p.346) simple connection with the hydrogen spectrum, as was believed possible at one time. If this connection had merely had the asymptotic character which one might expect from the correspondence principle, then we should not have been tempted to apply mechanics so crudely as we believed possible for some time. On the other hand, it was just these mechanical considerations that were helpful in building up the analysis of the optical phenomena which gradually led to quantum mechanics.134

Although the perception of a serious crisis was real, it was far from shared by all physicists. It was restricted to the relatively few who were actively engaged in foundational research related to atomic structure, perhaps a dozen or so. Even Sommerfeld, the leader of the important Munich school in atomic physics, did not identify the situation leading up to quantum mechanics as a crisis preceding a revolution. ‘The new development’, he wrote in 1929, ‘does not signify a revolution, but a joyful advancement [eine erfreuliche Weiterbildung] of what was already in existence, with many fundamental clarifications and sharpenings’.135 Another physicist who did not consider the turn to quantum mechanics revolutionary was Einstein. To him, the shift from the old to the new quantum theory was an evolution rather than a revolution. He did not consider quantum mechanics a fundamental theory but of the same phenomenological nature as the Bohr–Sommerfeld theory.

The Bohr orbital model was not abandoned because it was challenged by a new and better theory of the atom; witness that the Bohr model lost its credibility about a year before the emergence of quantum mechanics. The reason why physicists lost faith in the model and looked forward to a new theory to replace it, was a combination of empirical problems and theoretical shortcomings.136 For one thing, persistent discrepancies between the Bohr–Sommerfeld theory and experimental results called for more and more radical revisions of the basic assumptions of the theory. Heisenberg’s second core model had little in common with Bohr’s orbital theory of the chemical elements of 1922. Many of the experimental challenges came from attempts to account for spectroscopic observations in terms of quantum numbers naturally connected to the theory. The anomalous Zeeman effect and other observations suggested that l = k – 1 rather than k was of fundamental significance, yet the l quantum number had no natural interpretation within Bohr’s theory. Band spectra and complex spectra indicated, indeed required, half-integral quantum numbers, and sometimes the inner quantum number j needed to be represented by $j ( j + 1 )$. Such oddities had no place in the Bohr model and could only be grafted onto it.

It should not be forgotten that Bohr’s atomic model, plagued by difficulties as it came to be, was eminently fruitful in a vast domain and within a short span of time scored a large number of empirical successes. Table 8.1 lists some of these successes; that is, explanations or predictions which at the time were generally considered as confirmations or support of the theory.137 Most of them have been discussed in earlier chapters. Some of the empirical successes were evidently regarded as more important than others. While the explanation of the hydrogen atom was an impressive and unqualified (p.347)

Table 8.1. Partial list of experimental successes and failures in the history of the Bohr atom 1913–1925. There is no correlation between the left and right columns.

Successes

Failures

One-electron atoms (H, He+)

Helium

Spectrum, ionization energy etc.

Spectrum, ionization energy etc.

One-electron atoms (H, He+) fine structure

Anomalous Zeeman effect

Franck–Hertz experiments

Many-electron atoms, complex spectra

Stern–Gerlach experiments

Ramsauer effect

Space quantization

(Davisson–Kunsman experiment)

Stark effect

Paschen–Back effect in hydrogen

Normal Zeeman effect

Hydrogen molecule ion (H2 +)

Periodic system, explanation of

Zero-point energy (molecular spectra)

Prediction of element Z = 72 (Hf)

Behaviour of electrons in crossed electric and magnetic fields

X-ray spectra

Molecules; covalent bond

Optical dispersion

Superconductivity

success, one which was almost too good to be true, Bohr’s later theory of the periodic system was only temporarily successful. It was soon replaced by better theories (Stoner’s and Pauli’s) which were within the general framework of Bohr’s atomic theory but still differed from it in various respects. To repeat from Section 4.4, Sommerfeld’s explanation of the fine structure of one-electron atoms was considered a great triumph of the Bohr–Sommerfeld theory, yet it turned out that the triumph was illusory and the nearly perfect agreement between theory and data fortuitous. As Bertrand Russell moralized:

A theory which explains all the known relevant facts down to the minutest particular may nevertheless be wrong. There may be other theories, which no one has yet thought of, which account equally well for all that is known. We cannot accept a theory with any confidence merely because it explains what is known.138

Sommerfeld’s fine-structure formula was correct for the wrong reasons, but in the present context this is less relevant since it became recognized only after the emergence of quantum mechanics. Yet it is an instructive case, both from a historical and a philosophical point of view.139

The experiments on space quantization in a magnetic field that Otto Stern and Walther Gerlach performed in Frankfurt am Main in 1921–1922 have a certain similarity to Paschen’s experiments on the fine structure of the spectra of hydrogen and ionized helium. They both produced experimental evidence in support of a wrong theory. According to Sommerfeld’s theory of 1916, atoms with one valence electron (p.348) should possess a magnetic moment of one Bohr magneton, given by M = eh/4πmc; if placed in a magnetic field they should assume only two orientations, either aligned with the direction of the field or opposite to it. Stern and Gerlach reasoned that if space quantization were a reality, a beam of silver atoms passing a non-uniform magnetic field should split up into two separate beams and not, as would be expected from the classical Larmor theory, merely be broadened. This effect, wrote Stern in a preliminary paper, ‘unequivocally decides between the quantum-theoretical and classical interpretation’.140 In their experiment of 1922 they found a distinct separation into two beams. The splitting was only 0.2 mm, but it was enough to prove the space quantization expected from quantum theory and to determine the magnetic moment of the silver atom to one Bohr magneton.141

Of interest here is merely that the Stern–Gerlach experiment was considered a triumph of the Bohr–Sommerfeld quantum theory because it confirmed that an electron orbit can only assume discrete orientations in space. The results were not only a confirmation of Sommerfeld’s theory, they also agreed with the ideas of Bohr, who had theoretical reasons to believe that ‘the magnetic axis of the silver atom is always directed parallel to the field and never perpendicular to it’. When Gerlach informed Bohr of the results in a postcard of 8 February 1922 (Figure 8.3), he congratulated him ‘on the confirmation of your theory’.142 Paschen even claimed that the experiment ‘proves for the first time the reality of Bohr’s stationary states’.143

Fig. 8.3. Postcard from Gerlach to Bohr of 8 February 1922, informing him of the magnetic splitting of a beam of silver atoms.

Source: Friedrich and Herschbach 2003, p. 56 (Niels Bohr Archive).

(p.349) However, the situation was somewhat ambiguous, for a closer consideration of the Stern–Gerlach effect indicated that it presented difficulties for the ordinary understanding of the quantum theory, in particular the unrestricted validity of the adiabatic hypothesis. In a critical analysis of the experiment, Einstein and Ehrenfest discussed the difficulties and referred to ‘Bohr’s view—that in complicated fields there will be no sharp quantization at all’.144 Although known, the difficulties were not seen as posing a threat to the fundamental assumptions of the theory, and it was only after the emergence of spin quantum mechanics that it became clear that what Stern and Gerlach had observed did not really support the Bohr–Sommerfeld quantum theory. In retrospect they had discovered the two spin states of the electron.

Basically successful as Bohr’s theory was, its empirical success was limited and countered by a growing number of anomalies. The most important of the experimental problems were undoubtedly the helium atom and the anomalous Zeeman effect, but there were several other phenomena that the theory could not account for satisfactorily (Table 8.1). Anomalies are of course bad news for a theory, but they come in different kinds and are often evaluated in widely different ways by the scientific community. Some are regarded as crucially important—the kind of stuff that may cause a wholesale refutation of a theory—while others are hardly noticed and have almost no effect at all on how the theory is judged. In the case of the Bohr atom, physicists paid great attention to the anomalies related to the helium atom and the Zeeman effect, which, as we have seen, were characterized as nothing less than catastrophes. On the other hand, there were also anomalies that indicated flaws in the theory but nonetheless were ignored for all practical purposes or at least assigned very little weight.

One example was the evidence of a Paschen–Back effect in hydrogen, where according to the Bohr–Sommerfeld theory there should be none. The inconsistency between theory and experiment was recognized by leading physicists, but without considering it a problem that threatened the theory. The contradiction did not really disturb the physicists, who convinced themselves that it should not count as a genuine anomaly. We have a somewhat similar case in the failed attempt to understand the hydrogen molecule ion H2 + . This anomaly—and it clearly was one—was not much noticed at the time and did not contribute to the rise of the crisis that began in 1923. As a third and rather remarkable example of a quantum anomaly that was not taken seriously and scarcely recognized as a proper anomaly, one may mention the covalent bond that ties atoms together in molecules. As early as 1915 it was evident that Bohr’s theory could not account for the H2 molecule, yet the sustained impotence of the theory in the area of molecular constitution was of no importance in the crisis of the old quantum theory. It was of importance to the reputation of the theory in chemical circles, but this was something few physicists cared about.

The case of zero-point energy, or the corresponding notion of half-integral quantum numbers, was an anomaly only in the sense that it did not fit naturally with the Bohr–Sommerfeld quantum theory. As we have seen, half-quanta turned up in both theories of the anomalous Zeeman effect and in molecular spectroscopy. The zero-point energy (p.350) that Mulliken inferred in 1925, and which had been suggested before him, could not be theoretically justified by existing quantum theory, and this constituted a problem. However, it is worth pointing out that Mulliken’s ‘discovery’ of the zero-point energy was not much noticed in the community of atomic and quantum physicists, possibly because it was made in a chemical rather than physical context. In fact, the physicists in Copenhagen and Munich seem to have ignored his work or, more likely, been unaware of it. It did not contribute to the quantum crisis any more than the hydrogen molecule failure did. At any rate, Bohr much disliked the half-quanta, arguing that they were irreconcilable with the basic principles of quantum theory, while most physicists learned to live with them, if not love them. According to Van Vleck, they were necessary, yet ‘in many ways decidedly illogical’.145 It was only with Heisenberg’s quantum mechanics from the summer of 1925 that the problem of zero-point energy disappeared. As Heisenberg showed, it followed from his new theory that the energy of a harmonic oscillator is given by

$Display mathematics$

and not, as in the older theory, by W = nhν. This was not the end of the zero-point energy problem, but only the beginning of a new chapter, where it came to be seen as the energy density of vacuum. It was eventually interpreted in terms of the cosmological constant introduced by Einstein in 1917. But this new chapter, fascinating as it is, does not belong to the history of the Bohr atomic model.

