Vickers on "Historical Magic in Old Quantum Theory"

Vickers (2011)

This paper follows closely from another paper, co-written by the author with Juha Saatsi,"Miraculous Success? Inconsistency and Truth in Kirchhoff's Diffraction theory" (2010). In both papers, the aim is to present historical case studies that stand as counterexamples to scientific realism. The key realist commitment here is that there must be an acceptable explanation for the fact that some scientific theories are spectacularly successful in providing explanations and/or predictions of empirical phenomena. The preferred explanation, of course, is that the key assumptions underlying these theories are in fact true.

Vickers offers several examples drawn from old quantum theory, which was the dominant idea in atomic physics at the beginning of the twentieth century until Heisenberg's development of quantum mechanics in 1925. The basic Bohr model of the atom depicted it as something like a planetary system, with electrons in elliptical orbits around the nucleus. The "quantum" aspect of this theory was that only certain kinds of orbits, corresponding to particular energy levels, were allowed. There were two key successes associated with this model. Firstly, Bohr was able to theoretically derive the value of the Rydberg constant, which is used to give the frequencies of the emission spectrumof the hydrogen atom. This theoretical value agreed very precisely with the value measured experimentally. The second success was Bohr's prediction of the spectrum of ionised helium (He+).

However, according to Vickers, these successes were based the assumption that the atom is literally like a planetary system, which we now believe to be quite false. In particular, one of the key boundary conditions used in the derivation of the Rydberg constant is that at higher quantum numbers the frequency of the light emitted must be the same as that predicted by classical electrodynamics. But the classical result for the case of a charged particle revolving around another charged object is that the frequency of emission should match the frequency of the revolution itself. Similarly, the derivation of the He+ spectrum relies on the notion that both electron and nucleus in fact orbit around their joint centre of gravity (this can also be referred to as calculating the "reduced mass" of the system).

The other major example discussed is Sommerfeld's derivation of the "fine structure" of the hydrogen spectrum. This derivation introduced a second quantum number that further subdivided each of the Bohr energy levels. For Sommerfeld, these numbers were understood to characterise the eccentricity of each electron's orbit. Converting the angular momentum of each orbit into a relativistic angular momentum then gives him a formula that accurately predicts the frequencies of the various fine structure lines. As with Bohr, however, the assumption that electron's have "real" orbits is radically false by our lights; in quantum mechanics, the second quantum number is understood to represent spin-orbit coupling and relativistic effects are introduced by means of aHamiltonian (representing energy) rather than angular momentum.

In all of these cases, Vickers believes that any attempt to explain the success of the theory by reference to the truth of its underlying assumptions is blocked since, by our current lights, these assumptions are false. He argues that there are two possible responses to this (there is also the possibility of discarding realism altogether, but this is not considered). The first is that the realist relax the claim that truth is always the only explanation for success. Sometimes, as he puts it, success may be due merely to "lucky coincidence". This appears to be his favoured resolution to this dilemma, as it is in the paper co-written with Saatsi.

The other response is to opt for structural realism. The structural realist does not claim that a successful theory is true as such, but only that the equations of the theory 'latch onto' the "structure" of the world. A (somewhat) clearer example is found in Worrall's "Structural Realism: The Best of Both Worlds?", where he argues that we should believe the wave equations of light, although we may disbelieve claims about what is "doing the waving". In a footnote (#23), Vickers argues that this position isn't quite realist enough, as while it gives an account of why the equation is successful, it doesn't explain how this success was achieved. In particular, the realist should be able to argue not only that the equation is true (or structural adequate), but that the assumptions used in deriving it are also true. This seems like an extremely strong claim about what is required for realism.

Indeed, the structuralist would certainly not see herself as required to defend this latter claim. She would wish to apply the structural interpretation to both the successful equation and the assumptions used in deriving it. Indeed, she would explain the success of the equation by reference to the claim that the assumptions are themselves structurally sound. To take one example, it is certainly true that Sommerfeld was wrong to believe that electrons have elliptical orbits, and that their energy levels are constrained because only certain eccentricities are allowed. He was not wrong to believe, however, that the electrons are characterised by a second quantum number, and this attribute further constrains their energy levels. He was right about the constraint, although wrong about what was "doing the constraining"!