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Historical magic in old quantum theory?

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Abstract

Two successes of old quantum theory are particularly notable: Bohr’s prediction of the spectral lines of ionised helium, and Sommerfeld’s prediction of the fine-structure of the hydrogen spectral lines. Many scientific realists would like to be able to explain these successes in terms of the truth or approximate truth of the assumptions which fuelled the relevant derivations. In this paper I argue that this will be difficult for the ionised helium success, and is almost certainly impossible for the fine-structure success. Thus I submit that the case against the realist’s thesis that success is indicative of truth is marginally strengthened.

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Notes

  1. Eg. Kitcher’s ‘working posits’ versus ‘presuppositional posits’ (1993), Psillos’s ‘divide et impera’ distinction between ‘idle’ and ‘essentially contributing constituents’ (1999), Saatsi’s focus on ‘success-fuelling properties’ (2005), and Chakravartty’s ‘semirealism’ distinction between ‘detection properties’ and ‘auxiliary properties’ (2007).

  2. For example, between 1900 and 1913 Lenard, Thomson, Stark, and Wien had ideas about the origin of spectra, but really these were little more than speculations. As Pais (1986, p. 197) puts it, ‘So it was when Bohr came along. In his words, in those early days spectra were as interesting and incomprehensible as the colors on a butterfly’s wing.’

  3. Of course, the distinction between predictive success and ‘mere’ explanatory success is important here. See below for discussion.

  4. Any realist who is moved by explanatory success would be even more committed to Old Quantum Theory, because they would have to take into account explanations of Whiddington’s law, Bragg’s law, the covalent bond, etc. Such a realist would be even harder hit by the fact that the theory is now known to be false.

  5. Of course, the Bohr and Sommerfeld theories also had many failures (especially as regards atoms of greater complexity than hydrogen). However, these don’t need to be discussed here, because they don’t prevent a selective realist from making a commitment. A selective realist says ‘successes suggest truth’, regardless of any failures the theory has. Failures can be explained by the fact that a theory does contain false claims; what’s crucial for the realist, though, is that any successes are due to what the theory got right.

  6. Norton 2000, p. 83: ‘Quantization of energy levels ... contained an arbitrary deviation from the theory of Planck, justified only by its success in giving the right result. Planck’s theory required emission of light energy in integral multiples of hv, where v is the frequency of the emitting oscillator; Bohr based his theory on the supposition that stationary states with frequency w are formed by the emission of light energy in integral multiples not of hw but of hw/2, when an electron is captured by the nucleus.’ See Heilbron and Kuhn 1969 for the full story.

  7. The story is actually a little more complicated. Fowler doubted that the lines in question were due to helium at all, but that will not be important here.

  8. The term ‘naive optimist’ is introduced in Saatsi and Vickers (2010). See section 5, below, for discussion.

  9. In fact this is just the tip of the iceberg insofar as the success of Bohr’s theory is concerned. And there were also some predictive successes which would have been really impressive if only they had been noticed. One such success would have been the prediction of a band of spectral lines identical to hydrogen except shifted to a slightly shorter wavelength. These would have been predicted by Bohr if he had been aware of deuterium atoms—hydrogen atoms but with a neutron accompanying the proton to form a two-particle nucleus with the same charge but twice the mass. And this certainly would have counted as a novel prediction. In fact deuterium wasn’t discovered until 1932, but this doesn’t make the prediction any less significant so far as the realist is concerned: Bohr’s theory really does make this prediction (whether it was noted at the time or not), and the prediction really was empirically confirmed (even though this actually happened many years later). See Eisberg and Resnick 1985, p. 107 for further details here.

  10. See Kragh 1985, p. 69. In this section I draw heavily on Kragh’s paper, and also on Robotti (1986).

  11. Sommerfeld’s motivation for these developments was certainly not just the fine-structure: he was also interested in explaining other phenomena, such as the Stark effect. Cf. Jungnickel and McCormmach (1990), p. 351ff.

  12. For more quotations of commitment from the community see Kragh’s paper, and also Robotti (1986).

  13. Norton is keen to show that Bohr’s derivation is an example of demonstrative induction, an alleged phenomenon where theory is deductively inferred from the phenomena. He is thus keen to show that there is a common subset to Bohr’s theory and modern quantum theory, namely, that subset which can be inferred from the phenomena. For the realist, this subset is key to explaining Bohr’s success in realist terms, as we will see.

  14. Of course classically there is a continuous spectrum of possible orbits, but on Bohr’s theory if one makes n large enough the energies of neighbouring allowed orbits can be made as close as you like, so the predictions of the two theories really do coincide for large enough n. For hydrogen, at n = 10,000 the percentage difference between the two values is 0.015 (Eisberg and Resnick, p.118), and this gets smaller as n increases.

