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§ Phase ii · Matter Waves · Chapter 01

de Broglie's wavelength

In the autumn of 1923 a thirty-one-year-old French aristocrat, working in his older brother's private laboratory in Paris, wrote three short notes that turned all of physics inside out. If light could behave as particles, he asked, why should particles not behave as waves? The answer arrived four years later from a wiring mishap in a Manhattan basement.

By 1923 the wave nature of light was the oldest settled fact in physics. Thomas Young had pushed two beams of sunlight through paired slits in 1801 and seen the bright-and-dark fringes that only waves can make. Augustin Fresnel had then written the equations and predicted the strange bright spot at the centre of a circular shadow that Siméon Poisson had ridiculed as absurd and that François Arago had then promptly photographed. By 1865 James Clerk Maxwell had pulled four equations out of his head in Cambridge and showed that an electromagnetic disturbance must travel at exactly the measured speed of light, which meant light was an electromagnetic disturbance, end of story. Heinrich Hertz produced the radio waves in his Karlsruhe lab in 1887 to clinch it. Light is a wave. Anyone who said otherwise in the year 1900 was simply wrong.

Then in 1905 a young patent clerk named Albert Einstein read Max Planck’s recent paper on black-body radiation and decided to take Planck more seriously than Planck took himself. We met them both in the previous phase. Einstein wrote down a single equation, E = hν, declared that a beam of light was actually a stream of discrete energy packets which he called Lichtquanten (light quanta) and that this was how the photoelectric effect was to be understood. The wave theory was not wrong, exactly, but it was no longer the whole story. By the early 1920s, after Arthur Compton’s 1923 experiment showed X-ray photons bouncing off electrons with the clean billiard-ball mechanics of point particles, every working physicist had to admit that light was somehow both a wave and a stream of bullets.

This bothered a certain kind of theorist deeply. The discomfort was not philosophical, it was symmetric. Nature, the argument went, does not generally pull off a trick on one side of the ledger without pulling the same trick on the other. If you let waves carry momentum like particles, the symmetry of the equations almost begs for the reverse: particles ought to carry frequencies like waves. No one had yet had the nerve to write that down. In the summer of 1923, sitting in the family library of one of the oldest aristocratic houses in France, a young theorist finally did.

Louis de Broglie’s training had been unusual. He had taken his first university degree in 1910 in history, intending the diplomatic career expected of a younger son of the house. Only after that did he turn to physics, prodded by Maurice, taking a second degree in 1913. The First World War interrupted everything. He spent the next six years as a wireless operator at the foot of the Eiffel Tower, where the great French military radio station had been built, transmitting and receiving signals at frequencies that were just beginning to be understood as electromagnetic waves. The war ended, he came home, and he started to ask himself what a wave really was.

The picture in his head went something like this. Einstein had said light carried energy in lumps E = hν and momentum in lumps p = h/λ. The two relations were tied together by special relativity, since for a photon E = pc and λν = c. So a wave with frequency ν and wavelength λ comes with a particle of energy hν and momentum h/λ. Beautiful. But then turn it around. Take a particle of momentum p (an electron, say, or a baseball, or a grain of sand) and apply the same relation in reverse. There must be, attached to that particle, a wave whose wavelength is

λ = h / p

That is the entire content of de Broglie’s hypothesis. One symbol on the left, three on the right. Nothing else.

De Broglie published the idea in three short notes to the Comptes Rendus of the Paris Academy of Sciences, dated September 10, September 24, and October 8, 1923. Each note runs only a page or two. He then expanded them into his doctoral thesis, Recherches sur la théorie des quanta, which he defended at the Sorbonne in November 1924. The examining committee, chaired by Paul Langevin, was unsure what to make of it. The mathematics was clean. The physical idea was so far outside the consensus that they could neither endorse it nor dismiss it. Langevin sent a copy to Einstein in Berlin and asked for his opinion. Einstein read the thesis on the train, replied with a single line that has been quoted ever since (he had, said Einstein, “lifted a corner of the great veil”), and added in a separate letter that the work was important enough that Schrödinger in Zurich should be told to read it immediately. The degree was granted. Schrödinger read the thesis the following summer on a skiing holiday in the Alps and went home to write the wave equation we will meet in the next chapter.

There was one practical problem. Einstein’s photon hypothesis had been confirmed by Lenard’s photoelectric data (and would soon be nailed shut by Compton). De Broglie’s matter wave was, in 1924, a theoretical guess with no observational backing. If electrons really were waves, then a beam of electrons fired at a properly sized obstacle ought to show diffraction. That is how you tell a wave from a bullet. A bullet either hits the obstacle or misses it. A wave bends around the edge and interferes with itself on the far side. So how do you build that obstacle? An ordinary slit will not do. The wavelength of even a slow electron is fantastically small.

A few quick numbers make the point. Take an electron accelerated through a one-volt potential, the sort of mild kick you would get from a flashlight battery. Its kinetic energy is one electron-volt, its momentum is p = √(2mE), and its de Broglie wavelength comes out to about twelve angstroms (12 × 10⁻¹⁰ metres). That is roughly the spacing between adjacent rows of atoms in a metal crystal. Spin the electron faster and the wavelength shrinks: at one hundred electron-volts it is about one angstrom, comparable to the X-ray wavelengths that Maurice de Broglie’s lab had been bouncing off crystals for a decade. At one million electron-volts it is a few thousandths of an angstrom, finer than any atom. To diffract electrons you need a grating whose spacing is also angstrom-scale. The natural answer was the same answer Max von Laue had used for X-rays in 1912: a crystal. Atoms in a crystal are arranged in regular three-dimensional lattices whose plane spacings sit right around the angstrom mark. If de Broglie was right, an electron beam shot at a crystal ought to come out the other side with the same Bragg-peak diffraction pattern that X-rays did.

