Abstract
This chapter discusses the work of Niels Bohr and Hendrik Antoon Kramers on the quantum theory of the atom in relation to the correspondence principle. Within the main argument of this book, this chapter plays a pivotal role. It presents a historical analysis of the genesis of the principle and provides the baseline for the historical analysis in the following chapters. Following unpublished as well as published papers, it analyzes the origin, the formulation, and the consolidation of the correspondence principle from 1913 until 1923 and puts a special emphasis on the role of physical problems for the development of the principle.
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Notes
- 1.
This analysis builds on the extensive scholarship on the history of Bohr’s quantum theory of the atom in general given by Jammer (1966), Heilbron (1964), and Heilbron and Kuhn (1969) as well as the correspondence principle, which was discussed by Darrigol (1992), Assmus (1990, 1992b), Tanona (2002) and Bokulich (2009).
- 2.
This section follows the work of John Heilbron and Thomas S. Kuhn, Rud Nielsen and Olivier Darrigol (see Footnote 1). A previous version has been published as Jähnert (2013).
- 3.
- 4.
- 5.
Bohr (1913a).
- 6.
For the discussion of the 1913 trilogy see Heilbron (1964, 268–294), Jammer (1966, 70–88), Heilbron and Kuhn (1969, especially 266–283) and Darrigol (1992, 85–89). The quote given in the main text is taken from Heilbron and Kuhn (1969, footnote 98 on page 251); for other identifications of the correspondence idea in the trilogy see Jammer (1966, 80) and Darrigol (1992, 89).
- 7.
Heilbron and Kuhn (1969, 251).
- 8.
Heilbron and Kuhn (1969). In general, Heilbron and Kuhn have reconstructed the following pathway that led Bohr to his model: After working on his dissertation on the electron theory of metals, Bohr focused on the problem of mechanical stability in the Rutherford atom and developed a first quantum theoretical description of the atom. This description, as Heilbron and Kuhn have emphasized, focused on the idea that there are certain states of the atom which are assumed to be mechanically stable by definition. They are selected by a quantum condition which does not involve assumptions about radiation or even transitions between states. Instead Bohr assumed a constant ratio of the kinetic energy of the atom and its mechanical frequency. Bohr only introduced the idea of transitions between stationary states in reaction to the work of J. W. Nicolson, which led him to incorporate the Balmer formula into his considerations. Working his way backwards from the empirical Balmer formula, Heilbron and Kuhn have suggested, Bohr introduced the idea that radiation was emitted during transitions between stationary states, which he now reinterpreted as stable because they were radiation-free.
- 9.
Heilbron and Kuhn (1969, 251).
- 10.
Bohr (1913a, 4–5).
- 11.
Bohr (1913a, 15).
- 12.
The frequency of radiation is then: \( \nu =\frac {\pi ^2 me^2E^2}{2c^2h^3}\frac {2N-1}{N^2(N-1)^2}\) . For the orbiting frequency before and after emission, Bohr writes: \( \omega _N=\frac {\pi ^2 me^2E^2}{2c^2h^3N^3}~\text{and}~\omega _{N-1}=\frac {\pi ^2 me^2E^2}{2c^2h^3(N-1)^3}\) .
- 13.
Bohr (1913a, 13).
- 14.
Bohr (1913a, 14).
- 15.
- 16.
- 17.
The second one, Bohr thought, needed to be invoked for the Zeeman effect. As Darrigol (1992, 92) has pointed out, Bohr assumed that the frequencies of the Zeeman effect did not obey Ritz’s combination principle and that the magnetic field did not change the energy levels of the atom. Following these assumptions, Bohr thought, the only possibility to account for the Zeeman effect was to change the frequency condition and to assume that the energy of the transition was associated with different frequencies. These arguments did not involve the connection between radiation and motion and will therefore not be discussed in the following.
- 18.
Bohr to Rutherford, 31 December 1913 in Bohr (1981, 591).
- 19.
