Everything has a wavelength, and almost nothing shows it

By 1923 light had been shown to carry momentum in discrete amounts, with p = h/λ. De Broglie’s thesis proposed reading that relation the other way: any object with momentum p has a wavelength λ = h/p.

The proposal was not supported by any measurement and its author had no experiment to offer. What it had was a symmetry argument and one striking consequence: applied to an electron in a hydrogen atom, requiring a whole number of wavelengths to fit around the orbit reproduces exactly the quantisation condition Bohr had imposed by hand a decade earlier. A rule that had been stipulated became a rule about fitting.

Where the relation comes from #

There is no derivation in the ordinary sense, because the relation is a postulate. What there is instead is an argument that it is the only postulate available.

For light, p = h/λ is forced: the relativistic relation between energy and momentum gives E = pc for a massless object, and E = hf with f = c/λ then gives the momentum. So the relation between wavelength and momentum for light contains no free choice.

De Broglie’s step was to notice that the relation, written as λ = h/p, makes no reference to the speed of light or to masslessness. It is a statement about a momentum, and a momentum is something a slow massive object also has. Extending it costs nothing and predicts something immediately: an electron accelerated through a hundred volts has a momentum of 5.4 × 10⁻²⁴ kg m/s, so a wavelength of 123 picometres, which is smaller than visible light by four thousand and comparable to the spacing between atomic planes in nickel.

That numerical coincidence is what made the idea testable in 1927 rather than in 2027. A wavelength has to be comparable to some available spacing before it can be made to diffract, and nature happened to put the electron’s wavelength at accelerating voltages a laboratory can produce right on top of the lattice constant of every metal.

The experiment, which was an accident #

Davisson and Germer were not testing de Broglie. They were measuring how electrons bounce off a nickel target, as part of an industrial investigation into vacuum tubes, and getting a smooth featureless distribution.

Then a liquid-air bottle broke, air got into the apparatus, and the nickel oxidised. Cleaning it required prolonged heating, which annealed the polycrystalline target into a few large crystals. When the measurement resumed, the smooth distribution had been replaced by sharp peaks at particular angles.

The peaks sat where Bragg’s law puts them for a wave of the de Broglie wavelength, and their positions moved with the accelerating voltage exactly as λ ∝ V^(−1/2) requires. In the same year G. P. Thomson passed electrons through thin metal foils and got rings — the polycrystalline version of the same thing. The Nobel Prize went to both in 1937, thirty-one years after G. P. Thomson’s father received one for showing that the electron is a particle.

The orbit that had to fit #

The consequence de Broglie put in his thesis is still the quickest way to see why the idea was taken seriously before it had any experimental support.

Bohr’s 1913 model of hydrogen worked — it gave the right wavelengths for the spectral lines to four figures — and its central assumption was arbitrary. Bohr required the electron’s angular momentum to be a whole multiple of ħ, with no reason offered beyond that it produced the answer. A quantisation rule imposed to fit data is a repair, not an explanation, and Bohr said so.

Give the electron a wavelength and the rule stops being arbitrary. An electron circling at radius r has to come back into phase with itself after one circuit, or it interferes with itself destructively and there is nothing left. That requires the circumference to be a whole number of wavelengths, 2πr = nλ, and substituting λ = h/p gives pr = nh/2π — exactly Bohr’s condition, with the arbitrariness replaced by the same requirement that decides which notes a string will sound.

The orbit that had to fit is the other half of the idea, and it arrived first. Bohr had stipulated that angular momentum comes in multiples of $\hbar$ with no reason offered; de Broglie supplied one. If an electron in an orbit is a wave, the orbit must contain a whole number of wavelengths or the wave meets itself out of step and cancels — the same condition that puts a discrete list of frequencies on a clamped string. Bohr’s arbitrary integer becomes a count of wavelengths round a loop, which is not a derivation of the atom but is a great deal better than an assumption.

The Bohr orbit is not how an atom actually works — the electron has no orbit — and the argument above is superseded. What it did was make the case that quantisation is a fitting condition rather than a postulate, and that case survived the model it was made in.

Why it is not part of ordinary life #

The relation applies to everything. The reason it is invisible is arithmetic, and the arithmetic is worth drawing rather than asserting.

There is no aperture in the universe narrow enough to diffract a tennis ball, and there never will be, because the smallest structures that exist are twenty-four orders of magnitude too wide. This is not a statement about measurement difficulty. It is a statement that the wave behaviour of a macroscopic object has no observable consequence of any kind, at any energy, using any apparatus.

That is a different and much stronger kind of “negligible” than the site usually deals in. The small-angle approximation fails visibly at forty degrees; ray optics fails as soon as the aperture approaches the wavelength. Here the correction is not small, it is unreachable — and that is what makes classical mechanics not merely a good approximation for large objects but an exact one for every purpose that will ever exist.