In addition to recognized failures or anomalies, there were also a few consequences of the quantum theory that disagreed with experiments but were recognized only after the crisis had culminated and the old quantum theory been replaced by quantum mechanics. One may speak of a post hoc anomaly, although of course it does not count as an anomaly in the real, historical sense. Consider the fine structure, where Hansen’s identification of a IIc component in hydrogen (Section 4.4) was anomalous but only clearly recognized as such at the end of 1925. On the basis of the Sommerfeld–Bohr–Kramers theory there should exist in the He+ spectrum a IIIb component, which did not in fact exist (Section 5.3), and this anomaly or inconsistency could have been known as early as 1919. However, it went unnoticed until Goudsmit and Uhlenbeck pointed it out in their paper in Physica in which they introduced their new classification scheme. Since this paper only appeared in September 1925, their observation was of no consequence in the crisis of the old quantum theory.

Another interesting and little known case is provided by the theory of dielectric constants (Section 6.3), where Pauling in 1927 concluded that because of the space quantization, ‘the old quantum theory definitely requires that the application of a strong magnetic field to a gas such as hydrogen chloride produce a very large change in the dielectric constant of the gas’. Even a magnetic field of moderate strength should have a drastic effect on the dielectric constant, a prediction that was flatly contradicted by experiments. Therefore, ‘it provides an instance of an apparently unescapable and (p.351) yet definitely incorrect prediction of the old quantum theory’.146 The new quantum mechanics fortunately led to the result that a magnetic field does not influence the dielectric constant, in agreement with experiment. Pauling’s result could presumably have been derived several years earlier, and would then have constituted an anomaly. But this is not what happened.

After quantum mechanics had stabilized, a few physicists looked back to consider the problems that had faced the Bohr–Sommerfeld theory and made it a theory of the past. In April 1928, at a symposium of the American Chemical Society, Van Vleck presented a list of the inadequacies of the old quantum theory in order to illustrate to the chemists the superiority of the new quantum mechanics. As to the successes of the old theory, he emphasized that all of these were reproduced by the Heisenberg–Schrödinger–Dirac theory. He called attention to the parts of the old theory that survived in, and could be better understood using, the new theory:

Although we cannot attach as much reality and vividness to electron orbits in stationary states as previously, nevertheless the new theory has the vital feature of a discrete succession of energy levels, with spectroscopic frequencies proportional to their differences. Thus we can still keep our beloved energy level diagrams, selection principles, and the like.147

According to Van Vleck, the Bohr–Sommerfeld theory had failed on a number of counts, which may be compared with those listed in Table 8.1. The failures mentioned by Van Vleck were the following:

1. (1) The fine details of the hydrogen and ionized helium spectrum, to which he not only counted the Paschen–Back effect and the appearance of the forbidden IIc line, but also the arbitrary exclusion of pendulum orbits (k = 0) ‘on the ground that in them the electron would collide with the nucleus’.

2. (2) The spectrum of neutral helium.

3. (3) The intensity of spectral lines. While admitting that in some cases intensities could be estimated by means of the correspondence principle, Van Vleck found that the method ‘lacked adequate logic’ and did not result in unambiguous quantitative results. The old quantum theory was never able to treat intensity in the same rigorous way as frequency, in the way quantum mechanics could.

4. (4) The anomalous Zeeman effect.

5. (5) Half-quanta. Whereas the half-integral quantum numbers were introduced phenomenologically in the old quantum theory, and never provided with a rigorous justification on the basis of the theory, in the new quantum mechanics ‘half quantum numbers [are] no longer a bugaboo’.

6. (6) Dielectric constants and magnetic susceptibilities. On this inadequacy, see p. 350 and in Section 6.3.

7. (7) Dispersion and the emission and absorption of radiation.

8. (8) Collisions of electrons with atoms, including the Ramsauer effect.

(p.352) Another retrospective list of problems in the old quantum theory was made by Owen Richardson, who, like Van Vleck, used it to confront the now abandoned Bohr–Sommerfeld theory with the new quantum mechanics.148 Several of the problems in Richardson’s list also appeared in Van Vleck’s, including the helium atom, the anomalous Zeeman effect, half-integral quantum numbers, and the Ramsauer effect. In addition he mentioned experiments on soft X-rays and the reflection of electrons on metals studied by Davisson and Kunsman.

# 8.6. Observability and Umdeutung

There could be no question of rejecting the Bohr–Sommerfeld theory just because it faced some insurmountable empirical problems such as the anomalous Zeeman effect and the spectrum of helium. In the absence of an alternative theory it would have left the physicists with a wealth of unexplained phenomena, a drastic loss of knowledge instead of a gain in knowledge. By 1924 it was realized that the Bohr–Sommerfeld theory was inadequate, perhaps even fundamentally wrong, and that its basic architecture was a hopeless mess, a patchy combination of incomparable elements. But it was also realized that the theory after all held a strong empirical record and that any theory of the future would have to be at least as strong. According to Bohr, this future theory needed to contain the existing theory as a limiting case and also, like the existing theory, to keep the connections to classical theory. ‘It is necessary for us’, he wrote to Born, ‘to regard the usual formulation of the quantum theory, as it is elaborated for atomic structure, as a limiting case of a more general theory’.149

The experimental problems did not themselves show the way to the more general and superior theory the physicists dreamed of. To find the way it was necessary to identify the methodological and conceptual flaws in the existing theory, flaws that were supposedly reminiscences of its semi-mechanical origin. A small group of physicists consequently focused on the methodological and logical structure of the theory, asking themselves about its very foundation. In this process fundamental notions were questioned, not only about the existence of electron orbits but also of the use of differential equations in the basic theory and the possibility of a space-time description of the interior of the atom. Such radical ideas were not foreign to Bohr, who in a letter to the Danish philosopher Harald Høffding from September 1922 expressed his view of the situation in atomic physics as follows:

Here we are in the peculiar situation that we have acquired certain information about the structure of the atom which can be considered as being as certain as any one of the facts in the natural sciences. On the other hand we encounter difficulties of such a deeply-rooted nature that we do not even faintly see the road to their solution; in my personal opinion these difficulties are of such a nature that they hardly allow us to hope that we shall be able, in the world of the atom, to carry through a description in space and time of a kind which (p.353) corresponds to our ordinary sensory images. Under these circumstances one must naturally constantly bear in mind that one is operating with analogies, and this step, in which the application of these analogies is delimited in each case, is of decisive significance for progress.150

Among the methodological issues that played a role in the declining phase of the old quantum theory was the doctrine or desideratum that physical theories should build solely on concepts which refer to quantities that can, at least in principle, be observed. As mentioned, this so-called observability principle had been advocated since 1919 by Pauli and Born in particular, and in a mild version it also entered the dispersion theories of Kramers, Heisenberg, and Born. On the other hand, Bohr seems not to have appreciated the observability principle as a heuristic tool, although he advocated a limited use of it as early as 1913. In his Atommechanik completed in November 1924, Born stated about the ‘unknown and true quantum laws’ that ‘we must require that they involve only observable quantities such as energies, frequencies of light, intensities and phases’.151 About half a year later he wrote a paper together with 22-year-old Pascual Jordan, his new assistant, in which the observability principle was presented as ‘A fundamental axiom of large range and fruitfulness [which] states that the true laws of nature involve only such quantities as can be observed and determined in principle’.152 Although the operationalist observability criterion was thus assigned an important role in parts of the German physics community, it was not closely connected to the hostile attitude to electron orbits that Pauli expressed in particular.

Neither did the criterion motivate Heisenberg to eliminate the unobservable orbits, as he did in his seminal paper from the late summer of 1925 which marks the birth of quantum mechanics. In his earlier papers he did not refer to the observability principle, which he first mentioned in a letter to Pauli of late June in which he adopted the doctrine stated by Born and Jordan. In Heisenberg’s formulation the Grundsatz (basic axiom) read: ‘In calculating any quantities, like energy, frequency, etc., only relations between those quantities should occur, which can be controlled in principle’.153 Less than two weeks later, he confirmed his nearly complete agreement with Pauli’s views, not only with regard to observability but also with regard to orbital or other mechanical models. Referring to Bohr’s model of the hydrogen atom, Heisenberg now wrote: ‘It is really my conviction that an interpretation of the Rydberg formula in terms of circular and elliptical orbits…does not have the slightest physical significance. And all my wretched efforts are devoted to killing totally the concept of an orbit—which one cannot observe anyway—and replace it by a more suitable one’.154

Heisenberg’s observability criterion is sometimes, and not without reason, seen as a version of the operationalist philosophy developed by the American physicist-philosopher Percy Bridgman, a Nobel laureate of 1946 for his works in high-pressure physics. Yet, although Bridgman recognized the similarity of the quantum observability principle to operationalism, he refrained from endorsing Heisenberg’s use of it or considering it a vindication of the true operationalist method. Like some modern (p.354) historians and philosophers of physics, he did not consider Heisenberg’s method to be a constructive principle, but suspected that it was instead a philosophical rationale added after the fact. ‘I have always wondered’, he said in 1936, ‘whether perhaps this requirement of Heisenberg was not formulated after the event as a sort of philosophical justification for its success, rather than having played an indispensable part in the formulation of the theory’.155

The result of Heisenberg’s hard thinking and intense intellectual interaction with Pauli was a paper that appeared in the 18 September 1925 issue of Zeitschrift für Physik with the title ‘Über Quantentheoretische Umdeutung Kinematischer und Mechanischer Beziehungen’; that is, on the quantum-theoretical reinterpretation of kinematic and mechanical relations. Historians have pointed out that in an earlier paper on the anomalous Zeeman effect, Sommerfeld used the magical word Umdeutung in the title, and that the similarity between the two papers is not limited to the titles but extends to the applied methodologies. Whether this coincidence implies a real connection, an inspiration from Sommerfeld in 1922 to Heisenberg in 1925, seems more doubtful. At any rate, other German physicists spoke of Umdeutung in a quantum context as well. Thus, in the summer of 1925 the Breslau physicists Fritz Reiche and Willy Thomas wrote a paper that both terminologically and methodologically has a great deal of similarity to Heisenberg’s. ‘We try to arrive at a general relation’, they wrote, referring to dispersion, ‘by maintaining the reinterpretation [Umdeutung] of classical quantities into quantum-theoretical ones for all quantum numbers’.156

Umdeutung apart, Heisenberg introduced his paper by referring to some of the well-known problems that faced existing quantum theory, such as the treatment of many-electron atoms and the case of a hydrogen atom placed in crossed electric and magnetic fields. Having reminded his readers that ‘the Einstein–Bohr frequency condition’ ΔE = hν already represented a complete break with classical mechanics, he appealed to observability as a guiding principle:

In this situation it seems sensible to discard all hope of observing hitherto unobservable quantities, such as the position and period of the electron, and to concede that the partial agreement of the quantum rules with experience is more or less fortuitous. Instead it seems more reasonable to try to establish a theoretical quantum mechanics, analogous to classical mechanics, but in which only relations between observable quantities occur.157

In his symbolic translation or reinterpretation of classical mechanics into a new formalism of quantum mechanics, Heisenberg relied heavily on Kramers’ dispersion theories. No less heavily did he rely on Bohr’s correspondence principle, which played an important if mostly implicit role in his reasoning. ‘Heisenberg’s quantum mechanics’, writes Olivier Darrigol, ‘may in fact be seen as the ultimate stage of the evolution of the correspondence principle’.158 Heisenberg later spoke of his Umdeutung paper in words that support this view:

For me [it] represented in a certain sense the quintessence of our discussions in Copenhagen – a mathematical formulation of Bohr’s correspondence principle. I hoped that, by means of a (p.355) mathematical method which for me was still new and very strange, I had found a way to the remarkable relations, which had already been glimpsed from time to time during discussions with Bohr and Kramers.159

The development of quantum mechanics proper is however outside the scope of the present work, for which reason I shall not deal further with Heisenberg’s remarkable paper.