He took the reciprocal \(\frac {dn}{dA_n}\) and multiplied it by \(\frac {dA_n}{da}\), yielding:
$$\displaystyle \begin{aligned} \frac{dn}{dA_n}\frac{dA_n}{da}=\frac{dn}{da}=\frac{1}{h\omega_n}\frac{dA_n}{da}. \end{aligned}$$Calculating the derivative and introducing the expression for ω the right-hand side of the equation can be evaluated easily as:
$$\displaystyle \begin{aligned}\frac{dn}{da}=\frac{\pi e \sqrt{m}}{h\sqrt{a}}\frac{\left(1\mp4E\frac{a^2}{e}\right)}{\sqrt{\left(1\mp3E\frac{a^2}{e}\right)}}. \end{aligned}$$A Taylor expansion of the square root term then yields the given formula, which Bohr used for the quantization of his model. In the limiting case of a weak electric field, Bohr obtained his quantization condition by neglecting higher powers of a beyond the quadratic term:
$$\displaystyle \begin{aligned} \frac{dn}{da}=\frac{\pi e \sqrt{m}}{h\sqrt{a}}\left(1\mp\frac{5}{2}E\frac{a^2}{e}\right) \end{aligned}$$Integrating this equation leads to an expression of the major axis a in terms of n, so that Bohr could reexpress the energy as a function of the quantum number n.
- 20.
Bohr (1913b, 515–516). To justify this constraint, Bohr argued that the continuous classical electrodynamics and quantum theory should yield the same results in the limit of large quantum numbers. As it was implausible in classical radiation theory that an electron with the energy E = E Balmer + E Stark turn into an energy E = E Balmer − E Stark, Bohr argued, this should also be impossible in quantum theory.
- 21.
Bohr used the relation \(\overline {T}/{\omega }=\oint Tdt=1/2 nh\), where \(\overline {T}\), ω and n are the mean kinetic energy, the mechanical frequency and the associated quantum number, respectively. Later he adopted the Sommerfeld quantum condition: \(J=\oint pdq=nh\), where J is the action determined from the phase integral of the generalized momentum p, conjugated to a particular generalized coordinate q.
- 22.
Heilbron and Kuhn (1969, 279–280).
- 23.
Darrigol (1992, 123). Identifying the issue of selection rules as the starting point for Bohr’s formulation of the correspondence principle, Darrigol has argued that Bohr formulated “the first systematic generalization of the correspondence idea” in his discussion of the Stark and Zeeman effect. Although Bohr’s earlier arguments certainly qualify as less systematic and are not as clearly directed towards the formulation of selection rules, Assmus shows that Bohr’s arguments on the harmonic oscillator and rotator not only predated these considerations, but also presented Bohr’s first explanation of selection rules from the harmonic character of the motion in a stationary state.
- 24.
Assmus (1992b).
- 25.
Assmus (1992b, 229).
- 26.
Schwarzschild (1916, 568). “Es wird also hier die Anschauung nahegelegt, daß dieselben Rotationen des Moleküls, deren Frequenzen im Ultrarot unmittelbar in äquidistanten Absorptionsstreifen zum Ausdruck kommen, im Ultraviolett nach dem BOHRschen Ansatz wirkend Banden gemäß dem quadratischen DESLANDRESschen Gesetz erzeugen.”
- 27.
Assmus (1992b, 228). Though this interpretation appears to be plausible, it is based on a tentative and incorrect translation of a small comment at the end of Schwarzschild’s paper. Assmus translates the German key phrases “unmittelbar […] zum Ausdruck kommen” as “revealed directly,” creating the impression that Schwarzschild explicitly saw the rotations as producing radiation according to the classical radiation mechanism. Moreover, she mistranslates “nach dem Bohrschen Ansatz wirkend” as “according to the Bohr condition, effective bands.” As “wirkend” is not an adjective for bands but an adverb describing the Bohrian approach, it should be translated as “acting according to Bohr’s approach.”
- 28.
Schwarzschild to Sommerfeld, 21 March 1916 in Sommerfeld (2000, 543–544). “Ein Versuch zu den Bandenspektren: Elektronen umkreisen ein rotierendes Molekül vom Trägheitsmoment J. Energie der Elektronenbewegung A, der Rotation nach Planck \(\frac {h^2}{8\pi ^2J}n^2 (n=,1,2,3\ldots ).\) Daraus [folgt] nach Bohr die Frequenzserie:
$$\displaystyle \begin{aligned} \nu=\frac{A}{h}+\frac{h^2}{8\pi^2J}n^2 \end{aligned}$$Das ist die Deslandres’-Formel.”