What the mass is doing #

Two objects with the same energy do not have the same wavelength, and the difference matters for how each is used.

For a non-relativistic particle, p = √(2mE), so λ = h/√(2mE). Doubling the mass at fixed energy shrinks the wavelength by √2. A thermal neutron at room temperature carries about 25 milli-electronvolts and has a wavelength of 178 picometres; an electron with that same energy would have a wavelength of 7.8 nanometres, forty-four times larger, because it is 1,839 times lighter.

That is why neutrons and electrons are used for different jobs. A neutron with an atomic-scale wavelength has an energy in the milli-electronvolt range — the same range as lattice vibrations — so a neutron can exchange a useful fraction of its energy with a phonon and report on the dynamics. An electron with an atomic-scale wavelength carries a hundred electronvolts and is far too energetic for that, but it is charged, so it can be focused, which is what an electron microscope is.

The energy also sets the resolution. Rayleigh’s criterion says the finest detail an instrument can separate goes with its wavelength, so a 123-picometre electron beam beats a 500-nanometre light beam by four thousand. Modern transmission electron microscopes run at 300 kV, where the wavelength is 1.97 picometres and the resolution is limited by lens aberrations rather than by diffraction — the same aberrations a mirror suffers from, in a magnetic lens that cannot be figured to an aspheric shape.

Naming the approximation #

Two of them, both in use above.

The non-relativistic momentum. λ = h/√(2mE) uses p = √(2mE), which is the low-speed expansion. At a kilovolt an electron is moving at 0.063 c and the correction to the wavelength is a tenth of a per cent; at 300 kV it is 22 per cent, and an electron microscopist who ignores it gets the lattice spacing wrong by that much. The curve on the opening figure is drawn to a kilovolt for that reason and would be a misleading picture beyond about ten.

The single wavelength. An accelerated beam has a spread of energies and therefore a spread of wavelengths, so “the” de Broglie wavelength is the centre of a distribution. That spread is not a defect to be minimised out of existence — it is what makes a packet, and a packet is the only thing that can be localised anywhere. A wave with one exact wavelength is infinitely long, and an electron is not.

How large an object can be made to interfere #

The ladder figure suggests a hard boundary, and the boundary is softer and more interesting than “small things are quantum”.

Interference has been demonstrated with electrons, neutrons, atoms, C₆₀, and molecules of over 25,000 atomic mass units. The obstacle in each case is not the relation λ = h/p failing; it is holding the object still enough, cold enough and isolated enough that its wave stays coherent while it crosses the apparatus. Every stray photon scattering off a large molecule carries away information about where it was, and that is enough to destroy the fringes.

So the practical limit is set by decoherence rather than by mass. Cooling the source and improving the vacuum have pushed the record up by three orders of magnitude in twenty years, and no one has found a mass at which the physics changes character. What changes is how hard it is to keep the environment from measuring the object — which is a statement about what a measurement does rather than about wavelengths.

Interference with things that have mass #

The strongest confirmations are not diffraction off crystals but interferometers, in which a matter wave is split along two separated paths and recombined.

Interference with things that have mass has since been pushed a very long way. Neutrons, whole atoms, and molecules of several hundred atoms have all been sent through gratings and produced fringes, and in every case the wavelength that comes out of the fringe spacing is $h/p$ with the measured momentum put in. The results are not a demonstration that large objects are secretly waves; they are a demonstration that the relation has no clause about size in it, and that what keeps a cricket ball from diffracting is arithmetic rather than a change of law.

Neutron interferometry has produced results that no diffraction experiment could. Rotating the interferometer about a horizontal axis changes the gravitational potential difference between the two arms by a few centimetres’ worth, and the fringes shift by a measurable amount — gravity acting on a quantum phase, in a tabletop apparatus. Rotating a neutron’s spin through 360° and recombining it with an unrotated copy gives destructive rather than constructive interference, which is the direct demonstration that a spin-½ object needs 720° to come back to itself.

Atom interferometers have gone further and become instruments. Splitting a cloud of cold atoms, letting the two halves fall for a few hundred milliseconds and recombining them measures g to nine figures, with the wavelength serving as the ruler. Because λ = h/p and the atoms are heavy and slow, that wavelength is picometres — which is precisely why the measurement is so sensitive.

The phase velocity that embarrassed the idea #

De Broglie did not simply write down a wavelength. He wrote down a wave, which means he also had to say what its frequency was, and the natural choice was the one light already used: E = hf, with E the particle’s full relativistic energy γmc². That fixes both quantities, and the two together fix a speed.