  15. One can also make a correction for finite nuclear mass here. Cf Eisberg and Resnick, p.105: ‘If we evaluate RH [making corrections for the finite nuclear mass], using the currently accepted values of the quantities m, M, e, c, and h, we find RH = 10968100 m−1. Comparing this with the experimental value of RH given in Section 4-4 [10967757.6], we see that the Bohr model, corrected for finite nuclear mass, agrees with the spectroscopic data to within three parts in 100,000!’

  16. ‘[T]he atomic spectrum of He+ [ionised helium] is exactly the same as the hydrogen spectrum except that the reciprocal wavelengths of all the lines are almost exactly four times as great. This is explained very easily, in terms of the Bohr model, by setting Z2 = 4.’ (Eisberg and Resnick 1985, p. 103).

  17. There is no difficulty here: a ‘reduced demonstrative induction’ is a type of derivation. See Norton’s paper for details.

  18. The modification of assumptions 1–5 (above) required for the reduced derivation is straightforward. Simply put, the ‘orbits’ aren’t doing any work, and the derivation goes through just the same if one swaps talk of orbits for talk of ‘stationary states’ with certain energies (see Norton 2000, pp. 86–87 for the details).

  19. In addition, imagine what such a realist would have to say about current scientific theories. Even if we could determine the ‘working posits’ of our very best theories responsible for successful predictions (something doubted by anti-realists), such a realist would have to say that any one of these posits could be radically false. Thus our confidence in even the working posits of our best theories would be undermined, and it isn’t nearly as clear what such realists could claim to know about the unobservable world.

  20. In fact, Bohr was the first to introduce elliptical orbits, and even attempted to apply special relativity (see Nisio 1973, p. 54ff.). But Sommerfeld was the first to see these developments through rigorously.

  21. For further details of Sommerfeld’s derivation see eg. Arabatzis 2006, p. 158ff, or see Nisio 1973 for the full history. The extra details will not matter to my forthcoming argument that the selective realist strategy almost certainly won’t work here.

  22. Some structural realists might focus on the fact that we have the same equation and claim that this shows that Sommerfeld had latched onto the right ‘structure’ of the world, which explains the success. In fact, most realists wouldn’t count this as a realist position at all: what needs to be explained is how Sommerfeld managed to reach the fine-structure formula.

  23. Cf. Series (1988), p. 26.

  24. For the details see Burkhardt et al. (2006), p. 128ff. Cf. Eisberg and Resnick, p. 284ff. Burkhardt et al. consider a third term in the ‘final’ Hamiltonian, called the ‘Darwin Term’ (p. 130). However, this can be ignored for present purposes: at any rate, it plays a role ‘only when the orbital angular momentum is zero’ (Ibid.).

  25. Eg. Letokhov and Johansson (2009), p. 37: ‘The interaction between the spin and the electron’s orbit is called spin-orbit interaction, which contributes energy and causes the fine-structure splitting.’

  26. An appeal to approximate truth will not help. Any approximately true assumptions can be made true by an operation I have elsewhere called ‘internalising the approximation’ (Vickers forthcoming). For example, if x=y is approximately true, then (internalising the approximation) x≈y is true. I am assuming that all such assumptions are included within the ‘egg’ of truth on the RHS of Fig. 1.

  27. Although the precise nature of this ‘dropping out of the formalism’ is still debated. See Morrison (2004) for discussion.

  28. The slightly earlier, Schrödinger-Heisenberg QM therefore comes 3rd place insofar as the fine-structure formula is concerned, ranking behind not just the 1928 Dirac QM, but also the 1916 Sommerfeld theory! (See Biedenharn 1983 for further details.)

  29. There were certain developments of Sommerfeld’s theory in the early 1920 s (eg. new quantum numbers) that could be interpreted, perhaps, as surrogates for spin (thanks to an anonymous referee for pointing this out). However, this doesn’t help the realist explain why Sommerfeld’s theory was successful before 1920, and in particular when he made his derivation of the fine structure formula in 1916.

  30. Further qualifications are possible, of course. Couvalis (1997, p. 75) writes: ‘Whewell grasped that it is not the successful prediction of scattered novel facts which is good evidence for the approximate truth of a research program. The reason for this is that a research program which is not even approximately true can easily produce occasionally successful predictions of novel facts by coincidence.’ This suggests another ‘predictions not scattered’ qualification, although whether the realist can fully justify it is an open question.

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Acknowledgements

I am grateful to Juha Saatsi and John Norton for discussion and criticism, and also to three anonymous referees for helpful comments. Thanks also to audiences at the University of Leeds (2009), and at the 2nd biennial conference of the European Philosophy of Science Association in Amsterdam in 2009, where early versions of this paper were presented.

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Vickers, P. Historical magic in old quantum theory?. Euro Jnl Phil Sci 2, 1–19 (2012). https://doi.org/10.1007/s13194-010-0010-6

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