Matter Waves: de Broglie's Audacious Symmetry

Chapter 4 showed that light of frequency f is absorbed in whole packets, photons, each carrying energy E = hf, where h = 6.626×10⁻³⁴ J·s is Planck's constant. But a particle carries momentum as well as energy: the quantity of motion it hands over in a collision, for ordinary bodies mass times velocity, p = mv. Light has no mass, yet James Clerk Maxwell's electromagnetic theory predicted in 1873 that it presses on whatever absorbs it, and around 1900 Pyotr Lebedev in Moscow, and Ernest Nichols and Gordon Hull at Dartmouth College, measured this feeble radiation pressure.

Einstein's relativity explains how a massless thing can push. It links the energy, momentum and mass of anything through E² = (pc)² + (mc²)², where c is the speed of light. For a photon the mass term vanishes, leaving p = E/c. Combine that with E = hf and the wave relation c = fλ, and the momentum depends only on the wavelength: p = h/λ. Shorter waves hit harder. A green photon of 532 nm carries just 1.25×10⁻²⁷ kilogram-metres per second, which is why full sunlight at the ground, about 1,000 watts per square metre, presses on your skin with a pressure you never notice: about 3 millionths of a newton per square metre.

The proof that single photons carry momentum came in 1923. Arthur Compton scattered X-rays from the electrons in graphite and found that the scattered rays had longer wavelengths, by an amount that grew with the scattering angle. No classical wave theory could explain the shift, but a collision could: treat each photon as a particle of momentum h/λ recoiling from an electron like a billiard ball, and conservation of energy and momentum predict the shift exactly. Compton received the 1927 Nobel Prize in Physics. Today solar sails ride on photon momentum, optical tweezers grip living cells with it, and laser cooling brakes atoms with it, one kick at a time.

Louis de Broglie made an unlikely revolutionary. Born in 1892 into a French aristocratic family of soldiers and statesmen, he took a first degree in history before turning to physics, drawn in by his elder brother Maurice, an accomplished experimenter with X-rays. After wartime service in army radio, working at the transmitter on the Eiffel Tower, he pondered X-rays that behaved like waves in one experiment and like particles in the next, and came to suspect that this doubleness was not a peculiarity of light at all, but a property of everything.

In three short notes to the French Academy of Sciences in 1923, and in his doctoral thesis, defended at the University of Paris on 25 November 1924, he proposed a symmetry. If a light wave comes with particles of momentum p = h/λ, then every particle of matter should come with a wave of wavelength λ = h/p. Read it as a recipe: divide Planck's constant by a particle's momentum and you get the length of its wave, the distance from one crest to the next. Wavelength and momentum are inversely proportional, so doubling the momentum halves the wavelength, and for anything moving much slower than light, λ = h/(mv). This is the de Broglie wavelength, and the wave is called a matter wave.

His examiners were impressed and baffled in roughly equal measure. His supervisor, Paul Langevin, sent a copy to Einstein, who wrote that de Broglie had 'lifted a corner of the great veil'. By de Broglie's own later account, asked at the defence how the idea might be tested, he suggested scattering electrons from a crystal, whose regularly spaced atoms could act as a diffraction grating. Experiment answered three years later, and in 1929 he received the Nobel Prize in Physics 'for his discovery of the wave nature of electrons'.

A wave announces itself through diffraction, the spreading of a wave past an edge or through an opening, and through interference, the reinforcement and cancellation where waves overlap, both met in chapter 2. Both are dramatic only when the wavelength is comparable to the openings or obstacles involved: sound, about a metre long, bends round doorways, while light casts sharp shadows of them. So the question for any object is never whether it has a wavelength, since everything with momentum does, but whether that wavelength is comparable to anything it meets.

Take a baseball of mass 0.145 kg thrown at 40 m/s, about 90 miles per hour. Its momentum is 5.8 kilogram-metres per second, so its wavelength is 6.626×10⁻³⁴ J·s divided by 5.8: 1.1×10⁻³⁴ metres, about 10¹⁹ times smaller than a proton. No slit or crystal could ever diffract it. Slowing it down does not help: to stretch its wavelength to one millimetre, the ball would have to creep so slowly that crossing a single nanometre would take about 500 times the age of the universe. You, at 70 kg strolling at 1.4 m/s, have a wavelength near 7×10⁻³⁶ metres. Big objects obey quantum mechanics too; their waves are simply far too short to show.