- 29.
Sommerfeld to Schwarzschild, 24 March 1916 in Sommerfeld (2000, 545). Without going into details, Sommerfeld only made a short remark on Schwarzschild’s band spectrum: “Auf die Entwicklung ihres Banden-Spektr[ums] bin ich gespannt.”
- 30.
Sommerfeld to Schwarzschild, 29 March 1916 Sommerfeld (2000, 546). “Ihre Bandenformel ist ja sehr merkwürdig. Hier nehmen Sie, soviel ich sehe, die ganze Energie zur Bestimmung des ν, nicht den Energie-Übergang aus einer Bahn in eine andere. Ist das nicht vom Bohr’schen Standpunkt eine Inconsequenz?”
- 31.
Schwarzschild probably did so while proofreading his paper, which he sent one day after Sommerfeld’s letter.
- 32.
Burgers’ major contribution to the developing quantum theory of multiply periodic systems was to show that action-angle variables of a non-degenerate system were adiabatic invariants just like the energy of a system.
- 33.
Burgers (1917, 170–171).
- 34.
Burgers (1917, 171).
- 35.
Burgers’ work published in the Proceedings of the Royal Academy of Amsterdam during the war was cited only by Torsten Heurlinger (Heurlinger 1920).
- 36.
Burgers’ solution to the problem was to include the interaction between rotation and electronic motion in Schwarzschild’s quantization procedure. In this way, he found an additional term in the energy expression, depending linearly on the quantum number for the rotation n 4. With this new term, the frequency condition gave an additional term in the series formula and offered the possibility of recovering the equidistance of Bjerrum’s theory on the condition, which he did not interpret physically, that the quadratic term vanishes for transitions between certain initial and final states. Sommerfeld restricted the possibilities of transitions by imposing what he called “quantum inequalities.” According to these additional relations, each quantum number could only decrease during a transition.
- 37.
Bohr (1981, 433). With these presentations, Darrigol has argued, Bohr hoped to grasp the conceptual challenges for the further development of quantum theory.
- 38.
Bohr (1981, 433–461).
- 39.
See Nielsen (1976) for an overview of the publishing history and development of Bohr’s grand treatises.
- 40.
Bohr (1981, 433).
- 41.
Bohr (1981, 443).
- 42.
Bohr (1981, 443).
- 43.
- 44.
Ibid.
- 45.
Bohr (1981, 445).
- 46.
Ibid.
- 47.
Bohr (1981, 447).
- 48.
Note that Bohr tacitly assumed that the moment of inertia I was the same in both the initial and the final state.
- 49.
Bohr (1981, 448, my emphasis).
- 50.
Bohr (1981, 448).
- 51.
- 52.
Bohr (1976, 48–52).
- 53.
Bohr (1976, 50–51, emphasis MJ). See Darrigol (1992, 123–125) for his explanation of the selection rule argument. Darrigol identified this argument as the first systematic exposition of a selection rule in Bohr’s work and hinted at the possibility that Bohr might have remembered his earlier restrictions of possible transitions from 1915. Bohr’s explanation of selection rules was clearly different from his early arguments. In his admittedly preliminary account, Bohr had limited transitions for the Stark effect on the assumption that there were different series of stationary states between which no transitions occurred. The conception that the orbit of an atom in an electric field performed a precession within its plane, and by extension the harmonic character of such a motion, did not play a role. Bohr’s argument on the restriction of transitions became possible only with the new approach, which allowed him to quantize different motions separately and thereby to impose restrictions on each one independently.
- 54.
Bohr (1976, 49, annotations in the edition).
- 55.
Ibid.
- 56.
Ibid.
- 57.
Einstein (1916).
- 58.
Darrigol (1992, 126–127).
- 59.
Bohr (1918a).
- 60.
Bohr (1918a, 8).
- 61.