Bohr was not actively involved in the creation of Heisenberg’s new mechanics in the summer of 1925, a period in which there was actually no interaction between the two physicists. Yet Bohr should be given the last word. On 24 August and again on 31 August Bohr gave talks on the recent developments in quantum theory, the first time in Oslo and the second time in Copenhagen to the Sixth Scandinavian Mathematical Congress. On neither of these occasions did he mention Heisenberg’s theory, of which he was most likely unaware. A few days after his address in Copenhagen he received a letter from Heisenberg, then vacationing in Austria: ‘I committed the crime of writing a paper on quantum mechanics, about which I should like to hear your opinion. It will presumably appear in the next issue of the Zeitschrift’.160

A revised version of Bohr’s Copenhagen address was later published in Nature, and here Bohr did comment on Heisenberg’s work and its further development by Born and Jordan into a theory of matrix mechanics. In a manuscript for this publication he described Heisenberg’s theory as ‘probably of extraordinary scope’ and ‘a brilliant realization of the efforts which hitherto have served as a guideline in the development of the atomic theory’. He further noted that, ‘In the nature of matter, there is no longer question of an independent correspondence principle in the above-mentioned scheme [Heisenberg’s], but rather the whole formulation of quantum mechanics may well be regarded as a sharpening of the content of this principle’.161

As far as Bohr was concerned, the quantum revolution of 1925 was not a proper revolution. He tended to agree with Sommerfeld’s view that it was rather ‘a joyful advancement of what was already in existence’.

# Notes for Chapter 8

(1.) Van Vleck 1926, p. 287.

(2.) Pauli 1946, p. 214. On the history of the anomalous Zeeman effect, see Forman 1970, Massimi 2005, pp. 47–52, and Mehra and Rechenberg 1982a, pp. 445–84.

(3.) Born 1923, p. 541. As early as 1920 Sommerfeld had called the effect one of the two unsettled questions of atomic physics’, but at that time the question was not considered a burning one. Although he thought it called for ‘entirely new things’, Sommerfeld did not consider it more serious than another unsettled question, the one of the polyhedral structure of the atom. See Sommerfeld 1920a and also Sommerfeld 1920b, where he characterized the anomalous Zeeman effect as a Zahlenmysterium (number mystery).

(4.) Landé to Bohr, 4 February 1921, reproduced in Forman 1970, p. 238.

(p.356) (5.) Landé to Sommerfeld, 17 March 1921. Sommerfeld to Landé, late March 1921. Both letters are reproduced in Forman 1970, pp. 260–1.

(6.) Landé 1921, p. 239.

(7.) Sommerfeld to Landé, 25 February 1921, in Mehra and Rechenberg 1982a, p. 470.

(8.) Sommerfeld to Einstein, 17 October 1921, in Hermann 1968, p. 94.

(9.) Landé 1923, p. 198.

(10.) Heisenberg to Pauli, 21 February 1923, and Pauli to Landé, 10 March 1923, in Hermann et al. 1979, p. 81 and p. 83.

(11.) Bohr 1922a, p. 59. Bohr was kept informed about the recent works on the anomalous Zeeman effect by letters from Landé, who in February 1921 reported his latest results (see Mehra and Rechenberg 1982a, p. 469).

(12.) Bohr to Landé, 15 May 1922 (Archive for History of Quantum Physics, Bohr Scientific Correspondence).

(13.) Bohr 1923c, in Rud Nielsen 1977, p. 629.

(14.) Ibid., p. 646. Cassidy (1978 and 1979, p. 222) suggests that Bohr was stimulated by a non-mechanical ‘mystery force’ that Van Vleck had previously introduced in his attempt to understand the helium atom. According to Van Vleck, such a hypothetical force, as proposed by Langmuir and a few others, would be negligible except at atomic distances and ‘not have a mechanism based on the Maxwell field-equations’ (Van Vleck 1922, p. 851). However, he did not advocate the idea and I know of no evidence that it stimulated Bohr’s thinking. For Langmuir’s mystery force, see Section 6.1.

(15.) Bohr 1925a, p. 850.

(16.) Bohr 1923c, in Rud Nielsen 1977, p. 647.

(17.) Rud Nielsen 1976, p. 558. The manuscript is in German.

(18.) Bohr to Landé, 14 February 1923 (Archive for History of Quantum Physics, Bohr Scientific Correspondence).

(19.) Bohr to Landé, 3 March 1923, quoted in Richter 1979, p. 34. See also Heisenberg to Sommerfeld, 4 January 1923, in Eckert and Märker 2004, p. 133.

(20.) Pauli to Landé, 23 May 1923, in Hermann et al. 1979, p. 90. Pauli also reported about Bohr’s views in his letter to Sommerfeld of 6 June 1923.

(21.) Pauli 1923, p. 162.

(22.) Pauli to Sommerfeld, 6 June 1923, in Hermann et al. 1979, p. 97.

(23.) Pauli to Landé, 17 August 1923, in ibid., p. 110.

(24.) Pauli to Sommerfeld, 19 July 1923, in ibid., p. 105.

(25.) Heisenberg 1922. On this paper, see Cassidy 1978 and, for a more detailed and scholarly version, Cassidy 1979.

(26.) McLennan 1923, p. 58, who relied on Rosseland 1923b. For k = 1, the minimum distance to the uranium nucleus was found to be 16 × 10–12 cm, hence permissible.

(27.) Heisenberg to Pauli, 19 November 1921, in Hermann et al. 1979, p. 38. Stressing its opportunistic character, Cassidy 1978 calls Heisenberg’s first core model ‘Machiavellian’.

(28.) ‘Heisenberg…is very unphilosophical, he does not care about a clear elaboration of the fundamental assumptions and of their relation to the previous theories’. Pauli hoped that Heisenberg, after having stayed with Bohr in Copenhagen, would ‘return home with a philosophical orientation to his thinking’. Pauli to Bohr, 11 February 1924, in Hermann et al. 1979, p. 143.

(p.357) (29.) Sommerfeld to Einstein, 11 January 1922, in Hermann 1968, p. 96. On Sommerfeld as a quantum craftsman, see Seth 2010. In the third edition of Atombau, Sommerfeld added an appendix on ‘the riddle of the anomalous Zeeman effect’ in which he expounded his own and Heisenberg’s ideas on the subject (Sommerfeld 1922a, pp. 497–504).

(30.) Pauli to Sommerfeld, 6 June 1923, in Hermann et al. 1979, p. 95, and similarly in his letter to Landé of 17 August. On Pauli’s negative attitude to the core model, see Serwer 1977.

(31.) Heisenberg to Pauli, 9 October 1923, in Hermann et al. 1979, p. 125. Heisenberg also gave an account of his new theory in a letter to Bohr of 22 December 1923 (Archive for History of Quantum Physics, Bohr Scientific Correspondence). In a letter of 7 December 1923, Heisenberg sent his manuscript on Zeemangemüse mit Quantensosse (Zeeman vegetables with quantum sauce) to Pauli, informing him that he was hoping for Bohr’s approval—‘papal blessing’. He did receive the blessing, and after some modifications the paper was published in August 1924.

(32.) Heisenberg to Sommerfeld, 8 December 1923, in Eckert and Märker 2004, p. 157. The symbol J denotes an action variable.

(33.) Heisenberg 1924, p. 292. See Darrigol 1992, pp. 196–200.

(34.) In Hermann et al. 1979, pp. 126–7.

(35.) Heisenberg to Bohr, 22 December 1923, as quoted in Cassidy 1992, p. 171. The reference is to Bohr 1923c, which appeared in a special issue of Annalen der Physik in honour of the seventieth birthday of the spectroscopist Heinrich Kayser. On Bohr’s invitation, Heisenberg spent some time in Copenhagen in March 1924, where they discussed the theory and its formulation, including its relation to radiation theory.

(36.) Letter of 29 November 1923, quoted in Cassidy 1992, p. 171.

(37.) Pauli to Landé, 14 December 1923, in Hermann et al. 1979, p. 134. On Pauli’s evaluation of Heisenberg’s theory, see also Pauli 1926a, pp. 237–9.

(38.) Pauli to Bohr, 21 February 1924, in Hermann et al. 1979, pp. 147–8.

(39.) Pauli to Eddington, 20 September 1923, in Hermann et al. 1979, p. 116.

(40.) Born 1925, p. 34. Born’s remarks in Atommechanik echoed the introduction to his important paper on ‘quantum mechanics’ from the summer of 1924 (Born 1924, see also Section 8.4).

(41.) Weyl to Pauli, 9 December 1919, and Pauli to Eddington, 20 September 1923, in Hermann et al. 1979, p. 6 and p. 116. Pauli 1919, pp. 749–50. In his Enzyklopädie article on relativity theory of 1921, Pauli expressed the same criticism, that the field strength at a given point in the interior of an electron is ‘unobservable, by definition, and thus fictitious and without physical meaning’ (Pauli 1958, p. 206). For Pauli’s operationalism and his early views concerning unified field theories, see Hendry 1984.

(42.) Born to Pauli, 23 December 1919, in Hermann et al. 1979, p. 10. See also Darrigol 1992, p. 196.

(43.) Interview with Heisenberg by T. S. Kuhn on 19 February 1963. American Institute of Physics, Niels Bohr Library & Archives. http://www.aip.org/history/ohilist/4661_6.html.

