Stretched Until It Tore — Sommerfeld's Extension and the Impasse of the Old Quantum Theory

Bohr's 1913 model got the light given off by hydrogen strikingly right. So how far could the idea be stretched? Let the electron's circular path become an ellipse, bring in relativity, put the atom in a magnetic field — between 1916 and 1925, physicists added to the model piece by piece. Part 4 of this series on the old quantum theory is about a patchwork that fitted better with every patch, and then tore along its seams (Part 3 is here).

From circles to ellipses — Sommerfeld's extension of 1916

In December 1915 and January 1916, Arnold Sommerfeld of the University of Munich published two papers extending Bohr's model. In Bohr's model the electron went round in a circle (as noted in Part 3, this "path" is how the theory of the time pictured things). Sommerfeld allowed ellipses as well. He split the electron's motion in two — going round the centre, and moving closer to and farther from it — and applied to each the same kind of rule Bohr had used: only certain separated values are allowed. In both cases the size of the step was Planck's constant h, which we met in Part 1. The electron's state was now specified by two numbers instead of one.

The result was a surprise. As long as the two numbers add up to the same total, a circle and a thin ellipse have exactly the same energy. Bohr's formula survived untouched.

The difference appeared once relativity was brought in. An electron on an ellipse speeds up as it swings close to the centre. According to Einstein's theory of relativity, something moving fast becomes slightly heavier. How much this matters depends on how thin the ellipse is, so even with the same total, the energies shift apart by a tiny amount. A line that looks single turns out, on close inspection, to be split into fine components — the phenomenon called "fine structure" now had an explanation. The size of the splitting is set by one number: the speed of an electron on the innermost circle, as a fraction of the speed of light. It comes to about 1/137. It later became known as the "fine-structure constant", and it is still counted among the fundamental constants of physics.

The check came from Friedrich Paschen, a spectroscopist in Tübingen. In the helium ion (the helium with only one electron that we met in Part 3), the splitting is sixteen times wider than in hydrogen. Paschen measured the line at a wavelength of 468.6 nanometres, and in May 1916 he wrote to Sommerfeld: "My measurements are now finished and are in wonderful agreement with your fine structures. Without your theory these results would not have been found." In August Einstein wrote that only through Sommerfeld's analyses had Bohr's idea become entirely convincing.

When you have no answer, ask the old physics — the correspondence principle

The theory could give the wavelengths of the lines, but which lines come out strong and which weak? Which jumps happen, and which never do? To this question, left open at the end of Part 3, the theory had no answer.

What Bohr put forward in 1918 was not a head-on answer but a clever detour. Where the numbers are large — high up the ladder of energy steps — the steps crowd together, and the fact that the values come in separate steps hardly shows. There, the answers of the quantum theory ought to approach those of classical physics. And indeed, the frequency of the light (how many times it waves per second) given off in a jump between neighbouring steps comes closer and closer, as the numbers grow, to the frequency of light that an orbiting electron would give off in classical physics.

Taking this as his lead, Bohr broke the classical motion of the electron into components: the basic revolution, plus wobbles at two times, three times its rate, and so on. Each component corresponds to one jump. If a component is missing, that jump does not happen. If a component is strong, that line comes out strong. In 1920 Bohr named this way of thinking "correspondence". Today it is known as the correspondence principle.

This was less a theory than a way of estimating the right answer, without knowing it, from the conditions that answer had to satisfy. Even so, the estimates worked well, and a few years later they became the footing on which the young Heisenberg built a new mechanics.

With two electrons, nothing fits any more

A theory that fitted better the further it was stretched nevertheless ran aground in two places.

The first was the helium atom: just two electrons around a central charge, the simplest atom after hydrogen. Yet whatever combination of paths the two electrons were assumed to follow, the numbers would not come out. In 1922 the American physicist Van Vleck worked carefully through a leading model and found that the energy needed to pull one electron away came to 20.7 electronvolts (the electronvolt is the unit of energy used for atoms). The experimental value at the time was 25.4 electronvolts. In 1923 Born and Heisenberg attempted a more general calculation and wrote that "a consistent quantum-theoretical calculation of the helium problem leads to wrong values for the energy terms". Either the rules themselves were wrong, they concluded, or the electrons no longer obeyed the laws of mechanics.

The second was magnetism. Put a glowing atom between the poles of a magnet and a single line splits into several. The tidy case, a split into three, could be explained both by classical physics and by Bohr's model. In reality, though, most lines split in ways nobody could explain — into four, into six. This was called the "anomalous Zeeman effect". Spectroscopists found rules from experiment that matched the patterns of lines, but those rules contained "halves" — numbers that the theory gave no reason for.

In the autumn of 1922 the 22-year-old Pauli was walking gloomily through the streets of Copenhagen when an acquaintance remarked that he looked very unhappy. His reply: "How can one look happy when he is thinking about the anomalous Zeeman effect?" Pauli told the story himself in a talk in 1946.

A silver trace split in two, and the electron that "spins"

On the night of 7–8 February 1922, in Frankfurt, Otto Stern and Walther Gerlach succeeded with an experiment. They heated silver to make a thin stream of atoms, passed it between the poles of a specially shaped magnet, and let it land on a glass plate. Each silver atom behaves like a tiny magnet, so the magnetic field bends its path.

By classical physics, those tiny magnets should point in every possible direction, so the trace on the plate would simply be smeared out. Bohr's theory predicted that only certain orientations were allowed, and that the stream would divide in two. The result: two. The traces were about 0.1 millimetres apart, clearly distinguishable in a photograph through a microscope. Gerlach promptly sent Bohr a postcard with a photo of the result, saying that his theory had been confirmed.

What we know now, however, is that the two-way split was not produced by what Bohr's theory had in mind — the electron's motion around the atom. A property of the electron that nobody knew about at the time was giving the right answer.

That property took shape three years later. In 1925 Pauli announced a rule: each electron in an atom is specified by four numbers, and no two electrons in the same atom can share all four (the Pauli exclusion principle). Of the fourth number, he would say only that it was a "two-valuedness" of the electron "which cannot be described from the classical point of view".

That autumn, two young physicists at Leiden, Uhlenbeck and Goudsmit, proposed that the fourth number meant the electron was spinning. Their professor, Ehrenfest, is said to have told them: "Well, that is a nice idea, though it may be wrong. But you don't yet have a reputation, so you have nothing to lose." The paper appeared that autumn. Pauli at first strongly doubted the idea because of its classical, mechanical character, but, as he said later in his Nobel lecture, he was won over by Thomas's calculations. Today spin is understood not as a little ball actually rotating, but as a property the electron is simply born with.

What had been achieved by this point, and what had not

Quite a lot had been achieved. The fine splitting of lines. A working estimate of how strong lines are, and which jumps happen. The fact that an atom in a magnetic field can take only certain orientations. And the rule that electrons are specified by four numbers, with no two sharing the same set — a rule that opened the way to explaining why elements with similar properties turn up at regular intervals in the periodic table.

What had not been achieved was deriving all of this from a single principle. The rules for ellipses, the correction from relativity, the unexplained "halves", the rule of thumb called correspondence, and a fourth number added after the fact. Each of them fitted, but each had been patched on separately. Helium never came out right, and in the Stern–Gerlach experiment, even when the theory was right, the reason it was right was mistaken.

Born and Heisenberg had named two possibilities in 1923: either the rules were wrong, or the electrons did not obey the laws of mechanics. As it turned out, the answer lay closer to the second. What needed changing was not the seams, but the mechanics used to describe the electron's motion itself. Next time, in the final part, we follow how that new mechanics was born between 1924 and 1927, and what it carried over from the old quantum theory.