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Correspondence principle versus Planck-type theory of the atom

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Abstract

This article examines the problem of the origins of the correspondence principle formulated by Bohr in 1920 and intends to test the correctness of the argument that the essential elements of that principle were already present in the 1913 “trilogy”. In contrast to this point of view, moreover widely shared in the literature, this article argues that it is possible to find a connection between the formulation of the correspondence principle and the assessment that led Bohr to abandon the search for a Planck-type theory. In fact, a thorough examination of Bohr’s works shows that the birth of this principle coincided with the depletion of a research program whose origins may date back to Bohr’s stay at the Rutherford’s laboratory (summer 1912). Finally, this article argues that original program of research was abandoned when it became clear that Planck’s quantum hypothesis for the harmonic oscillator was not an adequate support for the theoretical architecture of atomic physics; namely, there was evidence enough to justify a most drastic conclusion, according to Bohr: “I do not think that a theory of the Planck type can be made logical consistent”.

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Notes

  1. Bohr (1913b, c, d).

  2. Bohr (1920). The article was published in English in Bohr (1922b, pp. 20–60); also in Bohr (1976, pp. 242–282) [henceforth CW3].

  3. CW3 282; my emphasis.

  4. Cfr Kragh (2012, pp. 189–225). Sommerfeld in the first edition of Atombau und Spektrallinien wrote: “Bohr has found a magic wand [Zauberstab] in his analogy principle [...] which without clearing up the conceptual difficulties allows him to make the results of the classical wave theory directly useful for the quantum theory” Sommerfeld (1919, p. 403). In a letter to Bohr dated November 11, 1920, Sommerfeld expressed his doubts on a “principle” whose origins were in his opinion “foreign to the quantum theory,” CW3 690.

  5. CW3 250.

  6. Ibid., pp. 243–244.

  7. Bohr (1916, unpublished paper), in Bohr (1981, pp. 433–461: 434) [henceforth CW2].

  8. Ibidem.

  9. According to the two expert spectroscopists, these lines had to be attributed to hydrogen and should correspond, according to Bohr’s theoretical formula, to semi-integer quantum numbers. On the contrary, Bohr considered these lines to be emitted by ionized helium atoms whose frequencies were given by an expression similar to that of hydrogen but with a value four times greater than the Rydberg constant. The two interpretations were therefore subjected by Evan Evans to a sort of experimentum crucis that required the repetition of Fowler’s measurements in a vacuum tube filled solely with helium and with ascertained absence of any traces of hydrogen; Evans (1913).

  10. See infra Sect. 2.1.

  11. This is what we read in a letter sent by G. von Hevesy to Bohr on October 23, 1913. Hevesy had met Einstein in Vienna on the occasion of the 85th Versammlung deutschen und Naturforscher Aertze and wrote to Bohr of his favorable impression, CW2 532.

  12. Heilbron e Kuhn (1969, p. 272).

  13. Rud Nielsen (1976, § 2), Darrigol (1997, p. 549), Kragh (2012, p. 63).

  14. Bohr (1916, unpublished paper), cit. CW2 445.

  15. Cfr. infra Sect. 2.3.

  16. Bohr develops the argument that will be taken up within the correspondence principle in Bohr (1918a, pp. 15–16). In this regard, see J. Rud Nielsen “Introduction” to CW3, in particular § 3.

  17. Bohr (1916, unpublished paper), cit.

  18. Bohr thanked Rutherford for the invitation received in a letter dated June 19, 1914, CW2 594–595. He had offered a one-year readership, and Bohr had reached Manchester in October 1914, a few weeks after the beginning of the war operations and the closure of the German borders. Bohr stayed on in Manchester until July 1916.

  19. Bohr to Oseen March 17, 1916, CW2 571–573.

  20. Ibid., p. 572.

  21. Bohr to Sommerfeld March 19, 1916, CW2 603–604: 603.

  22. Bohr to Oseen March 17, 1916, cit.

  23. N. Bohr, The Rutherford memorandum (1912) in CW2 136–143: 137.

  24. The manuscript is not dated, but it is reasonable to place its drafting in late June/early July 1912. Bohr sent the manuscript to Rutherford with a letter dated July 6, 1912, which states “I send the remarks concerning the structure and stability of molecules, for which you kindly asked” CW2 577. On the discovery, reconstruction and interpretation of this manuscript cf. Rosenfeld (1963) and Heilbron and Kuhn (1969).

  25. From a long letter written by Bohr to his brother Harald October 23, 1911, it can be deduced, inter alia, that Thomson had displayed a total lack of interest for Bohr’s criticism concerning some mistakes that he apparently committed in the calculation of the absorption, Bohr (1972) [henceforth CW1], original and English translation pp. 526–533.

