Abstract
The conceptual core of this chapter is Heisenberg’s discovery of quantum mechanics, considered as arising from certain fundamental principles of quantum physics and as established by giving these principles a mathematical expression. The chapter also considers Bohr’s 1913 atomic theory, a crucial development in the history of quantum theory ultimately leading to Heisenberg’s discovery, and Schrödinger’s discovery of wave mechanics, initially from very different physical principles. At the same time, Schrödinger had implicitly used some of the same principles that were expressly used by Heisenberg, thus meeting Heisenberg’s program, against Schrödinger’s own grain. After a general introduction given in Sect. 2.1, Sect. 2.2 considers some of the key aspects of Einstein’s and Bohr’s work in the old quantum theory, especially significant for the invention of quantum mechanics by Heisenberg and Schrödinger, discussed in Sects. 2.3 and 2.4, respectively. Sect. 2.5, by way of a conclusion, reflects on the new relationships between mathematics and physics established by quantum mechanics in nonrealist, RWR-principle-based, interpretations.
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Notes
- 1.
- 2.
Bohr’s 1913 postulates should not be confused with Bohr’s more general concept of “the quantum postulate,” introduced, along with the concept of complementarity, in 1927, following quantum mechanics, although the quantum postulate, too, concerned quantum phenomena themselves and did not depend on quantum mechanics (Bohr 1927, 1987, v. 1, pp. 52–53).
- 3.
Both Bohr’s theory and quantum mechanics predicted the probabilities or statistics of transitions between them, but unlike Bohr’s theory, which treated stationary states classically and hence also by representing them (as orbits), matrix mechanics did not treat the behavior of electrons in stationary states at all. Dirac’s q-number scheme and then Schrödinger’s equation were able to do so, but now also in probabilistically or statistically predictive terms, rather than representational terms (against Schrödinger’s initial hopes). As I said, by that time the concept of electron orbit was no longer possible to sustain even for stationary states.
- 4.
Bohr’s ad hoc but ingenious use of the correspondence principle in the old quantum theory is less germane to my argument in this study and will be put aside.
- 5.
The concept and principle of complementarity, as formulated here, are closer to the way they are presented in Bohr’s later works, from 1929 on, impacted by his debate with Einstein. In these works, the concept is exemplified by position and the momentum measurements. Such measurements are always mutually exclusive, and as such correlative to the uncertainty relations, but both possible to be performed on a given quantum object at different points of time and both necessary for a complete (Bohr-complete) account of the behavior of quantum objects, in Bohr’s ultimate, RWR-principle-based, interpretation, in terms of effects quantum objects can have on measuring instruments.
- 6.
This does not of course mean that Heisenberg’s invention of quantum mechanics was independent of or was not helped by preceding contributions, even beyond the key pertinent works in the old quantum theory by Einstein, Bohr, Sommerfeld, and others, discussed in Sect. 2.1. H. Kramers’s work on dispersion and his collaboration with Heisenberg on the subject were especially important for Heisenberg’s work (Kramers 1924; Kramers and Heisenberg 1925). See (Mehra and Rechenberg 2001, v. 2) for an account of this history.
- 7.
Bohr was not unprepared for this eventuality, as is clear from his letter to Heisenberg (Letter to Heisenberg, April 18, 1925, Bohr 1972–1996, vol. 5, pp. 79–80). The letter was written in the wake of the collapse of the so-called Bohr–Kramers–Slater (BKS) proposal, which, among other things, implied that the energy conservation law only applied statistically (Bohr et al. 1924), and shortly before Heisenberg’s discovery of quantum mechanics. Bohr’s article was in preparation as a survey of the state of atomic theory before Heisenberg’s discovery of quantum mechanics, but it was modified in view of this discovery and Born and Jordan’s work on casting Heisenberg’s mechanics into its proper matrix form. Bohr added a section from which I cite here. Bohr’s views expressed in this section are crucial, and I shall return to his argument there in closing this chapter.
- 8.
As noted earlier (note 3), matrix mechanics did not offer a treatment of stationary states, in which and only in which one could in principle speak of a position of an electron in an atom.
- 9.