Bohr to Sommerfeld, 27 July 1919 in Bohr (1976, 689). “Ich leide aber so sehr von Schwierigkeiten Abhandlungen in befriedender Form zu bringen und von einem unglücklichen Hand alle Resultate in systematischer Reihenfolge erscheinen zu lassen.”
- 62.
Whereas this equation can be obtained from Hamilton-Jacobi theory in action-angle-variables, Bohr derived it from Ehrenfest’s theorem in Paragraph 2 of his paper. Bohr immediately marked the importance of the “equation […] which will be often used in the following.” Bohr (1918a, 12). As Darrigol has pointed out, the rule and its generalization to systems of s degrees of freedom establishes the unambiguity of the quantum condition and other important properties of multiply periodic systems. See Darrigol (1992, 115).
- 63.
Bohr (1918a, 15).
- 64.
Bohr (1918a, 15–16. emphasis in the original).
- 65.
Bohr (1918a, 15–16).
- 66.
Bokulich (2009, 1). Bokulich has identified this argument as the core of the principle, but did not comment on the context in which Bohr developed his correspondence idea. For a short discussion of the problem of selection rules before Bohr’s correspondence principle and the relation to the principle, see Darrigol (1992, 123).
- 67.
Bohr (1918a, 16).
- 68.
Bohr (1918a, 16).
- 69.
Bohr (1918a, 32–35). For a discussion, see Darrigol (1992, 128–132), who argues that this last conclusion was based on Bohr’s and Kramers’ new way of dealing with perturbation theory. It allowed them to circumvent the problems of the perturbation techniques Sommerfeld, Schwarzschild, and Epstein had used, most of which were due to the fact that a perturbed system was no longer separable and hence not solvable in Hamilton-Jacobi theory.
- 70.
Bohr (1920, 427, emphasis in the original). “Ferner, obgleich es unmöglich ist, den Strahlungsvorgang, mit welchem ein Übergang zwischen zwei stationären Zuständen verbunden ist, in Einzelheiten zu verfolgen mit Hilfe der gewöhnlichen elektromagnetischen Vorstellungen, nach welchen die Beschaffenheit einer von einem Atom ausgesandten Strahlung direkt von der Bewegung des Systems und von ihrer Auflösung in harmonische Komponenten bedingt ist, hat es sich nichtsdestoweniger gezeigt, dass zwischen den verschiedenen Typen der möglichen Übergänge zwischen diesen Zuständen einerseits und den verschiedenen harmonischen Komponenten, in welche die Bewegung des Systems zerlegbar ist, andererseits eine weitgehende Korrespondenz stattfindet.”
- 71.
The historio-critical analysis would usually be part of Bohr’s introduction, in which he reviewed quantum theory and emphasized the results on which the correspondence principle could be based. See for example Bohr (1920, 424–427). The physical derivation would involve a shortened form of the argument made in “The Quantum Theory of Line Spectra”; see Bohr (1920, 430–432).
- 72.
Bohr (1923b, 142). “[Die] Möglichkeit des Auftretens eines von Strahlung begleiteten Übergangs zwischen zwei stationären Zuständen eines mehrfach periodischen Systems, deren Quantenzahlen bzw. gleich \(n^\prime _1\ldots n^\prime _u\) und \(n^{\prime \prime }_1\ldots n^{\prime \prime }_u\) sind, als bedingt ansehen von der Gegenwart derjenigen harmonischen Schwingungskomponente in dem durch (2) gegeben Ausdruck für das elektrische Moment des Atoms, für deren Schwingungszahl τ 1 ω 1 + … + τ u ω u die Gleichungen gelten:
$$\displaystyle \begin{aligned} \tau_1=n^\prime_1-n^{\prime\prime}_1, \ldots , \tau_u=n^\prime_u-n^{\prime\prime}_u. \end{aligned}$$Diese nennen wir deshalb die ‘korrespondierende’ Schwingungskomponente in der Bewegung, und den Inhalt der obigen Aussage bezeichnen wir als das ‘Korrespondenzprinzip’ für mehrfach periodische Systeme.”
- 73.