  26. Bohr (1961, p. 1086).

  27. Ibid.

  28. Ibid., p. 1085.

  29. Cf. Kragh (2012, p. 47). Bohr himself was to recall that while he considered the atomic number and the radioactive displacement laws as results that supported the Rutherford theory, Hevesy, Fajans and Soddy thought precisely the opposite and “they thought that [this] was completely against Rutherford,” Interview to Niels Bohr by Thomas S. Kuhn, Léon Rosenfeld, Aage Petersen and Erik Rüdinger on October 31, 1962, AIP.

  30. The Comité scientifique international chaired by Lorentz was composed of M. Curie, W.H. Bragg, M. Brillouin, H. Kamerlingh Onnes, M. Knudsen, A. Righi, E. Rutherford and E. van Aubel.

  31. J.J. Thomson, La structure de l’atome, in Solvay II, pp. 1–44. In this lecture, Thomson presented a new model of the atom (cf. Kragh 2012, pp. 108–110).

  32. Reserves and doubts were expressed by Bohr in a letter to Rutherford October 16, 1913, CW2 587–589.

  33. Solvay II, pp. 50–51 and 53–54.

  34. 76 references in 41 articles apart from those written by the same Rutherford, also in collaboration with other authors.

  35. Rutherford (1911). The quotations would be ten if one includes those contained in the articles of Rutherford himself: Rutherford (1912, p. 461) and Rutherford and Nuttal (1913, p. 702).

  36. Wilson (1912, p. 284). Wilson, a pupil of Thomson, was long his collaborator at the Cavendish Laboratory.

  37. Darwin (1912, p. 901, 1913, p. 201), Geiger (1912, p. 605), Geiger and Marsden (1913, p. 605), Marsden and Taylor (1913, p. 443), Moseley (1913, p. 1025), Bohr (1913a, p. 10, 1913b, p. 1.)

  38. Rutherford to Bragg, in Eve (1939, p. 208).

  39. Mayer (1913, p. 940).

  40. Interview to Niels Bohr by Thomas S. Kuhn, cit.

  41. Thomas Kuhn’s interview to Niels Bohr, planned by the project for the Archive for the History of Quantum Physics, took place in the fall of 1962. Over 18 days, from October 31, five sessions were held almost entirely devoted to young Bohr’s early years of scientific activity, the preparation of his doctoral thesis, his stay in England, first at Cambridge then in Manchester, as well as the publication in 1913 of his famous theory of the constitution of atoms and molecules. The interview was abruptly interrupted by a tragic event. The fifth session was held on November 17, and had concerned young Bohr’s cultural education, his philosophical interests and his relationship with Harald Høffding of whom he had been a pupil. Early in the afternoon of the following day, Bohr died from a heart attack.

  42. The article to which Bohr refers is Moseley (1913), cf Kragh (2012, § 3.4). In this regard, it is also significant to point out what Peter Debye replied to Kuhn who had asked “whether, when you had been in Zurich when the Rutherford atom was suggested, I wondered whether you had heard of it.” According to Debye “in Zurich we did not talk very much about models. [...] The model of Rutherford was compared to that of Thomson.” Interview to Peter Debye by Thomas S. Kuhn and G. Uhlenbeck on May 3, 1962, AIP.

  43. Interview to Niels Bohr by Thomas S. Kuhn ... on October 31, 1962, cit. p. 7.

  44. A first critical analysis of this document can be found in Rosenfeld (1963, “Introduction”). See also Heilbron and Kuhn (1969).

  45. Interview to Niels Bohr by Thomas S. Kuhn ... on October 31, 1962, cit. p. 10.

  46. Ibid., p. 8.

  47. Bohr to Rutherford, March 6, 1913, CW2 581–583: 582.

  48. Bohr (1913a).

  49. Bohr to Rutherford, March 6, 1913, cit.

  50. Rutherford to Bohr, March 20, 1913, CW2 583–584: 583.

  51. Bohr to Rutherford, March 6, 1913, cit. 581. Bohr had taken seriously the work that the astronomer John Nicholson had dedicated to the interpretation of some spectra also appropriately exploiting Planck’s quantum ideas, Nicholson (1912a, b). And though Bohr was engaged to highlight the relevant differences between his approach and that of Nicholson (see, for example, Bohr to Rutherford 31 January 1913 CW2 579–580), there were valid reasons for him to wish for a fairly rapid publication of his article. Heilbron, for example, speaks of the “alarming paper of John William Nicholson” in which Bohr found in the last weeks of 1912, while he was engaged in defining the conceptual framework of his theory, Heilbron (2013, p. 170).

