Matrix mechanics worked, but most physicists found it forbidding. Within months a very different theory arrived, built on mathematics every physicist already knew. For a short, intense period in 1926 there seemed to be two rival quantum mechanics, and a sharp argument about which one told the truth about nature.

A wave equation in the mountains

Erwin Schrödinger was an Austrian physicist teaching in Zurich. At 38 he was older than the Göttingen group, and he had read de Broglie's thesis with interest. According to a later recollection by Felix Bloch, Peter Debye remarked at a Zurich seminar that if electrons were waves, there ought to be a wave equation. Over Christmas 1925, staying at the Swiss Alpine resort of Arosa, Schrödinger found one.

Early in 1926 he published a series of papers titled "Quantization as an Eigenvalue Problem." In his theory the electron in an atom was described by a wave, and its allowed energies arose naturally, like the overtones of a vibrating string. A violin string can sound only certain notes because only certain waves fit between its ends. In the same way, only certain wave patterns fit around a nucleus, and the whole numbers that Bohr had imposed by hand simply appeared. Physicists were delighted. The equation used familiar tools, and it came with a picture.

Two theories, one result

The two theories looked utterly different: one used tables of numbers that ignored any picture, the other used smooth waves. Yet they gave the same answers. In 1926 Schrödinger showed they were mathematically equivalent, and others, including Pauli and the American Carl Eckart, reached similar results. They were two languages for one theory.

The equivalence did not end the quarrel. Schrödinger wrote in his 1926 paper that he had known of Heisenberg's theory but had felt "discouraged, not to say repelled," by its difficult algebra and lack of visualizability, as one translation puts it. Heisenberg, for his part, privately dismissed Schrödinger's picture in very rude terms. Behind the insults lay a real question: what, physically, was the wave?

What is the wave?

Schrödinger at first hoped the wave was a real physical thing, the electron's charge spread out like a cloud. That reading ran into trouble. A wave packet tends to spread out over time, while electrons are always detected in one place. Worse, for many particles the wave lived in an abstract space, not ordinary space: describing two electrons required a wave in six dimensions, three for each particle, and the count kept growing. Whatever the wave was, it did not seem to be a simple ripple in the world.

Born's footnote

Max Born offered a different answer in the summer of 1926, in a paper on collisions between particles. He proposed that the wave does not describe the particle's matter but tells us the chances of finding it in one place or another. Remarkably, he first stated that the square of the wave gives the probability in a footnote added at the proof stage, correcting his main text. In the same paper he summed up the new picture: "The motion of particles follows probability laws but the probability itself propagates according to the law of causality."

This was a large step. Classical physics had treated probability as a measure of our ignorance. In Born's reading, probability seemed built into the basic laws themselves. Many physicists accepted it as a working rule long before they agreed on what it meant, and Born's own recognition came late. He received the Nobel Prize in 1954, almost three decades later.

Uncertainty, 1927

Early in 1927, working at Bohr's institute in Copenhagen, Heisenberg published his uncertainty relations. To make them vivid he imagined trying to see an electron with a microscope using gamma rays, light of very short wavelength. Sharper images need shorter wavelengths, but shorter waves hit the electron harder, disturbing its motion. Bohr criticized the argument, insisting that the limit came from the dual wave and particle nature of things, not merely from clumsy measurement. Heisenberg added a note acknowledging the point. The debate over what this means belongs to the interpretations of quantum mechanics, which are a story of their own.

Worked example: two maps of one atom

Think of the hydrogen atom in 1926. Pauli's matrix calculation and Schrödinger's wave calculation produce the same list of energies. A matrix physicist would say: here is a table linking pairs of states, and from it we read off the frequencies of light. A wave physicist would say: here is a standing wave around the nucleus, and only certain patterns fit. Both predict the same red line in hydrogen's spectrum to the same precision.

The episode shows that two descriptions can look nothing alike and still be the same theory, much as one city can be drawn as a street map or as a subway diagram. It also shows that agreeing on predictions is not the same as agreeing on meaning. Schrödinger wanted a picture, Heisenberg distrusted pictures, and Born's probability rule worked inside either one. The physicists of 1926 settled how to calculate long before they settled, if they ever did, what the calculations describe.