Background Thought experiments: when they become real
The cascaded Stern–Gerlach experiments we have described — especially the one in which the two separated beams are brought back together — are thought experiments (in German Gedankenexperiment): they are a way to reason step by step, but we have not carried them out in the laboratory, nor are they easy to carry out.
This is not a technical detail. With a macroscopic apparatus, splitting the beams is easy, but recombining them coherently is almost impossible: one would have to bring the two wave packets of the same atom back together to within less than an atomic wavelength, controlling the magnetic fields to a degree that magnets and slits simply cannot reach. This is the so-called Humpty-Dumpty problem (Englert, Schwinger and Scully, 1988), after the nursery-rhyme character who, once fallen and broken, can never be put back together.
Only recently has the recombination actually been achieved. Between 2013 and 2021 the group of Ron Folman, at Ben-Gurion University of the Negev (Israel), built a complete, closed-loop Stern–Gerlach interferometer: it splits and then recombines the wave packet of a single atom. Not with magnets and ovens, but with ultracold atoms steered by atom chips — microcircuits that generate strong, exquisitely controlled magnetic fields. These are sophisticated and very expensive set-ups (ultra-high vacuum, laser cooling, Bose–Einstein condensates), a world away from a simple laboratory bench.
Why, then, have we relied on experiments that, in this form, are essentially never performed? Because they are a tool for reasoning: through them we introduced complex numbers in a natural way — the probability amplitudes — and we were led to suggest a formulation of Quantum Mechanics, which we then summarised in the principles. It is worth remembering, though, that those principles can be postulated independently of these experiments. The thought experiments make them plausible and motivate them, but they do not prove them.
A debt to Feynman. The approach of this opening part — Stern-Gerlach apparatuses used as filters, beams split and then recombined, probability amplitudes drawn from the experiments and summarised in a few principles — closely follows that of Richard Feynman in the third volume of the Feynman Lectures on Physics (in particular chapters 5 and 6), as already noted in the introduction: for the most part the text mirrors his treatment, recast here with a somewhat different formalism and with original graphical representations.
There are two differences. The first is the starting point: Feynman begins with spin one — atoms that split into three beams, that is three base states — and turns to spin one-half only later; here, instead, we use spin one-half from the very start, just two states as for silver, the simplest case. The second concerns the principles: those we summarised in the chapter contain the laws of amplitudes that Feynman states — among them the rule that the amplitude to go from one state to another is obtained by summing the contributions through each base state — but they add time evolution, which does not appear in those chapters. And it is precisely that addition which makes what follows possible: deriving the Schrödinger equation instead of postulating it, the road Feynman does not take and which is this thesis’ own contribution.
Even the Humpty-Dumpty image is his: in his idealised thought experiment the beams recombine and “Humpty Dumpty is put back together again”; here we have added that, in reality, it is anything but simple.