  52. Bohr (1913b, p. 12).

  53. Ibid., pp. 2–3.

  54. Ibid., p. 12.

  55. In a recent paper, Heilbron identifies four formulations of the theory, Heilbron (2013, pp. 175–177).

  56. Bohr (1913b, p. 4).

  57. Ibid., p. 4–5.

  58. Ibid., p. 5.

  59. Ibidem.

  60. Ibid., p. 9.

  61. Ibid., p. 12.

  62. Ibid., p. 7.

  63. Ibid., pp. 12–13.

  64. Ibid., p. 13.

  65. Ibid., p. 14. With reference to what I said in the introduction, this is the passage in which both Darrigol (1997, p. 549) and Kragh (2012, p 63) place the origins of the correspondence principle.

  66. Ibidem.

  67. CW2 583.

  68. Bohr (1913b, pp. 14).

  69. Ibidem.

  70. Ibidem.

  71. These words can be found in a letter written by Harald to Niels Bohr in the fall of 1913 while he was in Göttingen, CW1 567. The first reactions of Paul Ehrenfest, who would later become one of Bohr’s strongest supporters, were very negative. In a letter of August 25, 1913, he wrote to Sommerfeld “Bohr’s work on the quantum theory of the Balmer formula [...] has driven me to despair. If this is the way to reach the goal, I must give up doing physics,” quoted in Klein (1970, p. 278).

  72. Sommerfeld to Bohr, September 4, 1913, CW2 123.

  73. Jeans (1913, p. 379).

  74. Bohr (1914a), Eng. trans. in Bohr (1922b, pp. 1–19: 10) [also in CW2 283–301]. Here and hereinafter, the citations are taken from the English edition.

  75. Ibid., p. 14.

  76. Ibid., p. 10.

  77. Ibidem.

  78. Debye (1910).

  79. Ibidem.

  80. Ibidem.

  81. Bohr (1922b, p. v).

  82. Ibidem.

  83. Ibid., pp. v–vi. John Heilbron implicitly seems to resume this opinion of Bohr’s when he says that the same derivation of the Rydberg constant “which is only sketched in the trilogy, would become the most powerful, as it contains the seed of the ‘Correspondence Principle’ by which Bohr would seek to tease out the rules of the microworld by a comparison with calculations made using classical theory” Heilbron (2013, p. 176).

  84. Bohr considered this text to be an important stepping stone since, as he recalled years later, he was to use it as the basis for a number of important seminars held in Monaco and Göttingen in the summer of 1914. “I went to Göttingen in the summer of ’14 [...] and there I gave a talk about the spectra from the point of view, or at any rate something like the points of view that I had put a half a year before in this paper in the Physical Society.” Interview to Niels Bohr by Thomas S. Kuhn ... on November 7, cit., pp. 1–2.

  85. Rutherford to Bohr, December 11, 1913, CW2 589. Stark’s work, published by Preußische Akademie der Wissenschaften, was announced in a brief note appeared in Nature, Stark (1913).

  86. Bohr to Rutherford 31 December 1913, CW2 591.

  87. Ibidem.

  88. Bohr (1914b).

  89. “The considerations were of a preliminary nature and their main intention was to emphasize certain principal features of the explanation of the problem in question.” N. Bohr, Draft of a note to Phil. Mag. Concerning the Stark effect (unpublished), CW2 370–371: 371.

  90. Bohr (1914b, p. 510).

  91. Ibidem.

  92. Ibid., pp. 510–511.

  93. Bohr to Hansen, 12 May 1915, CW2 516. Bohr’s article dated July was published in the October issue: Bohr (1915b).

  94. Bohr (1915a). In this article, Bohr provides a correct interpretation of the famous Frank and Hertz experiment, around which an interesting debate developed, cfr. Kragh (2012, pp. 143–146).

  95. CW2 433.

  96. “These calculations have been criticized by Nicholson (1914), who has attempted to show that the configurations chosen for the electrons in the atoms are inconsistent with the main principles of the theory, and has also attempted to prove the impossibility of accounting for other spectra by help of assumptions similar to those used in the interpretation of the hydrogen spectrum. Although I am quite ready to admit that these points involve great and unsolved difficulties, I am unable to agree with Nicholson’s conclusions” Bohr (1915a, p. 399).

  97. Ibidem, my emphasis.

  98. Bohr to Oseen March 17, 1916, cit.

  99. CW2 434.

  100. Unlike what we find in his earlier articles, see for example Bohr (1915a, p. 396).

  101. CW2 434.

  102. Planck (1913). On this see Kuhn (1978, chap. X).

  103. CW2 434, my emphasis.

  104. “Of course my project can’t be carried through without resort to hypotheses, and I fear that your hatred of the zero-point energy extends to the electrodynamic emission hypothesis that I introduced and that leads to it. But what’s to be done? For my part, I hate discontinuity of energy even more that discontinuity of emission.” This Planck wrote to Paul Ehrenfest in the spring of 1915; cit. in Kuhn (1978, p. 253).