Now take an electron, about 10²⁹ times lighter than the ball. Accelerated from rest through 54 volts, it gains 54 electron-volts of kinetic energy and a momentum of 3.97×10⁻²⁴ kilogram-metres per second, so its wavelength is 0.167 nm, comparable to the spacing of atoms in a solid. A crystal is a ready-made diffraction grating for it. For electrons there is a handy shortcut: after acceleration through V volts, λ ≈ 1.226/√V nanometres, good to better than one percent up to about 10,000 volts. Featherweight particles with atom-sized wavelengths: that is why quantum physics rules chemistry and electronics while baseballs follow Newton's parabolas.

The decisive evidence arrived partly by accident. At the New York laboratory of Western Electric, renamed Bell Telephone Laboratories in 1925, Clinton Davisson had been bouncing electrons off metals since 1919. In 1925 a liquid-air bottle exploded, cracking the vacuum tube and oxidizing the hot nickel target. Davisson and Lester Germer restored it by prolonged heating, which recrystallized the nickel into a few large crystals, and the scattered electrons changed their pattern sharply. After discussions in England with Max Born and others in 1926, Davisson made the work a deliberate test of de Broglie's waves, and early in 1927 he and Germer found sharp electron beams leaving a single nickel crystal at particular angles, just as X-rays do.

Their famous result: electrons accelerated through 54 volts struck the crystal head-on, perpendicular to its surface, and a strong beam came back out at 50° from that perpendicular. The atoms on a nickel surface sit in rows 0.215 nm apart, and such rows act like the lines of a grating: waves from neighbouring rows reinforce one another when the row spacing times the sine of that angle equals a whole number of wavelengths. That gives a measured wavelength of 0.165 nm. De Broglie's formula predicts 0.167 nm. Nineteen beams in all confirmed λ = h/p within the accuracy of the measurements.

Independently, in Aberdeen, George Paget Thomson and his student Alexander Reid fired electrons of 20,000 to 60,000 volts straight through films only about 10⁻⁶ centimetres thick. A photographic plate recorded sharp concentric rings, like those X-rays make passing through a film of tiny crystals, with sizes matching de Broglie wavelengths of 5 to 9 picometres (a picometre is 10⁻¹² metres) to about one percent. When a magnet deflected the beam, the rings moved with it, proving that the pattern was made by charged electrons and not by stray X-rays. Davisson and Thomson shared the 1937 Nobel Prize in Physics 'for their experimental discovery of the diffraction of electrons by crystals'.

Here de Broglie's idea pays its first great dividend. In the last chapter Niels Bohr made hydrogen work by decree: the electron may circle the nucleus only where its angular momentum, mass times speed times radius for a circular orbit, is a whole-number multiple of ħ = h/2π. Bohr could not say why; a wave can. A wave running round a closed loop survives only if it joins up smoothly with itself after each lap. If the circumference holds a whole number of wavelengths, crest meets crest and the wave settles into a standing wave, a stable pattern that oscillates in place; if not, successive laps arrive out of step and cancel.

In symbols: if n whole wavelengths fit around an orbit of radius r, then nλ = 2πr. Substitute λ = h/(mv) to get n·h/(mv) = 2πr, and rearrange: m·v·r = n·h/2π = nħ. That is Bohr's rule, no longer a decree but the condition for the electron's wave to fit. In hydrogen's lowest Bohr orbit, of radius 0.0529 nm, the electron moves at 2.19×10⁶ m/s, giving λ = 0.332 nm; the circumference, 2π times 0.0529 nm, is also 0.332 nm, exactly one wavelength. The second orbit holds exactly two wavelengths and the third holds three. Quantization turned out to be the ordinary arithmetic of confined waves.

Matter waves are also how we see the very small. No microscope can resolve, that is, show as separate, two details much closer together than about half the wavelength it uses, a limit Ernst Abbe worked out in 1873. Visible light spans about 400 to 700 nm, so no conventional light microscope can separate points closer than about 200 nm, while atoms, a few tenths of a nanometre across, lie a thousand times below that. Electrons accelerated through 100,000 volts have a wavelength of about 0.0037 nm, some 150,000 times shorter than green light, and current-carrying coils focus them as glass lenses focus light.