(44.) Einstein to Born, 27 January 1920, in Born 1971, p. 21. Einstein published his idea of reconciling quanta and fields, or rather deriving quanta from fields, in Einstein 1923. His question of how to produce discontinuous solutions from differential equations would be answered with Schrödinger’s wave mechanics, if not quite to Einstein’s satisfaction. Later in life, Einstein considered on various occasions the possibility of replacing the continuous field concept with an ‘algebraic physics’ based on discrete space-time, but nothing came out of these speculations. See Stachel 1993.

(p.358) (45.) Heisenberg to Bohr, 26 February 1930 (Archive for History of Quantum Physics, Bohr Scientific Correspondence), translated and analysed in Carazza and Kragh 1995. Some modern theories of quantum gravity, in particular the theory of loop quantum gravity developed since about 1990, assumes space to be built up of smallest cells.

(46.) Heisenberg to Bohr, 22 December 1923 (Archive for History of Quantum Physics, Bohr Scientific Correspondence). The connection also appeared in Heisenberg’s published paper (Heisenberg 1924, p. 299 and p. 307).

(47.) Bohr 1923b, p. 33.

(48.) Compton 1923, p. 501. Remarkably, he did not explicitly connect his X-ray quanta with Einstein’s theory of light quanta and in fact did not mention Einstein’s name in his paper. For details about Compton’s experiments and theoretical ideas, see Stuewer 1975.

(49.) Debye 1923. On the relationship between the works of Compton and Debye, see Stuewer 1975, pp. 234–7.

(50.) Stuewer 1975. Hendry 1981 disagrees that Compton’s discovery had a decisive effect on physicists’ view on the nature of electromagnetic radiation.

(51.) Kramers and Holst 1923, p. 175. The two authors stressed that Einstein’s theory of the light quantum ‘in no way has sprung from the Bohr theory, to say nothing of its being a necessary consequence of it’.

(52.) Bohr’s notes for the Silliman lectures are included in Rud Nielsen 1976, pp. 581–601. The lectures were covered by The New York Times, which gave summaries of all of them. ‘Likened to solar system, he pictures the atom with nucleus corresponding to sun, and electrons to planets’, it informed its readers on 7 November 1923. See also the summary account in Science 58 (1923), 459–60. Earlier Silliman lecturers included J. J. Thomson (1905) and Rutherford (1906).

(53.) Swann’s suggestion is mentioned in a footnote on p. 290 in Compton and Simon 1925. See also Stuewer 1975, p. 301.

(54.) Duane to Bohr, 5 March 1924, and Bohr to Duane, 17 May 1924, in Stolzenburg 1984, p. 323 and p. 326. Duane believed at the time that the Compton shift was caused by photoelectrons emitted from the K and L levels by the primary X-rays. See also Bohr to Rutherford, 9 January 1924, in ibid., p. 487.

(55.) Quoted from the extracts in Stolzenburg 1984, pp. 13–14. Darwin’s unpublished manuscript exists in at least one other version (Archive for History of Quantum Physics), which is quoted and discussed in Jammer 1966, pp. 171–2.

(56.) Bohr to Darwin, July 1919 (draft), in Stolzenburg 1984, p. 16.

(57.) Darwin 1923, p. 771 and Darwin 1922. On Darwin and his works in quantum physics, see Navarro 2009. See also Konno 2000 for Darwin’s dispersion theory.

(58.) Sommerfeld 1922a, p. 311. The passage did not appear in the first edition of 1919.

(59.) Bohr to Darwin, 21 December 1922, in Stolzenburg 1984, p. 316. Darwin, in a letter to Bohr of 23 November 1922, had ‘left it open whether the upper states really are stationary, but in private [I] may say that I personally do not believe they are’ (ibid., p. 17).

(60.) Bohr 1924, p. 40, who referred to experiments reported in Gans and Miguez 1917.

(61.) Slater 1967, p. 6.

(62.) Bohr to Slater, 10 January 1925, in Stolzenburg 1984, p. 67.

(63.) Slater to Kramers, 8 December 1923, in Stolzenburg 1984, p. 492. See also Hendry 1981. Slater further explained that his ideas might explain the helium spectrum, if only ‘by making a whole lot of guesses’.

(p.359) (64.) Manuscript of 4 November 1923, quoted in Petruccioli 1993, p. 113. On the development of Slater’s views, see Konno 1983.

(65.) Slater 1967, p. 6.

(66.) Slater 1925b.

(67.) Slater 1975, pp. 12–14.

(68.) Slater 1923. Ladenburg and Reiche 1923, p. 590, who introduced classical Ersatzoszillatoren (substitute oscillators) in dispersion theory.

(69.) Bohr et al. 1924. The theory has attracted considerable historical attention. Apart from Hendry 1981 and Petruccioli 1993, see also Stolzenburg 1984, Duncan and Janssen 2007, pp. 597–617, and Dresden 1987, pp. 41–78, 159–215, who deals in particular with Kramers’ contribution to and view of the theory.

(70.) Bohr to Slater, 10 January 1925, quoted in Stuewer 1975, p. 293. On Bohr’s coupling principle and latent forces, see Bohr 1924, p. 36.

(71.) Bohr to Michelson, 7 February 1924, in Stolzenburg 1984, p. 405. See also Livingston 1973, pp. 301–2, according to whom (p. 292) Michelson had met Bohr at the 1921 Solvay congress, where he, ‘feeling somewhat lost’, listened to Bohr lecturing on atomic theory. However, Bohr did not attend the Solvay conference this year and he only met with Michelson during his trip to the United States in the autumn of 1923.

(72.) Bohr et al. 1924, p. 786.

(73.) Ibid. The treatise referred to was Bohr 1924. The BKS paper is reprinted in van der Waerden 1967, pp. 159–76.

(74.) Bohr et al. 1924, p. 795. Bohr 1924, pp. 29–30. The emphasis on the limited definability of motion and energy would later play an important role in Bohr’s complementarity philosophy. For this connection, see Tanona 2004.

(75.) Bohr et al. 1924, pp. 790–1. The BKS theory maintained statistical energy conservation. In 1929 Bohr went further, proposing that energy and momentum might not be conserved even on a macroscopic level. This much more radical hypothesis he thought justified in order to explain the continuous beta spectrum and also stellar energy production. See Jensen 2000, pp. 146–56.

(76.) Bohr et al. 1924, p. 799.

(77.) Kramers and Heisenberg 1925, p. 685.

(78.) Kramers and Holst 1925, pp. 123–40. The preface of the book was dated March 1925, the new chapter on the interaction of light and matter, including the BKS theory, being the work of Kramers. The chapter also appeared separately, as a paper in a Danish physics journal (Kramers 1925). On Kramers’ view of causality in relation to the BKS theory, see Radder 1983 and Dresden 1987, pp. 191–5.

(79.) Haber to Einstein, 1924, quoted in Stolzenburg 1984, p. 26.

(80.) Schrödinger to Bohr, 24 May 1924, in Stolzenburg 1984, p. 490. On Exner’s ideas of indeterminism and acausality, and the way they influenced Schrödinger’s thinking and response to the BKS theory, see Hanle 1979.

(81.) Schrödinger 1924, p. 724.

(82.) According to Paul Forman, it is ‘only by reference to the widespread acausal sentiment that one can understand the immediate and widespread assent which the [BKS] theory received in Germany’ (Forman 1971, p. 99). However, there was no such widespread assent among German physicists.

(83.) Pauli to Bohr, 21 February 1924, in Hermann et al. 1979, p. 147.

(p.360) (84.) Einstein to Born, 27 January 1920, and Einstein to Hedwig and Max Born, 29 April 1924, in Born 1971, p. 23 and p. 82. Einstein also objected to the theory by means of more technical arguments based on thermodynamics. See Stolzenburg 1984, pp. 23–38, where the reactions of Einstein and other physicists are recounted.

(85.) Einstein to Ehrenfest, 31 May 1924, quoted in Stolzenburg 1984, p. 27.

(86.) Bohr to Born, 1 May 1925, a reply to Born’s letter of 24 April, both in Stolzenburg 1984, pp. 308–11.

(87.) Bothe and Geiger 1925.

(88.) Compton and Simon 1925, p. 299.

(89.) Einstein’s manuscript, dated 7 May 1925, is reproduced in Tolmasquim and Moreira 2002, pp. 240–2. The final results of the Bothe–Geiger experiments were published in the 15 May issue of Naturwissenschaften with a detailed report following in June in Zeitschrift für Physik.

(90.) On this question and the later philosophical interest in it, see the appendix (Chapter 9).

(91.) Bohr to Fowler, 21 April 1925, in Stolzenburg 1984, p. 82.

(92.) The name ‘photon’ was proposed by G. N. Lewis in December 1926, but in a sense widely different from Einstein’s light quantum. Lewis’s photon was a ‘hypothetical new atom, which is not light but plays an essential part in every process of radiation’; also contrary to the light quanta, his photons were conserved quantities, ‘uncreatable and indestructable’ (Lewis 1926b; see also Stuewer 1975, pp. 323–6). In spite of the speculative nature of Lewis’s ideas of radiation, Bohr found them interesting. See Bohr to Richardson, 16 February 1926, Archive for History of Quantum Physics.

(93.) Bohr to Geiger, 21 April 1925, in Stolzenburg 1984, p. 79. Much later, in 1936, Paul Dirac advocated a reintroduction of a BKS-like theory in order to explain experiments by Robert Shankland that seemed to imply energy nonconservation in individual atomic processes. Interestingly, in this case Bohr argued against Dirac’s proposal, while Einstein was in favour of it. Referring to the old BKS theory, Bohr pointed out that the situation was now ‘quite different’ from what it had been in the days of the old quantum theory (Bohr 1936). For this episode, see Kragh 1990, pp. 169–73.

(94.) Kramers to Urey, 16 July 1925, in Stolzenburg 1984, p. 86.

(95.) Slater 1925b.

(96.) Slater 1925a. Slater 1967, p. 8.

(97.) Slater 1925a, p. 398. On Bohr’s response to Slater’s paper, see Bohr to Slater, 10 January 1925, in Stolzenburg 1984, pp. 66–8.

(98.) Bohr 1925b, p. 156. See also Darrigol 1992, pp. 249–51, and Bohr’s correspondence with Fowler, Born, and Franck concerning collision and capture processes, as quoted in Stolzenburg 1984, pp. 71–4.