Bohr (1923b, footnote on page 142–143). “In Q.d.L. [Quantentheorie der Linienspektren] wird diese Bezeichung noch nicht benutzt, sondern der Inhalt des Prinzips ist dort als eine formale Analogie zwischen Quantentheorie und klassischer Theorie bezeichnet. Eine solche Ausdrucks-weise könnte jedoch Missverständnisse veranlassen, da ja […] das Korrespondenzprinzip als ein rein quantentheoretisches Gesetz betrachtet werden muss, das in keiner Weise den Kontrast zwischen den Postulaten und der elektrodynamischen Theorie zu vermindern vermag.”
- 74.
Darrigol (1992, 138).
- 75.
Bohr (1923b).
- 76.
See Darrigol (1992, 138) for a discussion of Bohr’s idea of a “rational generalization.”
- 77.
Darrigol (1992, 151). The most frequent example for this use of the principle as an interpretational device was the explanation of selection rules and the effect of external fields on the spectrum discussed above. Bohr came to regard selection rules and the effect of external fields as different aspects of the same phenomenon. On the one hand, the appearance of new harmonic components due to the perturbation of the external field explained why new lines occurred; on the other hand, the fact that only two new fundamental frequencies without overtones entered into the Fourier series explained why this splitting was limited to two additional lines. See Bohr (1920, 444–452), Bohr (1921), and Bohr (1923b, 146). In addition to these paradigmatic examples, Bohr began to incorporate the research of other physicists into his approach by giving explanations of their results in terms of the correspondence principle. For example, Bohr incorporated Stern and Voelmer’s work on the broadening of spectral lines (Stern and Volmer 1919), and Ehrenfest and Breit’s work on weak quantization (Ehrenfest and Breit 1922).
- 78.
Bohr (1918a, 16). Bohr stated that “we cannot without a detailed theory of the mechanism of transition obtain an exact calculation” of the transition probabilities.
- 79.
Debye to Bohr, 6 June 1918 in Bohr (1976, 607). “Insbesondere ist Ihr Ansatz zur Berechnung der Intensitäten offenbar von grösster Wichtigkeit! Ein kleines unbefriedigendes Gefühl bleibt mir noch, wenn ich sehe, dass Sie die Intensität in Beziehung setzen zu den Fourier Coefficienten einer einzigen Bahn. Es scheint mir doch so zu liegen, dass wenn ein System von einer Bahn n 1, n 2, n 3… auf eine andere Bahn \(n^\prime _1,~n^\prime _2,~n^\prime _3 \ldots \) geht, die fraglichen Fourier Coefficienten für die erste Bahn etwa C und für die zweite Bahn davon verschieden etwa C ′ sein werden. Würde es nicht dem Sinne der Ueberlegungen besser entsprechen, wenn die Wahrscheinlichkeit des Ueberganges durch (C + C ′)∕2 oder vielleicht \(\sqrt {CC^\prime }\) gemessen würde? Oder liegt die Sache so, dass ich Sie nur nicht richtig verstanden habe?”
- 80.
For this particular period in Kramers’ biography, see Dresden (1987, 97–110). The young Dutch physicist had become Bohr’s assistant in 1916. He did most of the calculations for Bohr, especially on helium, which was one of the most important research topics in the old quantum theory. For his dissertation in Copenhagen, Bohr had assigned Kramers to work out an estimate of the intensities of spectral lines. Kramers ’ dissertation was concerned mostly with the hydrogen atom. In his dissertation he used the by then standard derivations for the hydrogen atom with and without external fields by means of celestial mechanics and dealt with the problem of intensities for spectral lines (Dresden 1987, 101). Dresden indicates that even the technicalities of the original “Bohr” argument were already worked out by Kramers, as he was the expert on Hamilton-Jacobi theory upon which Bohr relied.
- 81.
Kramers (1919, 327).
- 82.
Ibid.
- 83.
Ibid.
- 84.
Kramers (1919, 327).
- 85.
Kramers (1919, 330).
- 86.
Ibid.
- 87.
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Jähnert, M. (2019). The Correspondence Principle in Copenhagen 1913–1923: Origin, Formulation and Consolidation. In: Practicing the Correspondence Principle in the Old Quantum Theory. Archimedes, vol 56. Springer, Cham. https://doi.org/10.1007/978-3-030-13300-9_2
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