In 1931, at the Technical College in Berlin, the young engineer Ernst Ruska and Max Knoll built the first electron microscope, and they had not yet heard of matter waves. When Ruska learned of de Broglie's thesis that summer, he was at first dismayed that electron microscopes, too, would be limited by a wavelength, then heartened to find that electron waves were about five orders of magnitude shorter than light. By the end of 1933 his second instrument magnified 12,000 times, more than any light microscope could. Siemens began series production in 1939, and Ruska shared the 1986 Nobel Prize in Physics.

Oddly, the wavelength is not what limits electron microscopes. Magnetic lenses suffer aberrations, imperfections that blur the focus, which for decades kept the resolution some fifty to a hundred times coarser than the electron wavelength. Aberration correctors, practical since the late 1990s, changed that: the best instruments now resolve details smaller than 0.05 nm, about 4,000 times finer than the optical limit. Cryo-electron microscopy of flash-frozen biological molecules won the 2017 Nobel Prize in Chemistry, and in February 2020 it revealed the atomic structure of the new coronavirus's spike protein.

Every particle ever tested obeys λ = h/p. Neutrons, the uncharged particles of the nucleus, were first diffracted by crystals in 1936. Slowed in a reactor to the thermal energy of their surroundings, about 0.025 eV, they move near 2,200 m/s with a wavelength of 0.18 nm, ideal for crystals. Because neutrons scatter from nuclei rather than electrons, they reveal hydrogen atoms that X-rays barely see, and because they are tiny magnets they map magnetism; Clifford Shull shared the 1994 Nobel Prize in Physics for developing neutron diffraction. In 1975 Roberto Colella, Albert Overhauser and Samuel Werner even watched gravity shift the interference of neutron waves travelling at different heights.

Whole atoms followed. In 1930 Immanuel Estermann and Otto Stern diffracted helium atoms from a lithium fluoride crystal, and since 1991 atom interferometers have become exquisitely precise sensors of gravity and rotation. Then Vienna went big. In 1999 Anton Zeilinger's group, with Markus Arndt, heated C₆₀, buckminsterfullerene, a football-shaped cage of 60 carbon atoms, to 900 to 1,000 kelvin and sent the molecules at about 220 m/s through a grating of slits 100 nm apart. Their wavelength, 2.5 pm, is almost 400 times smaller than the molecule itself, yet 1.25 metres downstream the detector found a clear diffraction pattern, with side peaks about 30 micrometres from the centre.

The giants kept growing. In 2019 Arndt's group reported interference of molecules made of up to 2,000 atoms, with masses beyond 25,000 atomic mass units. In January 2026 it reported in Nature the interference of sodium clusters containing more than 7,000 atoms each, above 170,000 atomic mass units and about 8 nm across, split by gratings made of ultraviolet light. As of 2026 these are the most massive objects ever sent through a matter-wave interferometer. The push continues, partly to test collapse models, proposals in which superpositions of heavy enough objects fade away on their own.

A water wave is a wave of matter: water itself rises and falls. So is sound, in which air is squeezed and stretched. A matter wave, despite its name, is neither. Nothing material ripples. In every experiment in this chapter, each electron, neutron or molecule arrives at the detector whole, at a single place, and the pattern appears only as many arrivals pile up; chapter 7 will watch electrons build one dot by dot. In 1926 Max Born proposed the reading used ever since: the matter wave is a probability amplitude, a quantity whose squared size, |ψ|², gives the probability of detecting the particle at each place. Chapter 9 makes this precise.

What the amplitude is physically, a real guiding field as de Broglie came to believe or only a tool for predicting outcomes, depends on your interpretation of quantum mechanics, and physicists still debate it; the standard interpretations all make the same predictions. One practical rule holds regardless: interference survives only while nothing records which path a particle took. That is why these experiments run in high vacuum, where a single collision with a gas molecule could betray the path, and why hot C₆₀ still interfered: the two or three infrared photons each molecule emitted in flight had wavelengths around ten micrometres, far too long to reveal which 100-nanometre slit it passed through.