In a 1925 conversation in Göttingen, the German theorist Walter Elsasser pointed out exactly this. Elsasser was twenty-one years old, an assistant to Max Born, and he had read de Broglie’s thesis the moment it arrived from Paris. Try electron diffraction on a nickel crystal, he suggested in a short note. Born forwarded the idea to his colleagues in England, but no one in Europe was set up to do the experiment immediately. The result, when it came, would arrive from somewhere unexpected.

The confirmation came from Bell Labs in Manhattan, through an accident no one had been hoping for. In April 1925, the American physicist Clinton Davisson and his assistant Lester Germer were studying how electrons scattered off the surface of nickel. They had a small evacuated chamber with an electron gun on one side and a movable detector on the other, and they were rotating the detector around to log how many electrons came off at each angle. Then a liquid-air bottle exploded next to the apparatus, the vacuum failed, and the hot nickel target oxidised badly. To clean it up, Davisson baked the nickel under hydrogen for hours and let it slowly cool. The cycle re-crystallised the polycrystalline nickel into a few large single crystals. When they fired the electron gun again, the angular intensity plot had developed sharp peaks where before it had been smooth.

Davisson initially thought he had a contamination problem. He measured for two more years. In late 1926 he attended a meeting of the British Association in Oxford and showed the curves to a German speaker named Born, who looked at them, blinked, and explained calmly that Davisson had just produced electron diffraction. The peaks were at angles given by Bragg’s law (nλ = 2d sin θ) using a wavelength of h / p, exactly as de Broglie had predicted. The paper, “Diffraction of Electrons by a Crystal of Nickel,” appeared in the December 1927 issue of the Physical Review. The wavelength they extracted from the data agreed with the de Broglie value to within a couple of percent.

That same year, on the other side of the Atlantic, an independent experiment was running. George Paget Thomson, son of J.J. Thomson, was professor at Aberdeen. His father had spent his career proving that the cathode ray was a particle, a tiny chip of negative electricity now called the electron, and had won the 1906 Nobel Prize for it. The son, working with a research student named Alexander Reid, fired faster electrons (a few tens of thousands of volts) through thin films of celluloid, gold, and aluminium foil and photographed the result on a plate behind the foil. What he got was a series of concentric rings, exactly the kind of pattern that X-rays make when they pass through a polycrystalline solid. The wavelength extracted from the ring spacings, again, was the de Broglie value. The 1937 Nobel Prize in Physics was awarded jointly to Davisson and the younger Thomson. De Broglie’s own Nobel had come earlier, in 1929, awarded for the thesis alone, eight years before the experiments had been fully digested.

p = E/c = hν/c = h/λ

E = γ m c²       p = γ m v       γ = 1 / √(1 − v²/c²)

λ = h / p = h / (γ m v)

λ = h / √(2 m K)

It is worth pausing on what the symmetry has bought us. Before 1924 the world had two kinds of stuff. Light was a wave, full stop; matter was a collection of point particles, full stop. After 1927 it had one kind of stuff that wore two costumes depending on what you asked it. An electron in a cathode-ray tube draws a sharp dot on the screen, behaving as a particle. An electron in a Davisson-Germer chamber draws fringes, behaving as a wave. A photon in a Compton scattering experiment bounces like a billiard ball; a photon in Young’s two-slit setup makes bright and dark bands. The two faces are no longer attached to two different kinds of object. They are two faces of one thing, and which face you see is set by what you measure.

That is the philosophical scandal at the bottom of quantum mechanics, and the rest of this book is essentially the long task of working out what it actually means. We will see, in the next chapter, Erwin Schrödinger sit down in a guest house in the Swiss Alps over Christmas 1925 with de Broglie’s thesis open on his desk, and write down the partial differential equation that any matter wave obeys. We will then meet, in the third phase, the strange probability rule that turns the wave amplitude into the chance of finding the particle at a particular spot when you look. And we will spend the rest of the book working out the consequences: the orbitals of atoms, the structure of the periodic table, the spectra of stars, the colours of dyes, the rigidity of solids, the inevitability of chemistry. All of it falls out of one equation. And that equation could not have been written without the small heuristic step that a thirty-one-year-old French aristocrat took, on a summer afternoon in 1923, when he asked himself why nature should not be symmetric.

De Broglie himself spent the rest of his life in a quiet revolt against the conclusions that followed from his own equation. He never quite accepted the probabilistic Copenhagen reading of the wavefunction. From the 1950s onward he worked with David Bohm on a “pilot wave” interpretation in which a real wave guides a real particle along a real trajectory, restoring something like classical determinism beneath the statistics. The mainstream of physics moved on without him. He inherited the family dukedom in 1960, on the death of his elder brother Maurice, and died unmarried in Louveciennes in 1987 at the age of ninety-four. The Sorbonne preserved his original 1924 thesis in its archives. It is roughly the length of a long magazine essay. Every page has been thumbed into transparency by half a century of physicists checking, line by line, the place where matter first turned into something that could interfere with itself.