It is true that quantum data may present itself in terms of interferometry, which is seen in the graphical representation of counting rates (proportional to the probabilities in question) that are typically oscillatory. In referring to this data, one could speak more intuitively, albeit still metaphorically, of “amplitudes” of these oscillations, just as one speaks of “interference” in referring to the (discrete) interference pattern observed in the double-slit experiment in the corresponding set-up (with both slits opens and no devices installed allowing one to establish through which slit each quantum object passes). I am indebted to G. Jaeger for pointing out this aspect of the quantum-mechanical situation. However, these amplitudes (which are related to real measurable quantities) are not the same as the “symbolic” amplitudes in question. The latter amplitudes are complex quantities enabling us to predict the probabilities relating to the oscillations in question. This is why these amplitudes are seen as “symbolic” by Bohr and Heisenberg, that is, as symbols borrowed from classical physics without having the physical meaning they have there. To cite Bohr: “The symbolic character … of the artifices [of the quantum-mechanical formalism] also becomes apparent in that an exhaustive description of the electromagnetic wave fields leave no room for light quanta and in that, in using the conception of matter waves, there is never any question of a complete description similar to that of the classical theories. Indeed, … the absolute value of the so-called phase of the waves never comes into consideration when interpreting the experimental results. In this connection, it should also be emphasized that the term ‘probability amplitude’ for the amplitude function of the matter waves is part of a mode of expression which, although often convenient, can, nevertheless, make no claim to possessing general validity [as concerns what is observed]” (Bohr 1929b, 1987, v. 1, p. 17).
- 10.
For these reasons, quantum probabilities are sometimes referred to as non-additive. For a classic account of quantum probability amplitudes, see (Feynman et al. 1977, v. 3, pp. 1–11). Feynman has an excellent earlier article on the subject (Feynman 1951). See also (Gillies 2000; Hájek 2014; Khrennikov 2009).
- 11.
A quantum-theoretical interpretation refers here to the change of classical variables to quantum variables, rather than a physical interpretation of the resulting mathematical model (matrix mechanics), although this change implies certain physical features, specifically a predictive rather than representational nature of the model.
- 12.
In general, as noted in Chap. 1, their views were always somewhat different and, especially, diverged more from the 1930s on, without, however, ever losing some affinities. These affinities position both views within the spirit of Copenhagen.
- 13.
It is not my aim to offer a comprehensive account of Schrödinger’s work on wave mechanics, which has received several extended treatments. Mehra and Rechenberg give Schrödinger more space than to any other founding figure, and my analysis here is indebted to their historical discussion (Mehra and Rechenberg 2001, v. 5). I am less in accord with their philosophical argumentation, and indeed part with it nearly altogether. Another major study is (Bitbol 1996). Schrödinger’s collected papers on wave mechanics are assembled in (Schrödinger 1928). For his other important papers on quantum mechanics, see (Schrödinger 1995).
- 14.
It is worth keeping in mind that by the time of writing this paper Schrödinger already knew his equation, which, as indicated above, he discovered differently, by directly using de Broglie’s formulas, rather than in his first published paper. Accordingly, his introduction of this variable is not as unmotivated or sudden as it might appear.
- 15.
Cf., however, Schrödinger’s argument in (Schrödinger 1995), mentioned above.
- 16.
Such traces are dot-like only at a low resolution, which “disguises” a very complex physical object, composed of millions of atoms, and is particle-like only in the sense that a classical object idealized as a particle would leave a similar trace.
- 17.
Cf. J. B. Barbour’s concept of “Platonia,” an underlying reality without change and motion (Barbour 1999), the idea apparently originating with Parmenides, who inspired Plato. Barbour’s conception appears to derive from the idea that it does not appear possible by means of quantum theory to describe or represent the motion of the ultimate constituents of nature. From the present viewpoint, however, while this is true, it does not follow that everything “stands still” at that level, since, as just explained, the latter concept would not apply any more than that of “motion” (or “object” and “quantum”) to quantum objects.
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Plotnitsky, A. (2016). Bohr, Heisenberg, Schrödinger, and the Principles of Quantum Mechanics. In: The Principles of Quantum Theory, From Planck's Quanta to the Higgs Boson. Springer, Cham. https://doi.org/10.1007/978-3-319-32068-7_2
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