In the mid 1920s the old quantum theory was failing, and the people who fixed it were strikingly young. Within about three years a French prince, a handful of German and Austrian students and a shy Englishman built a new mechanics from the ground up. Wolfgang Pauli, Werner Heisenberg, Paul Dirac and Pascual Jordan were all in their early or middle twenties, and the joke went round that the new theory was Knabenphysik, "boys' physics," because its creators were mostly in their twenties.

A prince proposes matter waves

Louis de Broglie came from a French noble family and had studied history before turning to physics. During the First World War he served in radio work at the Eiffel Tower. In his doctoral thesis of 1924 he reversed Einstein's reasoning. If light, long known as a wave, also behaves like particles, then perhaps particles such as electrons also behave like waves. He gave each particle a wavelength that shrinks as its momentum grows.

The idea offered a striking explanation of Bohr's allowed orbits. In de Broglie's picture, each allowed orbit holds a whole number of wavelengths, so the wave closes on itself smoothly, like a vibrating ring. Orbits in between would make the wave interfere with itself and cancel. The examiners in Paris were unsure what to make of this. Paul Langevin sent a copy to Einstein, and Einstein endorsed the thesis, judging that it touched something real. That support mattered, and it brought de Broglie's work to the attention of Erwin Schrödinger.

Exclusion and spin

Meanwhile Pauli, in Hamburg, was puzzling over how electrons fill the shells of atoms. In early 1925 he proposed a rule that has held ever since: no two electrons in an atom can share all their quantum numbers. The exclusion principle explained why electrons do not all crowd into the lowest state, and so why the elements have such different chemistry.

Pauli's rule needed a fourth quantum number that could take only two values. Later in 1925 George Uhlenbeck and Samuel Goudsmit, two students in Leiden, suggested that the electron behaves as if it spins, with two possible directions. They had second thoughts and asked their teacher Paul Ehrenfest to hold the paper, but he had already sent it, telling them they were young enough to afford a foolish idea. Ralph Kronig had thought of spin earlier that year, but Pauli had discouraged him. Spin turned out to be real, though not literally a spinning ball.

Helgoland

Heisenberg, aged 23 and working in Göttingen under Max Born, had been trying to calculate the intensities of spectral lines. In June 1925 he was recovering from a severe attack of hay fever and went to Helgoland, a nearly treeless island in the North Sea. There he made a radical decision. Nobody could see an electron's orbit; what experiments actually measured were the frequencies and intensities of the light atoms gave off. So he would build the theory only from quantities that could be observed.

His paper, published that autumn, announced the aim in its opening line. In translation, it sought to establish quantum mechanics "founded exclusively upon relationships between quantities which in principle are observable." Working with tables of numbers linking pairs of states, Heisenberg found a consistent scheme that conserved energy. It had one strange feature: when he multiplied two quantities, the answer depended on which came first.

Matrices and a third route

Born recognized the oddity. Heisenberg's rule was matrix multiplication, in which order matters, a technique from mathematics that few physicists then knew. Born and his student Jordan quickly recast the theory in matrix form, and in November 1925 Born, Heisenberg and Jordan published a full account, later called the three-man paper. Matrix mechanics was born.

In Cambridge, Dirac read Heisenberg's work in proof and saw a deep link to a classical technique called the Poisson bracket. He built his own version, with quantities he called q-numbers. Then early in 1926 Pauli used matrix mechanics to derive the hydrogen spectrum, the same result Bohr had reached by fiat, now from a consistent theory. The new mechanics had passed its first real test.

Worked example: Heisenberg's discipline

Consider the choice Heisenberg made on Helgoland. Everyone had pictured electrons moving in orbits, and the old quantum theory was built around those orbits. Yet no experiment had ever observed one. What physicists measured were lines in a spectrum: their frequencies, which depend on pairs of states, and their brightness. Heisenberg asked what would happen if he stopped trying to describe orbits at all and kept only those measured quantities, arranged by pairs of states.

The step was risky. It meant giving up a picture of what goes on inside the atom, something many physicists, Schrödinger among them, considered essential. But it freed Heisenberg from assumptions that had been leading the old theory astray. A rule of thumb lies inside his choice: when a theory keeps failing, check whether some of its basic terms refer to things nobody has observed. Sometimes progress begins by dropping a familiar picture rather than adding a new piece.