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Specification and Application: Two Case Studies from the History and Philosophy of Quantum Mechanics

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The Material Realization of Science

Part of the book series: Boston Studies in the Philosophy of Science ((BSPS,volume 294))

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Abstract

In this chapter, I will present two detailed cases from modern physics, which throw more light on the philosophical discourse of the previous chapter. The two cases vary in kind. First (in Sect. 5.2), I discuss several historical aspects of the genesis of quantum mechanics, stressing the part played by the so-called correspondence principle. In this way, I intend to specify the philosophical argumentation given so far by illustrating it with a concrete historical case. Then, in Sect. 5.3, I will address the so-called “measurement problem in quantum mechanics”. This problem is analyzed as an application of the philosophical arguments. Here I intend to demonstrate that the philosophical conclusions obtained in the previous chapters permit an alternative view of the measurement problem. The main result is that this problem is much less problematic than has usually been supposed in the relevant literature.

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Notes

  1. 1.

    See the discussion of conceptual discontinuity in Sect. 4.3 and of coreference in Sect. 4.4.

  2. 2.

    Thus, this contrasts to the approach taken by Klaus Meyer-Abich, who attempts to give an (unequivocal) interpretation of all Bohr has said about correspondence and the correspondence principle. See Meyer-Abich (1965, especially Chap. 2).

  3. 3.

    See Hund (1974, 65). For the events leading up to Bohr’s atomic theory, see Heilbron and Kuhn (1969, 211–266).

  4. 4.

    See Bohr (1913), 874–875 (I leave aside the assumptions 4 and 5 mentioned there).

  5. 5.

    For this development, see Heilbron and Kuhn (1969, 266–277).

  6. 6.

    See Bohr (1914), which is the translated text of a lecture given in December 1913.

  7. 7.

    For later usage, I note down the harmonic component τω n as ω n,τ .

  8. 8.

    Bohr (1913, 14). Note that, more generally, we have (in this region) the correspondence between ω n,τ and \( v_n^{n - \tau } \)for τ > 1.

  9. 9.

    For a systematic exposition of these results, see e.g. Kramers (1919, 1–45).

  10. 10.

    See Bohr (1918, 125–130). For a lucid review of this proof, see Tomonaga (1962, 142–146).

  11. 11.

    Kramers (1919, 48). He argues for this correspondence by referring to the fact that, for low frequencies, the intensity distributions implied by the radiation laws of Planck and of Rayleigh–Jeans also correspond with each other.

  12. 12.

    Kramers (1919, 61 and 72). For the sake of brevity, I will not deal any further with the correspondence regarding the polarizations. It is remarkable, though, that in some cases of polarization an exact correspondence occurs, also for small quantum numbers. For this point, see Bohr (1918, 110–111 and 130–133); Tomonaga (1962, 147–150); Jammer (1966, 113–114).

  13. 13.

    Letter from Bohr to Rutherford, 27 December 1917, published in Bohr (1976, 682–683). At about the same time, the first results of Kramers’s work were achieved (see the same letter and also Radder 1982c, 226).

  14. 14.

    Compare Epstein’s reaction to Bohr’s article “On the Quantum Theory of Line Spectra”: “It really seems that the discrepancy between the quantum mechanical and classical approach is not at all as large as was assumed until now.” (Letter from Epstein to Bohr, 14 May 1918, quoted in Bohr 1976, 637).

  15. 15.

    Kramers (1919), 47–50 (see Eq. 5.14 in Sect. 5.2.3).

  16. 16.

    Here I do not consider the rather opaque use of the correspondence principle in Bohr’s so-called “second atomic theory” (in the years 1920–1923). For some reviews of this theory, see Kramers (1923, 554–559); Jammer (1966, 133–155); Kragh (1979, especially 155–159).

  17. 17.

    At least, this holds good for the more systematic applications of the correspondence principle. For different applications, see note 16 and the references mentioned there.

  18. 18.

    This probabilistic aspect of the virtual field model has been treated too scantily in the historical literature (see Radder 1982c, 242–243).

  19. 19.

    Heisenberg (1925). For extensive discussions of this article, see Van der Waerden (1967, especially 25–35); MacKinnon (1977, especially 161–188).

  20. 20.

    Quoted in Serwer (1977, 242). See also Pauli’s letter to Bohr of 21 February 1924, quoted in Serwer (1977, 230).

  21. 21.

    Meyer-Abich (1965, 72–93) does try to give such a reconstruction of Bohr’s view, on the basis of much written material. However, the final result of this exercise (86–87) is both too vague (and thus not very illuminating) and unsatisfactory, because it is still phrased in terms of the harmonic components of the motion of the electrons.

  22. 22.

    This goes against Meyer-Abich (1965, 85). See also Pauli’s views, as quoted in Serwer (1977, 242).

  23. 23.

    For clarity’s sake, I describe Born’s equation in my own notation.

  24. 24.

    Letter of Bohr to C.W. Oseen, January 1926 (quoted in Rosenfeld and Rüdinger 1967, 73).

  25. 25.

    An example is the substitution of Poisson brackets by commutators. See, e.g., Messiah (1969, Vol. 1, 317–318).

  26. 26.

    For the calculation of these energies, see Pauli (1926).

  27. 27.

    See also Hanson (1972, 149–157), who speaks of “logical” discontinuities.

  28. 28.

    One might question whether the correspondence language is a real (in the sense of a fully-fledged) language. My response is that, in this context, I do not use the term “language” in a strict sense, as is not uncommon in philosophy. By way of alternative, we could speak of a “correspondence terminology”.

  29. 29.

