Letting Go of the Orbit — The Birth of Quantum Mechanics and What the Old Quantum Theory Left Behind

By 1925 the old quantum theory had fitted better with every patch, and then torn along its seams. In the three years that followed, up to 1927, physicists let go of the very idea of an electron's "path" and gained a new mechanics: quantum mechanics. This final part of the series on the old quantum theory is about that leap, and about what was still standing once it had been made (Part 4 is here).

Electrons have waves too — de Broglie's matter waves, 1924

The first step came from an unexpected direction. In a short paper in 1923 and in his doctoral thesis of 1924, the French physicist Louis de Broglie reasoned like this: if light is a wave that also behaves like particles (Einstein's light quanta, which we met in Part 2), then perhaps electrons, which everyone took to be particles, also behave like waves.

As Part 2 showed, light quanta were resisted for so long because light shows interference and diffraction — things only waves can do. De Broglie took that "particle or wave" problem and extended it from light to matter. The wavelength of an electron, he proposed, gets shorter the harder the electron is moving (the greater its momentum).

This idea gives a reason for Bohr's condition, which in Part 3 had been left as a rule with no reason behind it. If an electron's wave goes once around the nucleus and comes back to its starting point after exactly a whole number of wavelengths, it joins smoothly onto itself and can keep its shape. With a fractional number, the crests and troughs fail to line up on every lap. The allowed paths come in steps because the lengths into which a wave fits exactly come in steps.

That electrons really do behave as waves was confirmed in 1927. In the United States, Davisson and Germer fired electrons at a crystal of nickel; in Britain, G. P. Thomson and Reid sent them through a thin film. Both caught the diffraction patterns that only waves can make.

Using nothing that cannot be seen — Heisenberg's "reinterpretation", 1925

The decisive step came by another route, from Werner Heisenberg, then 23 and working in Göttingen. At the start of a paper submitted on 29 July 1925, he wrote that it sought to lay the foundations of a quantum-theoretical mechanics based solely on relations between quantities that are observable in principle.

What cannot be observed is the electron's path. Nobody has ever seen where an electron goes round inside an atom. What can be seen is only the colour (frequency) and the strength of the light the atom gives off — and those two were exactly what the correspondence principle of Part 4 dealt with. Heisenberg took the quantities that in classical physics described "the shape of the path" and replaced them with quantities attached to pairs of states: which state jumps to which. He turned the correspondence principle directly into the framework of a mechanics.

When he worked out a rule for multiplying these quantities together, something strange happened. Swap the order of multiplication and the answer changes. With ordinary numbers 2×3 and 3×2 are the same, but here they are not. Born, who received the paper, recognised this as the multiplication of what mathematicians call "matrices" (tables of numbers arranged in rows and columns). In September, with his student Jordan, and in November, with Heisenberg as well, he set out the new mechanics as a system: matrix mechanics.

There is a well-known story that Heisenberg, fleeing hay fever, went to the small North Sea island of Helgoland and made the calculations come out in a single night, giving birth to quantum mechanics. But that story comes from the memoir he published in 1969. His letters from the time show that he kept working both before and after the island, was not confident in the finished manuscript, and left it to Born to decide whether it should be published. The paper itself is notoriously hard to follow; the physicist Steven Weinberg later called it "pure magic". The single night of inspiration is a story tidied up afterwards. In reality the theory was assembled by many hands over several months.

An equation for the waves — Schrödinger, 1926

Early in 1926, Erwin Schrödinger at the University of Zurich took de Broglie's matter waves as his clue and wrote down the equation that an electron's wave obeys. Just as a plucked string can sound only certain notes, the wave of an electron held by a nucleus can only vibrate in certain patterns. When he calculated the energy of each pattern, the sequence of energies for hydrogen that Bohr had found in 1913 came out exactly — this time without any rules added after the fact.

What guided Schrödinger was an analogy with light. Light is a wave, but when its wavelength is short enough, the old optics of straight-line rays is good enough. In the same way, classical mechanics is an approximation to the new wave mechanics. Historians who have studied his research notebooks point out that he grasped this while preparing his second paper. What the correspondence principle of Part 4 had demanded — that where the numbers are large, quantum answers should approach classical ones — had here become the very structure of the theory.

Calculation with matrices, and an equation for waves: two theories that looked nothing alike were shown by Schrödinger himself, within the same year of 1926, to be the same thing underneath.

The new mechanics also got past the wall the old quantum theory could not. In papers submitted in June and July 1926, Heisenberg took on two-electron helium. He explained why helium's light falls into two separate families of lines through something with no counterpart in classical physics: the two electrons trading places with each other. The energies agreed well enough that he judged the match acceptable, given the limits of his model and the accuracy of the measurements of the day. Helium, which in Part 4 never came out right, finally became something that could be calculated once the mechanics itself had been replaced.

What the waves tell us is the chance of finding the electron

So what is an electron's wave a wave of? Schrödinger wanted to believe that the electron itself was spread out like a wave. But on 25 June 1926 Born submitted a calculation of electrons bouncing off atoms, and offered another reading. What the wave decides is not which way the electron will fly off, but the probability that it flies off in each direction. In the same paper Born wrote: "I myself am inclined to give up determinism in the atomic world. But that is a philosophical question." In time this settled into the rule that the square of the wave's height at a place gives the probability of finding the electron there.

In a paper submitted on 23 March 1927, Heisenberg went further. The position and the momentum of an electron cannot both be pinned down as precisely as you like at the same time. To find where it is, you shine light of short wavelength on it, and the kick from the light disturbs its momentum. The more precisely you fix one, the more blurred the other becomes, and the two blurs multiplied together come to roughly Planck's constant h. This is the uncertainty relation. In the same paper he wrote: "the 'path' comes into being only because we observe it."

This is where the sentence that Part 3 set down and Part 4 recalled — that the picture of orbits does not hold in today's quantum mechanics — comes from. The picture of an electron circling the nucleus along a fixed line was itself given up at this point.

What the old quantum theory left behind

Was the old quantum theory, then, thrown away as a wrong theory? Looking back, it is hard to say so.

The first thing that remained was the step size h. The number Planck brought in in 1900 as "an act of desperation" appears in every equation of the new mechanics, and even sets the size of the uncertainty relation. Since 2019 it has also been used to define the kilogram.

The second was the answers themselves. Hydrogen's sequence of energies came straight out of Schrödinger's equation. Pauli's rule that each electron in an atom is specified by four numbers survived too, now given a meaning within the new mechanics.

The third was a way of thinking. Working out an answer you do not yet know from how it must correspond to classical physics: the correspondence principle was the scaffolding of Heisenberg's paper, and lived on in changed form in Schrödinger's analogy with light. A new theory must reproduce the old theory's answers wherever the old theory was right. That demand is still the yardstick physicists use to test a new theory today.

What was thrown away was the picture of electrons circling along fixed paths. But it was precisely because that picture was used to get the numbers right, and to pin down the places where it could never get them right, that it became clear what had to be thrown away. The old quantum theory may have been the theory that showed, more precisely than any other, "how far it could go, and where it could not".

The mechanics gained in 1927 remains, nearly a hundred years later, the foundation of technology, from the design of semiconductors to quantum computers that put superposition and interference to work in calculation. Yet on what probability really is, and what happens when we observe — the questions Born called "philosophical" — physicists still disagree today.