(99.) Pauli to Kramers, 27 July 1925, where he speaks of ‘the Scylla of the number-mystical Munich school and the Charybdis of the reactionary Copenhagen Putsch [coup or revolt]’ (Stolzenburg 1984, pp. 443–4). Pauli was at the time aware of Heisenberg’s manuscript that marked the beginning of a new quantum mechanics and which he praised in his letter to Kramers.

(100.) Heisenberg 1929, p. 492. According to Jammer 1966, p. 187, ‘It is hard to find in the history of physics a theory which was so soon disproved after its proposal and yet was so important for the future development of physical thought’ as the BKS theory. On the other hand, Duncan and Janssen (2007) argue that while the idea of virtual oscillators was important in the (p.361) development that led to quantum mechanics, the BKS theory played only a minor role. Significantly, in his 1951 historical review of the development of quantum theory, Sommerfeld did not even mention the BKS theory (Sommerfeld and Bopp 1951).

(101.) Dresden 1987, p. 244. Duncan and Janssen 2007, pp. 613–16.

(102.) For electromagnetic dispersion theories at the end of the nineteenth century and in the early twentieth century, see Buchwald 1985 and Loria 1914.

(103.) Bohr, ‘On the Application of the Quantum Theory to Periodic Systems’ (1916), in Hoyer 1981, p. 449. Bohr’s views of dispersion theories are surveyed in Konno 2000.

(104.) Ladenburg 1921, p. 451, translated in van der Waerden 1967, pp. 139–57. Duncan and Janssen (2007, p. 584) wrongly assert that the correspondence principle is not mentioned anywhere in Ladenburg’s paper. It appears in fact on p. 459.

(105.) Bohr, ‘Application of the Quantum Theory to Atomic Problems in General’ (1921), in Rud Nielsen 1976, p. 414. For Bohr’s appreciation of Ladenburg’s work, see also Bohr 1925a, p. 851.

(106.) Bohr 1924, p. 39.

(107.) Ladenburg and Reiche 1923. Ladenburg communicated his and Reiche’s results to Bohr in a letter of 14 June 1923 (Stolzenburg 1984, p. 400). Ladenburg’s two papers are analyzed in Konno 1993. See also interview with Reiche of 30 March 1962, by T. S. Kuhn and G. Uhlenbeck: http://www.aip.org/history/ohilist/4841_1.html. American Institute of Physics, Niels Bohr Library & Archives.

(108.) Slater to Van Vleck, 27 July 1924, quoted in Duncan and Janssen 2007, p. 588. See also Slater 1967, p. 7. The details of Kramers’ original reasoning, probably dating from November 1923, are not known, but Darrigol (1992, pp. 225–8) offers a reconstruction of it. See also Radder 1982a and Dresden 1987, pp. 150–9. In a detailed analysis of Kramers’ papers, Konno (1993) suggests that the BKS theory was in fact instrumental in his derivation of the dispersion formula.

(109.) Kramers 1924b, p. 674. The first paper, submitted on 25 March, was Kramers 1924a.

(110.) Van Vleck 1924, p. 344.

(111.) Breit 1924.

(112.) Kramers 1924b, p. 311.

(113.) Ladenburg to Kramers, 2 April 1924, in Duncan and Janssen 2007, p. 589.

(114.) Van Vleck 1924, the first part of which is reproduced in van der Waerden 1967, pp. 203–22 (quotation on p. 219). On Van Vleck’s contributions to radiation theory and their relation to Kramers’ theory of dispersion, see Duncan and Janssen 2007 and Konno 1993.

(115.) Van Vleck to Kramers, 22 September 1924, in Konno 1993, p. 146.

(116.) Heisenberg to Landé, 6 July 1924, quoted in Cassidy 1992, p. 180. Hendry (1984, p. 46) calls Kramers’s theory of 1924 ‘the Bohr–Kramers dispersion theory’.

(117.) Born 1924, pp. 386–7, with English translation in van der Waerden 1967, pp. 181–98.

(118.) Born 1924, p. 380.

(119.) Kramers 1924b. Dresden 1987, p. 222. Jammer 1966, p. 193, calls the translation formula ‘Born’s correspondence rule’.

(120.) Born 1925, pp. 113–14.

(121.) Kramers and Heisenberg 1925, with English translation in van der Waerden 1967, pp. 223–52. Smekal 1923. The Kramers–Heisenberg dispersion theory is discussed in Dresden 1987, pp. 215–22, and in Mehra and Rechenberg 1982b, pp. 170–90.

(122.) Kramers and Heisenberg 1925 (van der Waerden 1967, p. 234).

(p.362) (123.) Bohr 1932, p. 369. The memorial lecture, presented in the Kamerlingh Onnes Laboratory in Leiden, is reprinted in Aaserud 2007, pp. 355–60 (quotation on p. 358).

(124.) Conversation with Jagdish Mehra, quoted in Mehra and Rechenberg 1982b, p. 189. See also Born’s appreciation of Kramers’ (and Heisenberg’s) work on dispersion theory, according to whom ‘It was the first step from the bright realm of classical mechanics into the still dark and unexplored underworld of the new quantum mechanics’ (Born 1978, p. 216).

(125.) Raman 1928, who made his discovery while searching for an optical analogue of the Compton effect. For details about Raman and his work, see Singh 2002. Bohr was aware of the work done by Raman and his collaborators at the University of Calcutta. On Raman’s instigation, in 1924–1925 his student Bidhubnusan B. Ray did postdoctoral work in X-ray spectroscopy in Copenhagen. See Raman to Bohr, 21 March 1923, and Bohr to Raman, 18 May 1923 (Archive for History of Quantum Physics, Bohr Scientific Correspondence).

(126.) Heisenberg 1969, p. 87.

(127.) Eckert and Märker 2004, p. 157.

(128.) Pauli to Sommerfeld, 6 December 1924, in Eckert and Märker 2004, p. 177.

(129.) Coster 1923. Kramers 1923b. The figures were reproduced from Kramers and Holst 1923. See Section 7.1.

(130.) Born 1923, p. 542.

(131.) Pauli to Bohr, 12 December 1924, and Pauli to Kramers, 27 July 1925, in Stolzenburg 1984, p. 427 and p. 443. Kamerlingh Onnes 1924 contained pictures of atomic structures provided by Kramers. For Kamerlingh Onnes’s interest in atomic models, see Sauer 2007, p. 203. For Pauli’s view on Kramers as a believer in electron orbits, see also Pauli to Bohr, 21 February 1924, as quoted in Section 8.2.

(132.) For two different evaluations of the crisis in the old quantum theory, see Seth 2007 and Hentschel 2009. According to a classical study by Paul Forman from 1971, the very possibility of the crisis in the old quantum theory depended on the particular Zeitgeist that permeated German culture in the early 1920s and predisposed physicists toward a symbolic and acausal theory of the atom (Forman 1971). In Kragh 1999, pp. 153–4, I have listed some arguments against the strong version of Forman’s thesis in so far as it concerns the crisis in atomic theory in the years 1923–1925. As mentioned above, the reception of the BKS theory in Germany does not support the Forman thesis. On perspectives on Forman’s thesis, see Carson et al. 2011.

(133.) Heisenberg to Pauli, 21 June 1925, in Hermann et al. 1979, p. 219. In 1929, looking back at the development, Heisenberg remarked that the quantitative understanding of the hydrogen atom ‘appeared more and more as an accidental and incomprehensible result of the theory’ (Heisenberg 1929, p. 491).

(134.) Bohr to Oseen, 29 January 1926, in Stolzenburg 1984, pp. 238–9.

(135.) Sommerfeld 1929, p. v, who referred to both Schrödinger’s wave mechanics and Heisenberg’s quantum mechanics. For Sommerfeld’s relaxed attitude to the problems in atomic theory, see Sommerfeld 1924a. And for his emphasis on the need to base atomic theory on empirical regularities rather than models, Seth 2009.

(136.) For an early analysis of the decline of Bohr’s atomic theory, see Margenau and Wightman 1944.

(137.) The list is based on Kragh 2002b.

(138.) Russell 1927 (originally published 1923), p. 88, in connection with Sommerfeld’s relativistic theory of the hydrogen atom.

(p.363) (139.) Kragh 1985a. Vickers 2011 provides a philosophical perspective on Sommerfeld’s strikingly successful but nonetheless wrong theory.

(140.) Stern 1921, p. 252. English translation in Zeitschrift für Physik D 10 (1998), 114–16.

(141.) Gerlach and Stern 1922. For the experiments and their interpretations, see Mehra and Rechenberg 1982a, pp. 433–45, and Friedrich and Herschbach 1998 and 2003. Weinert 1994 analyzes the experiments from a philosophical perspective, suggesting that Stern and Gerlach ‘had done the right experiment, but on the wrong theory’.

(142.) Bohr to Gerlach, 18 February 1922 (Archive for History of Quantum Physics, Bohr Scientific Correspondence). Gerlach to Bohr, 8 February 1922, reproduced in Friedrich and Herschbach 2003. For Bohr’s view on the Stern–Gerlach effect, see Bohr 1924, p. 27.

(143.) Paschen to Gerlach 1922, quoted in Mehra and Rechenberg 1982a, p. 443. Karl Darrow concurred. Space quantization, he said in 1925, is ‘the most spectacular of all the remarkable consequences of Bohr’s interpretation of the stationary states, [and] also the only one which has ever been directly verified’ (Darrow 1925–1926, p. 684).

(144.) Einstein and Ehrenfest 1922, p. 34.

(145.) Van Vleck 1926, p. 230.

(146.) Pauling 1927, p. 145. See also Van Vleck 1932, pp. 113–15.

(147.) Van Vleck 1929, p. 490.

(148.) Richardson 1927. Yet another list of shortcomings in the old quantum theory appeared in Lindsay and Margenau 1936 (pp. 392–3), who mentioned many-electron atoms, dispersion, anomalous Zeeman effect, half-quanta, and the Ramsauer effect.

(149.) Bohr to Born, 9 December 1924, in Stolzenburg 1984, p. 73.

(150.) Bohr to Høffding 22 September 1922. The letter is reproduced in full in Faye 1991, p. 108.

(151.) Born 1925, p. 114. The role of the observability principle in the birth of quantum mechanics is discussed in, for example, Mehra and Rechenberg 1982b, pp. 273–90 and Camilleri 2009, pp. 17–31.

(152.) Born and Jordan 1925, p. 493.

(153.) Heisenberg to Pauli, 24 June 1925, in Hermann et al. 1979, p. 227. Mehra and Rechenberg 1982b, Camilleri 2009, Darrigol 1992, and MacKinnon 1977 all agree that the observability criterion did not play the crucial role in Heisenberg’s thinking that it is often ascribed and which Heisenberg himself expressed at some occasions.