  105. CW2 436.

  106. Starting from the finding that “\(\bar{T}/ \omega \) will remain constant for any small variation of motion, produced by external forces, for which \(W\) is unaltered” Bohr devoted ample space to the discussion on the role played by the invariant quantities in the quantum theory; however, they represented “a necessary condition for the application of ordinary mechanics to the stationary states of periodic systems”(ibid.). Of course, Bohr recognized the merits of Paul Ehrenfest in having achieved this result and for his formulation of the adiabatic principle. The latter, to which Bohr preferred to always refer to as the principle of mechanical transformability, played a crucial role together with the correspondence principle in the construction of the atomic theory up to the birth of quantum mechanics.

  107. CW2 443.

  108. Ibidem.

  109. Ibidem.

  110. Ibid., p. 268.

  111. Ibid., p. 269.

  112. Ibidem.

  113. Ibid. 268.

  114. Bohr to Oseen, March 17, 1916, CW2 571–573: 572.

  115. Bohr (1921, p. iv).

  116. In fact, there was a lacuna and, in light of what we have so far gathered, it can be argued that it had not been a trivial oversight. In that collection, the conference of December 1913 was absent, a text that was the antithesis of the theoretical perspective of the withdrawn paper and where not surprisingly, as it will become clear in the next section, Bohr located the “first germ” of the correspondence principle.

  117. Ibid., p. v.

  118. The quotes are just two. The first one concerns Plank’s oscillator brought forth as an example of a periodic system in the framework of the exam for Ehrenfest adiabatic principle (ibid. p. vi). The second one concerns a reference to the withdrawn paper in which “a preliminary attempt was made in the section in question to extend the assumption made by Planck about the a priori probability of a harmonic oscillator to systems of several degrees of freedom” (ibid. p. vii).

  119. Bohr to Richardson, August 1918 (CW3, pp. 14–15: 14).

  120. CW3 244.

  121. Ibidem.

  122. Ibidem.

  123. Ibid. 245.

  124. Ibidem.

  125. Ibid. 246.

  126. Ibidem.

  127. Ibid. 247, my emphasis.

  128. Ibid. 248.

  129. Ibid. 249, italics in the original.

  130. Ibidem.

  131. Ibidem.

  132. Bohr uses this expression in Bohr (1925, p. 849): “The correspondence principle expresses the tendency to utilize in the systematic development of the quantum theory every feature of the classical theories in a rational transcription appropriate to the fundamental contrast between the [quantum] postulates and the classical theories.”

  133. Ibid. 250.

  134. In fact, in some later works, Bohr would try to solve the obvious difficulty arising from the fact that, outside of the limiting region, the “corresponding” amplitudes can be quite different in the two stationary states involved in the transition, coming also to suggest that the sought frequency was the mean value of the corresponding vibrations, calculated on a continuous series of hypothetical “intermediate states.” It was the same ploy used by Ehrenfest to make it somewhat easier to explain the idea of correspondence to participants at the Solvay Conference. “This is the simplest way that can be imagined” claimed Ehrenfest during the discussion in reply to W. L. Bragg, who had asked how it was possible to define the average value of the corresponding frequency \(\omega \) for a transition; Solvay III pp. 255–156), CW3 388–89.

  135. CW3 377, italics in the original.

  136. Bohr (1923, in CW3 p. 376).

  137. In the first lines of the opening paragraph entitled “Quantum postulate and causality” Bohr asserts: “The quantum theory is characterized by the acknowledgment of a fundamental limitation in the classical physical ideas when applied to atomic phenomena. The situation thus created is of a peculiar nature, since our interpretation of the experimental material rests essentially upon the classical concepts. Notwithstanding the difficulties which hence are involved in the formulation of the quantum theory, it seems, as we shall see, that its essence may be expressed in the so-called quantum postulate, which attributes to any atomic process an essential discontinuity, or rather individuality, completely foreign to the classical theories and symbolized by Planck’s quantum of action” Bohr (1928, p. 580).

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Authors and Affiliations

  1. Università dell’Aquila, L’Aquila, Italy

    Sandro Petruccioli

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  1. Sandro Petruccioli
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Correspondence to Sandro Petruccioli.

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Communicated by: Jed Buchwald.

I wish to thank Helge Kragh for his critical comments and suggestions that were particularly useful in the revision of the first draft of this article.

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Petruccioli, S. Correspondence principle versus Planck-type theory of the atom. Arch. Hist. Exact Sci. 68, 599–639 (2014). https://doi.org/10.1007/s00407-014-0137-5

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