Physics 1924

Researches on the Theory of the Quanta

Louis de Broglie

If light can be a particle, then matter can be a wave.

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In depth · the introduction

A young aristocrat asked a simple, daring question: if light — a wave — can act like a particle, why can't a particle act like a wave?

The big idea

By 1924 physicists had reluctantly accepted that light is two-faced: a wave that spreads and interferes, but also a hail of tiny packets called photons. Louis de Broglie noticed that the deal was one-sided. Light, the wave, had been handed particle-like behaviour — but matter, made of particles, had never been handed wave-like behaviour. He proposed to balance the books.

His claim: every moving object has a wavelength, given by a strikingly simple formula — wavelength equals Planck's constant divided by momentum. The faster and heavier something moves, the shorter its wave. For a baseball the wave is unimaginably tiny and you never notice it. But for an electron, light and quick, the wave is about the size of an atom — big enough to matter, and big enough to measure.

How it came about

Louis de Broglie came from one of France's grand families and had first studied history. Drawn to physics by the puzzles of the new quantum theory, and influenced by his older brother Maurice, an experimental physicist working on X-rays, he turned the question of light's dual nature over in his mind for years.

His answer became his 1924 doctoral thesis, defended at the Sorbonne. The idea was so unusual that his examiners weren't sure it could be true. They asked Albert Einstein's opinion; Einstein read it and wrote back that de Broglie had "lifted a corner of the great veil." That single sentence changed everything — and within three years, experiments fired electrons at crystals and saw them ripple and interfere, exactly as waves do.

Why it mattered

De Broglie's wavelength is the hinge on which modern physics turned. It told Erwin Schrödinger what kind of equation to look for, and the wave equation he found in 1926 became the engine of all of chemistry and quantum mechanics — the reason we can explain why atoms bond, why materials conduct or insulate, why the periodic table has the shape it does. The idea that matter has a wavelength is also why electron microscopes exist, letting us see viruses and individual atoms.

A way to picture it

Think of a long jump-rope. Hold the ends and whip it: the wave travels along the rope, but the rope itself only moves up and down — the bulge of energy moves faster than any single point of rope. De Broglie's matter wave works like that. The electron is the slow-moving bulge — the "wave packet" — that carries it from here to there, while the fast little ripples inside (the phase wave) race ahead. He proved the two stay perfectly in step, like the spokes and the rim of a rolling wheel turning together.

Where it sits

This idea is a bridge. Behind it stand Max Planck and Albert Einstein, who first found that light comes in quantised packets; ahead of it stand Schrödinger and Werner Heisenberg, whose 1926 wave and matrix mechanics turned de Broglie's hint into a full theory. De Broglie won the 1929 Nobel Prize for the thesis. He spent his later years defending a "pilot-wave" picture that most physicists set aside — though it was revived decades later by David Bohm, and the debate over what the wave really is has never quite closed.

The original document

Original source text

Introduction — a symmetry to restore

Louis de Broglie · Recherches sur la théorie des quanta · Thesis, Paris, 1924 · Annales de Physique (10) 3 (1925): 22–128

[Light, since Einstein's 1905 light-quanta, had been granted a particle aspect alongside its wave aspect. De Broglie's thesis asks whether the converse holds for matter.]

After long reflection in solitude and meditation, I suddenly had the idea, during the year 1923, that the discovery made by Einstein in 1905 should be generalised by extending it to all material particles and notably to electrons.

Chapter I — The phase wave

An internal periodic phenomenon

We shall assume the existence of a certain periodic phenomenon of a yet to be determined character, which is to be attributed to each and every isolated parcel of energy, and which depends on its proper mass through the Planck–Einstein equation.

[For a particle of rest mass m₀, de Broglie sets the rest-frame frequency by hν₀ = m₀c². Seen from the laboratory, the moving particle's internal clock runs slow (time dilation) at ν₁ = ν₀√(1−β²), yet an accompanying wave runs at the higher frequency ν = ν₀ ⁄ √(1−β²) and at phase velocity V = c²/v.]

We are then inclined to admit that any moving body may be accompanied by a wave and that it is impossible to disjoin motion of body and propagation of wave.

The theorem of the harmony of phases

[De Broglie proves that the slow internal clock of the particle and the fast external wave, though they have different frequencies, stay perpetually in step: at the particle's position the phase of the wave always agrees with the phase of the internal vibration. This is the harmony of phases that the wave's velocity V = c²/v was chosen precisely to secure.]

[ … ]

The phase wave guides the displacement of energy; its group velocity is equal to the velocity of the particle, while its phase velocity exceeds the velocity of light without, since it carries no energy, contradicting relativity.

The wavelength of matter

[Combining the relations gives the result the thesis is remembered for: a moving body of momentum p has an associated wavelength.]

λ = h / p

[For an electron this wavelength is comparable to the spacing of atoms in a crystal — which is why de Broglie foresaw that a stream of electrons ought to be diffracted by a crystal lattice, just as X-rays are.]

Bohr's orbits, re-read

[De Broglie shows that Bohr's mysterious quantum condition for the allowed electron orbits is simply the requirement that the phase wave close on itself — that a whole number of wavelengths fit around the orbit, a standing wave rather than a wave that interferes with itself and dies away.]

The stability conditions of the trajectories in Bohr's theory are interpretable as the resonance condition of the phase wave along the closed path.