(154.) Heisenberg to Pauli, 9 July 1925, in Hermann et al. 1979, p. 231.

(155.) Bridgman 1936, p. 65. Bridgman’s sceptical attitude with regard to admitting quantum mechanics as an operationalist theory is discussed in Walter 1990, pp. 203–7.

(156.) Reiche and Thomas 1925, p. 512, as pointed out in Duncan and Janssen 2007. Sommerfeld 1922b. For the Sommerfeld–Heisenberg similarity, see Cassidy 1979 and Seth 2009.

(157.) Heisenberg 1925, p. 880, translated in van der Waerden 1967, pp. 261–76. There are several good analyses of Heisenberg’s work, such as MacKinnon 1977, Darrigol 1992, pp. 263–76, and Mehra and Rechenberg 1982b, pp. 290–305. For those aspiring to understand the mathematical details of the paper, Aitchison et al. 2004 is a helpful source.

(158.) Darrigol 1997, p. 558.

(159.) Heisenberg 1967, p. 100.

(160.) Heisenberg to Bohr, 31 August 1925, in Stolzenburg 1984, p. 366.

(161.) Bohr 1925a. Manuscript on ‘Atomic Theory and Mechanics’, in Stolzenburg 1984, pp. 264–5.

## Notes:

(1.) Van Vleck 1926, p. 287.

(2.) Pauli 1946, p. 214. On the history of the anomalous Zeeman effect, see Forman 1970, Massimi 2005, pp. 47–52, and Mehra and Rechenberg 1982a, pp. 445–84.

(3.) Born 1923, p. 541. As early as 1920 Sommerfeld had called the effect one of the two unsettled questions of atomic physics’, but at that time the question was not considered a burning one. Although he thought it called for ‘entirely new things’, Sommerfeld did not consider it more serious than another unsettled question, the one of the polyhedral structure of the atom. See Sommerfeld 1920a and also Sommerfeld 1920b, where he characterized the anomalous Zeeman effect as a Zahlenmysterium (number mystery).

(4.) Landé to Bohr, 4 February 1921, reproduced in Forman 1970, p. 238.

(p.356) (5.) Landé to Sommerfeld, 17 March 1921. Sommerfeld to Landé, late March 1921. Both letters are reproduced in Forman 1970, pp. 260–1.

(6.) Landé 1921, p. 239.

(7.) Sommerfeld to Landé, 25 February 1921, in Mehra and Rechenberg 1982a, p. 470.

(8.) Sommerfeld to Einstein, 17 October 1921, in Hermann 1968, p. 94.

(9.) Landé 1923, p. 198.

(10.) Heisenberg to Pauli, 21 February 1923, and Pauli to Landé, 10 March 1923, in Hermann et al. 1979, p. 81 and p. 83.

(11.) Bohr 1922a, p. 59. Bohr was kept informed about the recent works on the anomalous Zeeman effect by letters from Landé, who in February 1921 reported his latest results (see Mehra and Rechenberg 1982a, p. 469).

(12.) Bohr to Landé, 15 May 1922 (Archive for History of Quantum Physics, Bohr Scientific Correspondence).

(13.) Bohr 1923c, in Rud Nielsen 1977, p. 629.

(14.) Ibid., p. 646. Cassidy (1978 and 1979, p. 222) suggests that Bohr was stimulated by a non-mechanical ‘mystery force’ that Van Vleck had previously introduced in his attempt to understand the helium atom. According to Van Vleck, such a hypothetical force, as proposed by Langmuir and a few others, would be negligible except at atomic distances and ‘not have a mechanism based on the Maxwell field-equations’ (Van Vleck 1922, p. 851). However, he did not advocate the idea and I know of no evidence that it stimulated Bohr’s thinking. For Langmuir’s mystery force, see Section 6.1.

(15.) Bohr 1925a, p. 850.

(16.) Bohr 1923c, in Rud Nielsen 1977, p. 647.

(17.) Rud Nielsen 1976, p. 558. The manuscript is in German.

(18.) Bohr to Landé, 14 February 1923 (Archive for History of Quantum Physics, Bohr Scientific Correspondence).

(19.) Bohr to Landé, 3 March 1923, quoted in Richter 1979, p. 34. See also Heisenberg to Sommerfeld, 4 January 1923, in Eckert and Märker 2004, p. 133.

(20.) Pauli to Landé, 23 May 1923, in Hermann et al. 1979, p. 90. Pauli also reported about Bohr’s views in his letter to Sommerfeld of 6 June 1923.

(21.) Pauli 1923, p. 162.

(22.) Pauli to Sommerfeld, 6 June 1923, in Hermann et al. 1979, p. 97.

(23.) Pauli to Landé, 17 August 1923, in ibid., p. 110.

(24.) Pauli to Sommerfeld, 19 July 1923, in ibid., p. 105.

(25.) Heisenberg 1922. On this paper, see Cassidy 1978 and, for a more detailed and scholarly version, Cassidy 1979.

(26.) McLennan 1923, p. 58, who relied on Rosseland 1923b. For k = 1, the minimum distance to the uranium nucleus was found to be 16 × 10–12 cm, hence permissible.

(27.) Heisenberg to Pauli, 19 November 1921, in Hermann et al. 1979, p. 38. Stressing its opportunistic character, Cassidy 1978 calls Heisenberg’s first core model ‘Machiavellian’.

(28.) ‘Heisenberg…is very unphilosophical, he does not care about a clear elaboration of the fundamental assumptions and of their relation to the previous theories’. Pauli hoped that Heisenberg, after having stayed with Bohr in Copenhagen, would ‘return home with a philosophical orientation to his thinking’. Pauli to Bohr, 11 February 1924, in Hermann et al. 1979, p. 143.

(p.357) (29.) Sommerfeld to Einstein, 11 January 1922, in Hermann 1968, p. 96. On Sommerfeld as a quantum craftsman, see Seth 2010. In the third edition of Atombau, Sommerfeld added an appendix on ‘the riddle of the anomalous Zeeman effect’ in which he expounded his own and Heisenberg’s ideas on the subject (Sommerfeld 1922a, pp. 497–504).

(30.) Pauli to Sommerfeld, 6 June 1923, in Hermann et al. 1979, p. 95, and similarly in his letter to Landé of 17 August. On Pauli’s negative attitude to the core model, see Serwer 1977.

(31.) Heisenberg to Pauli, 9 October 1923, in Hermann et al. 1979, p. 125. Heisenberg also gave an account of his new theory in a letter to Bohr of 22 December 1923 (Archive for History of Quantum Physics, Bohr Scientific Correspondence). In a letter of 7 December 1923, Heisenberg sent his manuscript on Zeemangemüse mit Quantensosse (Zeeman vegetables with quantum sauce) to Pauli, informing him that he was hoping for Bohr’s approval—‘papal blessing’. He did receive the blessing, and after some modifications the paper was published in August 1924.

(32.) Heisenberg to Sommerfeld, 8 December 1923, in Eckert and Märker 2004, p. 157. The symbol J denotes an action variable.

(33.) Heisenberg 1924, p. 292. See Darrigol 1992, pp. 196–200.

(34.) In Hermann et al. 1979, pp. 126–7.

(35.) Heisenberg to Bohr, 22 December 1923, as quoted in Cassidy 1992, p. 171. The reference is to Bohr 1923c, which appeared in a special issue of Annalen der Physik in honour of the seventieth birthday of the spectroscopist Heinrich Kayser. On Bohr’s invitation, Heisenberg spent some time in Copenhagen in March 1924, where they discussed the theory and its formulation, including its relation to radiation theory.

(36.) Letter of 29 November 1923, quoted in Cassidy 1992, p. 171.

(37.) Pauli to Landé, 14 December 1923, in Hermann et al. 1979, p. 134. On Pauli’s evaluation of Heisenberg’s theory, see also Pauli 1926a, pp. 237–9.

(38.) Pauli to Bohr, 21 February 1924, in Hermann et al. 1979, pp. 147–8.

(39.) Pauli to Eddington, 20 September 1923, in Hermann et al. 1979, p. 116.

(40.) Born 1925, p. 34. Born’s remarks in Atommechanik echoed the introduction to his important paper on ‘quantum mechanics’ from the summer of 1924 (Born 1924, see also Section 8.4).

(41.) Weyl to Pauli, 9 December 1919, and Pauli to Eddington, 20 September 1923, in Hermann et al. 1979, p. 6 and p. 116. Pauli 1919, pp. 749–50. In his Enzyklopädie article on relativity theory of 1921, Pauli expressed the same criticism, that the field strength at a given point in the interior of an electron is ‘unobservable, by definition, and thus fictitious and without physical meaning’ (Pauli 1958, p. 206). For Pauli’s operationalism and his early views concerning unified field theories, see Hendry 1984.

(42.) Born to Pauli, 23 December 1919, in Hermann et al. 1979, p. 10. See also Darrigol 1992, p. 196.

(43.) Interview with Heisenberg by T. S. Kuhn on 19 February 1963. American Institute of Physics, Niels Bohr Library & Archives. http://www.aip.org/history/ohilist/4661_6.html.

(44.) Einstein to Born, 27 January 1920, in Born 1971, p. 21. Einstein published his idea of reconciling quanta and fields, or rather deriving quanta from fields, in Einstein 1923. His question of how to produce discontinuous solutions from differential equations would be answered with Schrödinger’s wave mechanics, if not quite to Einstein’s satisfaction. Later in life, Einstein considered on various occasions the possibility of replacing the continuous field concept with an ‘algebraic physics’ based on discrete space-time, but nothing came out of these speculations. See Stachel 1993.

(p.358) (45.) Heisenberg to Bohr, 26 February 1930 (Archive for History of Quantum Physics, Bohr Scientific Correspondence), translated and analysed in Carazza and Kragh 1995. Some modern theories of quantum gravity, in particular the theory of loop quantum gravity developed since about 1990, assumes space to be built up of smallest cells.

(46.) Heisenberg to Bohr, 22 December 1923 (Archive for History of Quantum Physics, Bohr Scientific Correspondence). The connection also appeared in Heisenberg’s published paper (Heisenberg 1924, p. 299 and p. 307).

(47.) Bohr 1923b, p. 33.

(48.) Compton 1923, p. 501. Remarkably, he did not explicitly connect his X-ray quanta with Einstein’s theory of light quanta and in fact did not mention Einstein’s name in his paper. For details about Compton’s experiments and theoretical ideas, see Stuewer 1975.