    We should keep in mind that this and other intertheoretical relations do not always connect the complete formalisms of the relevant theories. For instance, not only in Born’s Eq. 5.15, but also in the case of the Ehrenfest theorem, we are dealing with a relation that only applies to certain situations, to certain kinds of functions (see also Bunge 1970, 288–293). Concerning the Ehrenfest theorem, see Albert Messiah (1969, 217), Vol. 1, who writes: “It is generally not correct to state that the mean values <q i > and <p i > follow the laws of Classical Mechanics”. For this reason, Thomas Nickles’s account of the theorem stating, “that any relation that appears in classical mechanics must be valid as a relation between quantum theoretical expectation values”, is not correct (Nickles 1973, 194).

  30. 30.

    Note that the numerical correspondence between the Fourier components and the transition amplitudes is established indirectly (in contrast with that between the classical and quantum frequencies), for instance, on the basis of a theoretical interpretation of experiments on the dispersion of light by atoms. See also the explanation of the role of theory in the criterion of reference at the beginning of Sect. 4.4.

  31. 31.

    See also Kripke’s conception of proper names, in Kripke (1980, especially 47–60).

  32. 32.

    MacKinnon (1982, 457–460). Note that his examples of intertheoretical relations are different from Born’s relation that I have discussed here.

  33. 33.

    See also Wigner (1970), articles 1–5. Interestingly, MacKinnon (1982, 453) relates these geometrical invariances to measurement and measurability.

  34. 34.

    MacKinnon (1982, 459). Heinz Post (1971, 226–228) also points out the significance of such transtheoretical symmetries and invariances. However, he does not use these in an argumentation for realism, but in a plea for a rational heuristics.

  35. 35.

    One can express this difference in terms of “possessed” versus “latent observables” (see, e.g., Park 1968, 210–212).

  36. 36.

    I do not intend to give an extensive review of the measurement problem. For such reviews, see the literature, for instance Park (1968), Jammer (1974), D’Espagnat (1976), Fransen (1981). A complete understanding of the entire argumentation with all its details will certainly require technical-physical knowledge. I have aimed, however, at presenting the subject in such a way that the outline of the argument will also be understandable to readers without much education in physics.

  37. 37.

    See, e.g., Park (1968, 215); Belinfante (1975, 58–62); Fransen (1981, 10–12). In Sect. 5.3.2, Ishall come back to this objection.

  38. 38.

    For a discussion of several possibilities, see D’Espagnat (1976), Fransen (1981).

  39. 39.

    See his remarks on induction in D’Espagnat (1979).

  40. 40.

    This claim is quite plausible (see for instance the phenomenon of superconductivity).

  41. 41.

    See, e.g., Daneri et al. (1962), Machida and Namiki (1980); for some surveys of these approaches, see Jammer (1974, 471–521); D’Espagnat (1976, 161–226); Fransen (1981, 41–80).

  42. 42.

    Note that this does not imply that the systems described by equation (5.23) now possess the properties of a “classical ensemble”, that is to say, an ensemble built up from subensembles consisting of elements that can be unambiguously described (as far as M is concerned) by the states |g i >. The reason is that the decomposition of \( {\bar{\rho }_f} \) is not unique; hence, there will always be other observables of M that are not dispersion-free on the sub-ensembles with pointer positions g i . On this point, see Park (1968, 212–217); Fransen (1981, 10–12).

    However, this technical point does not raise any insurmountable problems for the measurement problem as it has been discussed here. In order to have numerical correspondence it is not necessary to show that every M is in a unique state |g i >. It suffices to show that, for the domain in question, a physically acceptable description exists that assigns the correct numerical values to the relevant classical quantities (such as the pointer position) for any apparatus M at any moment. For this purpose, a description such as (5.23) is satisfactory.

  43. 43.

    See also the remark at the end of Sect. 5.3.1. Of course, in practice we may not always succeed in supplying physical reasons for the equivalence of (5.22) and (5.23). I think that Machida and Namiki (1980) have come closest to achieving this. Daneri et al. (1962), however, have assumed this equivalence for nonphysical reasons (see also Sect. 5.3.3).

  44. 44.

    For some detailed reviews, see e.g. Stapp (1972), Scheibe (1973, 9–49); Radder (1979), Honner (1982).

  45. 45.

    This statement explicitly contradicts Jeffrey Bub’s assertion that Daneri, Loinger and Prosperi claim to use only purely quantum-mechanical arguments (see Bub 1968, 505 and 515). Apart from this point, Bub’s criticism seems to agree with mine.

  46. 46.

    However, in Radder (1979), my view of the measurement problem was closer to Bohr’s.

  47. 47.

    Honner (1982) interprets this relation as “transcendental”. And indeed, there does exist a clear analogy with the way Habermas establishes a relationship between intersubjectivity on the basis of instrumental controllability, the description of sensory experience with the help of fundamental cognitive schemata, and the limits these schemata impose on the basic predicates of internal-theoretical ontologies (see Sect. 2.3.2, point (5.5)). For this reason, my criticism of Bohr’s account will be analogous to that of Habermas^\primes views.

  48. 48.

    An important focus of this discussion is on the role played by the notions of nonlocality or nonseparability in the theorem of Einstein, Podolski and Rosen, and in the Bell theorem (see, e.g., D’Espagnat 1976, 1979; Selleri and Tarozzi 1981).

  49. 49.

    In Habermas’s view, this is the “language of things and events” (see Sect. 1.4.1). However, it is even questionable whether “things and events” form an adequate ontology for the classical physical theories (see the discussion of Bhaskar 1978 in Sect. 3.4).

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    Hans Radder

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Radder, H. (2012). Specification and Application: Two Case Studies from the History and Philosophy of Quantum Mechanics. In: The Material Realization of Science. Boston Studies in the Philosophy of Science, vol 294. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4107-2_5

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