(49.) Debye 1923. On the relationship between the works of Compton and Debye, see Stuewer 1975, pp. 234–7.

(50.) Stuewer 1975. Hendry 1981 disagrees that Compton’s discovery had a decisive effect on physicists’ view on the nature of electromagnetic radiation.

(51.) Kramers and Holst 1923, p. 175. The two authors stressed that Einstein’s theory of the light quantum ‘in no way has sprung from the Bohr theory, to say nothing of its being a necessary consequence of it’.

(52.) Bohr’s notes for the Silliman lectures are included in Rud Nielsen 1976, pp. 581–601. The lectures were covered by The New York Times, which gave summaries of all of them. ‘Likened to solar system, he pictures the atom with nucleus corresponding to sun, and electrons to planets’, it informed its readers on 7 November 1923. See also the summary account in Science 58 (1923), 459–60. Earlier Silliman lecturers included J. J. Thomson (1905) and Rutherford (1906).

(53.) Swann’s suggestion is mentioned in a footnote on p. 290 in Compton and Simon 1925. See also Stuewer 1975, p. 301.

(54.) Duane to Bohr, 5 March 1924, and Bohr to Duane, 17 May 1924, in Stolzenburg 1984, p. 323 and p. 326. Duane believed at the time that the Compton shift was caused by photoelectrons emitted from the K and L levels by the primary X-rays. See also Bohr to Rutherford, 9 January 1924, in ibid., p. 487.

(55.) Quoted from the extracts in Stolzenburg 1984, pp. 13–14. Darwin’s unpublished manuscript exists in at least one other version (Archive for History of Quantum Physics), which is quoted and discussed in Jammer 1966, pp. 171–2.

(56.) Bohr to Darwin, July 1919 (draft), in Stolzenburg 1984, p. 16.

(57.) Darwin 1923, p. 771 and Darwin 1922. On Darwin and his works in quantum physics, see Navarro 2009. See also Konno 2000 for Darwin’s dispersion theory.

(58.) Sommerfeld 1922a, p. 311. The passage did not appear in the first edition of 1919.

(59.) Bohr to Darwin, 21 December 1922, in Stolzenburg 1984, p. 316. Darwin, in a letter to Bohr of 23 November 1922, had ‘left it open whether the upper states really are stationary, but in private [I] may say that I personally do not believe they are’ (ibid., p. 17).

(60.) Bohr 1924, p. 40, who referred to experiments reported in Gans and Miguez 1917.

(61.) Slater 1967, p. 6.

(62.) Bohr to Slater, 10 January 1925, in Stolzenburg 1984, p. 67.

(63.) Slater to Kramers, 8 December 1923, in Stolzenburg 1984, p. 492. See also Hendry 1981. Slater further explained that his ideas might explain the helium spectrum, if only ‘by making a whole lot of guesses’.

(p.359) (64.) Manuscript of 4 November 1923, quoted in Petruccioli 1993, p. 113. On the development of Slater’s views, see Konno 1983.

(65.) Slater 1967, p. 6.

(66.) Slater 1925b.

(67.) Slater 1975, pp. 12–14.

(68.) Slater 1923. Ladenburg and Reiche 1923, p. 590, who introduced classical Ersatzoszillatoren (substitute oscillators) in dispersion theory.

(69.) Bohr et al. 1924. The theory has attracted considerable historical attention. Apart from Hendry 1981 and Petruccioli 1993, see also Stolzenburg 1984, Duncan and Janssen 2007, pp. 597–617, and Dresden 1987, pp. 41–78, 159–215, who deals in particular with Kramers’ contribution to and view of the theory.

(70.) Bohr to Slater, 10 January 1925, quoted in Stuewer 1975, p. 293. On Bohr’s coupling principle and latent forces, see Bohr 1924, p. 36.

(71.) Bohr to Michelson, 7 February 1924, in Stolzenburg 1984, p. 405. See also Livingston 1973, pp. 301–2, according to whom (p. 292) Michelson had met Bohr at the 1921 Solvay congress, where he, ‘feeling somewhat lost’, listened to Bohr lecturing on atomic theory. However, Bohr did not attend the Solvay conference this year and he only met with Michelson during his trip to the United States in the autumn of 1923.

(72.) Bohr et al. 1924, p. 786.

(73.) Ibid. The treatise referred to was Bohr 1924. The BKS paper is reprinted in van der Waerden 1967, pp. 159–76.

(74.) Bohr et al. 1924, p. 795. Bohr 1924, pp. 29–30. The emphasis on the limited definability of motion and energy would later play an important role in Bohr’s complementarity philosophy. For this connection, see Tanona 2004.

(75.) Bohr et al. 1924, pp. 790–1. The BKS theory maintained statistical energy conservation. In 1929 Bohr went further, proposing that energy and momentum might not be conserved even on a macroscopic level. This much more radical hypothesis he thought justified in order to explain the continuous beta spectrum and also stellar energy production. See Jensen 2000, pp. 146–56.

(76.) Bohr et al. 1924, p. 799.

(77.) Kramers and Heisenberg 1925, p. 685.

(78.) Kramers and Holst 1925, pp. 123–40. The preface of the book was dated March 1925, the new chapter on the interaction of light and matter, including the BKS theory, being the work of Kramers. The chapter also appeared separately, as a paper in a Danish physics journal (Kramers 1925). On Kramers’ view of causality in relation to the BKS theory, see Radder 1983 and Dresden 1987, pp. 191–5.

(79.) Haber to Einstein, 1924, quoted in Stolzenburg 1984, p. 26.

(80.) Schrödinger to Bohr, 24 May 1924, in Stolzenburg 1984, p. 490. On Exner’s ideas of indeterminism and acausality, and the way they influenced Schrödinger’s thinking and response to the BKS theory, see Hanle 1979.

(81.) Schrödinger 1924, p. 724.

(82.) According to Paul Forman, it is ‘only by reference to the widespread acausal sentiment that one can understand the immediate and widespread assent which the [BKS] theory received in Germany’ (Forman 1971, p. 99). However, there was no such widespread assent among German physicists.

(83.) Pauli to Bohr, 21 February 1924, in Hermann et al. 1979, p. 147.

(p.360) (84.) Einstein to Born, 27 January 1920, and Einstein to Hedwig and Max Born, 29 April 1924, in Born 1971, p. 23 and p. 82. Einstein also objected to the theory by means of more technical arguments based on thermodynamics. See Stolzenburg 1984, pp. 23–38, where the reactions of Einstein and other physicists are recounted.

(85.) Einstein to Ehrenfest, 31 May 1924, quoted in Stolzenburg 1984, p. 27.

(86.) Bohr to Born, 1 May 1925, a reply to Born’s letter of 24 April, both in Stolzenburg 1984, pp. 308–11.

(87.) Bothe and Geiger 1925.

(88.) Compton and Simon 1925, p. 299.

(89.) Einstein’s manuscript, dated 7 May 1925, is reproduced in Tolmasquim and Moreira 2002, pp. 240–2. The final results of the Bothe–Geiger experiments were published in the 15 May issue of Naturwissenschaften with a detailed report following in June in Zeitschrift für Physik.

(90.) On this question and the later philosophical interest in it, see the appendix (Chapter 9).

(91.) Bohr to Fowler, 21 April 1925, in Stolzenburg 1984, p. 82.

(92.) The name ‘photon’ was proposed by G. N. Lewis in December 1926, but in a sense widely different from Einstein’s light quantum. Lewis’s photon was a ‘hypothetical new atom, which is not light but plays an essential part in every process of radiation’; also contrary to the light quanta, his photons were conserved quantities, ‘uncreatable and indestructable’ (Lewis 1926b; see also Stuewer 1975, pp. 323–6). In spite of the speculative nature of Lewis’s ideas of radiation, Bohr found them interesting. See Bohr to Richardson, 16 February 1926, Archive for History of Quantum Physics.

(93.) Bohr to Geiger, 21 April 1925, in Stolzenburg 1984, p. 79. Much later, in 1936, Paul Dirac advocated a reintroduction of a BKS-like theory in order to explain experiments by Robert Shankland that seemed to imply energy nonconservation in individual atomic processes. Interestingly, in this case Bohr argued against Dirac’s proposal, while Einstein was in favour of it. Referring to the old BKS theory, Bohr pointed out that the situation was now ‘quite different’ from what it had been in the days of the old quantum theory (Bohr 1936). For this episode, see Kragh 1990, pp. 169–73.

(94.) Kramers to Urey, 16 July 1925, in Stolzenburg 1984, p. 86.

(95.) Slater 1925b.

(96.) Slater 1925a. Slater 1967, p. 8.

(97.) Slater 1925a, p. 398. On Bohr’s response to Slater’s paper, see Bohr to Slater, 10 January 1925, in Stolzenburg 1984, pp. 66–8.

(98.) Bohr 1925b, p. 156. See also Darrigol 1992, pp. 249–51, and Bohr’s correspondence with Fowler, Born, and Franck concerning collision and capture processes, as quoted in Stolzenburg 1984, pp. 71–4.

(99.) Pauli to Kramers, 27 July 1925, where he speaks of ‘the Scylla of the number-mystical Munich school and the Charybdis of the reactionary Copenhagen Putsch [coup or revolt]’ (Stolzenburg 1984, pp. 443–4). Pauli was at the time aware of Heisenberg’s manuscript that marked the beginning of a new quantum mechanics and which he praised in his letter to Kramers.

(100.) Heisenberg 1929, p. 492. According to Jammer 1966, p. 187, ‘It is hard to find in the history of physics a theory which was so soon disproved after its proposal and yet was so important for the future development of physical thought’ as the BKS theory. On the other hand, Duncan and Janssen (2007) argue that while the idea of virtual oscillators was important in the (p.361) development that led to quantum mechanics, the BKS theory played only a minor role. Significantly, in his 1951 historical review of the development of quantum theory, Sommerfeld did not even mention the BKS theory (Sommerfeld and Bopp 1951).

(101.) Dresden 1987, p. 244. Duncan and Janssen 2007, pp. 613–16.

(102.) For electromagnetic dispersion theories at the end of the nineteenth century and in the early twentieth century, see Buchwald 1985 and Loria 1914.

(103.) Bohr, ‘On the Application of the Quantum Theory to Periodic Systems’ (1916), in Hoyer 1981, p. 449. Bohr’s views of dispersion theories are surveyed in Konno 2000.

(104.) Ladenburg 1921, p. 451, translated in van der Waerden 1967, pp. 139–57. Duncan and Janssen (2007, p. 584) wrongly assert that the correspondence principle is not mentioned anywhere in Ladenburg’s paper. It appears in fact on p. 459.

(105.) Bohr, ‘Application of the Quantum Theory to Atomic Problems in General’ (1921), in Rud Nielsen 1976, p. 414. For Bohr’s appreciation of Ladenburg’s work, see also Bohr 1925a, p. 851.

(106.) Bohr 1924, p. 39.

(107.) Ladenburg and Reiche 1923. Ladenburg communicated his and Reiche’s results to Bohr in a letter of 14 June 1923 (Stolzenburg 1984, p. 400). Ladenburg’s two papers are analyzed in Konno 1993. See also interview with Reiche of 30 March 1962, by T. S. Kuhn and G. Uhlenbeck: http://www.aip.org/history/ohilist/4841_1.html. American Institute of Physics, Niels Bohr Library & Archives.

(108.) Slater to Van Vleck, 27 July 1924, quoted in Duncan and Janssen 2007, p. 588. See also Slater 1967, p. 7. The details of Kramers’ original reasoning, probably dating from November 1923, are not known, but Darrigol (1992, pp. 225–8) offers a reconstruction of it. See also Radder 1982a and Dresden 1987, pp. 150–9. In a detailed analysis of Kramers’ papers, Konno (1993) suggests that the BKS theory was in fact instrumental in his derivation of the dispersion formula.

(109.) Kramers 1924b, p. 674. The first paper, submitted on 25 March, was Kramers 1924a.

(110.) Van Vleck 1924, p. 344.

(111.) Breit 1924.

(112.) Kramers 1924b, p. 311.

(113.) Ladenburg to Kramers, 2 April 1924, in Duncan and Janssen 2007, p. 589.

(114.) Van Vleck 1924, the first part of which is reproduced in van der Waerden 1967, pp. 203–22 (quotation on p. 219). On Van Vleck’s contributions to radiation theory and their relation to Kramers’ theory of dispersion, see Duncan and Janssen 2007 and Konno 1993.

(115.) Van Vleck to Kramers, 22 September 1924, in Konno 1993, p. 146.

(116.) Heisenberg to Landé, 6 July 1924, quoted in Cassidy 1992, p. 180. Hendry (1984, p. 46) calls Kramers’s theory of 1924 ‘the Bohr–Kramers dispersion theory’.

(117.) Born 1924, pp. 386–7, with English translation in van der Waerden 1967, pp. 181–98.

(118.) Born 1924, p. 380.

(119.) Kramers 1924b. Dresden 1987, p. 222. Jammer 1966, p. 193, calls the translation formula ‘Born’s correspondence rule’.

(120.) Born 1925, pp. 113–14.

(121.) Kramers and Heisenberg 1925, with English translation in van der Waerden 1967, pp. 223–52. Smekal 1923. The Kramers–Heisenberg dispersion theory is discussed in Dresden 1987, pp. 215–22, and in Mehra and Rechenberg 1982b, pp. 170–90.

(122.) Kramers and Heisenberg 1925 (van der Waerden 1967, p. 234).

(p.362) (123.) Bohr 1932, p. 369. The memorial lecture, presented in the Kamerlingh Onnes Laboratory in Leiden, is reprinted in Aaserud 2007, pp. 355–60 (quotation on p. 358).

(124.) Conversation with Jagdish Mehra, quoted in Mehra and Rechenberg 1982b, p. 189. See also Born’s appreciation of Kramers’ (and Heisenberg’s) work on dispersion theory, according to whom ‘It was the first step from the bright realm of classical mechanics into the still dark and unexplored underworld of the new quantum mechanics’ (Born 1978, p. 216).

(125.) Raman 1928, who made his discovery while searching for an optical analogue of the Compton effect. For details about Raman and his work, see Singh 2002. Bohr was aware of the work done by Raman and his collaborators at the University of Calcutta. On Raman’s instigation, in 1924–1925 his student Bidhubnusan B. Ray did postdoctoral work in X-ray spectroscopy in Copenhagen. See Raman to Bohr, 21 March 1923, and Bohr to Raman, 18 May 1923 (Archive for History of Quantum Physics, Bohr Scientific Correspondence).

(126.) Heisenberg 1969, p. 87.

(127.) Eckert and Märker 2004, p. 157.

(128.) Pauli to Sommerfeld, 6 December 1924, in Eckert and Märker 2004, p. 177.

(129.) Coster 1923. Kramers 1923b. The figures were reproduced from Kramers and Holst 1923. See Section 7.1.

(130.) Born 1923, p. 542.

(131.) Pauli to Bohr, 12 December 1924, and Pauli to Kramers, 27 July 1925, in Stolzenburg 1984, p. 427 and p. 443. Kamerlingh Onnes 1924 contained pictures of atomic structures provided by Kramers. For Kamerlingh Onnes’s interest in atomic models, see Sauer 2007, p. 203. For Pauli’s view on Kramers as a believer in electron orbits, see also Pauli to Bohr, 21 February 1924, as quoted in Section 8.2.

(132.) For two different evaluations of the crisis in the old quantum theory, see Seth 2007 and Hentschel 2009. According to a classical study by Paul Forman from 1971, the very possibility of the crisis in the old quantum theory depended on the particular Zeitgeist that permeated German culture in the early 1920s and predisposed physicists toward a symbolic and acausal theory of the atom (Forman 1971). In Kragh 1999, pp. 153–4, I have listed some arguments against the strong version of Forman’s thesis in so far as it concerns the crisis in atomic theory in the years 1923–1925. As mentioned above, the reception of the BKS theory in Germany does not support the Forman thesis. On perspectives on Forman’s thesis, see Carson et al. 2011.

(133.) Heisenberg to Pauli, 21 June 1925, in Hermann et al. 1979, p. 219. In 1929, looking back at the development, Heisenberg remarked that the quantitative understanding of the hydrogen atom ‘appeared more and more as an accidental and incomprehensible result of the theory’ (Heisenberg 1929, p. 491).

(134.) Bohr to Oseen, 29 January 1926, in Stolzenburg 1984, pp. 238–9.

(135.) Sommerfeld 1929, p. v, who referred to both Schrödinger’s wave mechanics and Heisenberg’s quantum mechanics. For Sommerfeld’s relaxed attitude to the problems in atomic theory, see Sommerfeld 1924a. And for his emphasis on the need to base atomic theory on empirical regularities rather than models, Seth 2009.

(136.) For an early analysis of the decline of Bohr’s atomic theory, see Margenau and Wightman 1944.

(137.) The list is based on Kragh 2002b.

(138.) Russell 1927 (originally published 1923), p. 88, in connection with Sommerfeld’s relativistic theory of the hydrogen atom.

(p.363) (139.) Kragh 1985a. Vickers 2011 provides a philosophical perspective on Sommerfeld’s strikingly successful but nonetheless wrong theory.

(140.) Stern 1921, p. 252. English translation in Zeitschrift für Physik D 10 (1998), 114–16.

(141.) Gerlach and Stern 1922. For the experiments and their interpretations, see Mehra and Rechenberg 1982a, pp. 433–45, and Friedrich and Herschbach 1998 and 2003. Weinert 1994 analyzes the experiments from a philosophical perspective, suggesting that Stern and Gerlach ‘had done the right experiment, but on the wrong theory’.

(142.) Bohr to Gerlach, 18 February 1922 (Archive for History of Quantum Physics, Bohr Scientific Correspondence). Gerlach to Bohr, 8 February 1922, reproduced in Friedrich and Herschbach 2003. For Bohr’s view on the Stern–Gerlach effect, see Bohr 1924, p. 27.

(143.) Paschen to Gerlach 1922, quoted in Mehra and Rechenberg 1982a, p. 443. Karl Darrow concurred. Space quantization, he said in 1925, is ‘the most spectacular of all the remarkable consequences of Bohr’s interpretation of the stationary states, [and] also the only one which has ever been directly verified’ (Darrow 19251926, p. 684).

(144.) Einstein and Ehrenfest 1922, p. 34.

(145.) Van Vleck 1926, p. 230.

(146.) Pauling 1927, p. 145. See also Van Vleck 1932, pp. 113–15.

(147.) Van Vleck 1929, p. 490.

(148.) Richardson 1927. Yet another list of shortcomings in the old quantum theory appeared in Lindsay and Margenau 1936 (pp. 392–3), who mentioned many-electron atoms, dispersion, anomalous Zeeman effect, half-quanta, and the Ramsauer effect.

(149.) Bohr to Born, 9 December 1924, in Stolzenburg 1984, p. 73.

(150.) Bohr to Høffding 22 September 1922. The letter is reproduced in full in Faye 1991, p. 108.

(151.) Born 1925, p. 114. The role of the observability principle in the birth of quantum mechanics is discussed in, for example, Mehra and Rechenberg 1982b, pp. 273–90 and Camilleri 2009, pp. 17–31.

(152.) Born and Jordan 1925, p. 493.

(153.) Heisenberg to Pauli, 24 June 1925, in Hermann et al. 1979, p. 227. Mehra and Rechenberg 1982b, Camilleri 2009, Darrigol 1992, and MacKinnon 1977 all agree that the observability criterion did not play the crucial role in Heisenberg’s thinking that it is often ascribed and which Heisenberg himself expressed at some occasions.

(154.) Heisenberg to Pauli, 9 July 1925, in Hermann et al. 1979, p. 231.

(155.) Bridgman 1936, p. 65. Bridgman’s sceptical attitude with regard to admitting quantum mechanics as an operationalist theory is discussed in Walter 1990, pp. 203–7.

(156.) Reiche and Thomas 1925, p. 512, as pointed out in Duncan and Janssen 2007. Sommerfeld 1922b. For the Sommerfeld–Heisenberg similarity, see Cassidy 1979 and Seth 2009.

(157.) Heisenberg 1925, p. 880, translated in van der Waerden 1967, pp. 261–76. There are several good analyses of Heisenberg’s work, such as MacKinnon 1977, Darrigol 1992, pp. 263–76, and Mehra and Rechenberg 1982b, pp. 290–305. For those aspiring to understand the mathematical details of the paper, Aitchison et al. 2004 is a helpful source.

(158.) Darrigol 1997, p. 558.

(159.) Heisenberg 1967, p. 100.

(160.) Heisenberg to Bohr, 31 August 1925, in Stolzenburg 1984, p. 366.

(161.) Bohr 1925a. Manuscript on ‘Atomic Theory and Mechanics’, in Stolzenburg 1984, pp. 264–5.