The speed is fλ = E/p, and for a massive particle E/p = c²/v. Since v is less than c, that phase velocity is greater than c — and it grows without limit as the particle slows down, so a stationary electron is accompanied by a wave whose crests move infinitely fast.

That was the first objection raised against the thesis and it is a serious-looking one. The resolution is the distinction this collection makes elsewhere between the speed of a crest and the speed of a feature: differentiate the frequency with respect to the wavenumber rather than dividing one by the other, and the group velocity comes out to be exactly v, the speed of the particle. The crests outrun the particle; the envelope keeps pace with it, and the envelope is where the particle is.

De Broglie saw this and said so in the thesis, which is the part of the work that most deserves the credit. He had to invent the reading before there was an equation to justify it — the relation between group velocity and a wave equation was supplied by Schrödinger two years later — and he took a result that looked like a refutation and turned it into the first appearance of the idea that a particle is a packet.

When the wavelength catches up with the spacing #

The ladder figure compares each object’s wavelength with the spacing between atoms in a solid, and treats the comparison as a question about whether an experiment is possible. There is a second reading of the same comparison in which it is a question about what a substance does on its own.

The relevant length for a gas is the thermal de Broglie wavelength — the wavelength of a particle with a typical thermal momentum, which works out to Λ = h/√(2πmkT). It grows as the gas is cooled, because a colder particle has less momentum. For rubidium atoms in air at room temperature it is about a hundredth of a nanometre, several hundred times smaller than the distance between neighbours, and the gas behaves as a collection of separate objects with definite positions.

Cool the same atoms to a hundred nanokelvin and Λ becomes about six tenths of a micrometre — longer than a wavelength of visible light, and in a trapped cloud of the usual density, comparable to the spacing between one atom and the next. At that point the wave descriptions of neighbouring atoms overlap, the question of which atom is which stops having an answer, and the gas has to be treated as one object.

That is Bose–Einstein condensation, and the criterion for it is exactly the comparison just made: the number density times Λ³ must exceed about 2.6. Einstein worked it out in 1925, citing de Broglie’s thesis before the diffraction experiments existed, and it took seventy years and laser cooling to reach the temperature the arithmetic demanded.

So the wavelength is not only a tool for making pictures of crystals. It is the length that decides when a collection of particles stops being a collection — and the reason a hot gas is describable atom by atom is that the very same number that is 10⁻³⁴ metres for a tennis ball is a hundredth of a nanometre for an atom of rubidium in this room.

What the wavelength is not #

Three misreadings are common enough to be worth heading off, and all three come from taking the word “wave” too literally.

It is not a wave in anything. A sound wave is a displacement of air and a wave on a string is a displacement of string; the electron’s wave is not a displacement of a medium, and the nineteenth-century instinct to look for one — an aether for matter — has no counterpart here. What oscillates is a complex amplitude defined at every point in space, and the only thing measurable about it is the square of its magnitude.

It is not the electron spread out. An electron is never found in two places, and a detector never registers a fraction of one. What the wave describes is where a whole electron is likely to arrive, and the arrival is always at one point — which is exactly the tension the single-particle double slit exists to display and does not resolve.

And it is not a property the electron has independently of its motion. λ = h/p means the wavelength changes when the momentum does, so an electron falling through a potential has a wavelength that varies from place to place — which is what makes the wave refract, and which turns an electrostatic lens into a lens in exactly the sense a piece of glass is one. An electron microscope’s optics are built on that correspondence, aberrations and all.

What the picture cannot show #

The wavelength curve draws λ against V and asserts that the electron has that wavelength, and it cannot show what is waving. It is not the electron’s shape and it is not a vibration of any medium; the quantity with the wavelength is a complex amplitude whose squared magnitude gives the probability of finding the electron somewhere, and no plot of a wavelength conveys that.

The ladder figure hides a related sleight. It places a tennis ball at 10⁻³⁴ metres as though the ball had a single wavelength, when a tennis ball is 10²⁵ atoms each with its own momentum, and the centre-of-mass wavelength drawn is a property of an idealised rigid body rather than of the object. The number is right and the reasoning behind it is a much longer argument than the dot suggests.

Where the ladder goes next #

The rungs from here: the double slit done one particle at a time, which is this relation’s sharpest test; the Schrödinger equation, which is what happens when the wave is given a rule for evolving instead of just a wavelength; the boundary conditions that turn a wavelength into a spectrum; the Aharonov–Bohm effect, in which a matter wave’s phase responds to a potential in a region it never enters; and neutron interferometry, where the two arms of a wave are separated by centimetres and the object doing the interfering has a mass.

There is one more thing the ladder cannot show, and it is the reason the boundary in it is drawn with a dashed line rather than a wall. Every object on that axis obeys the same relation, and the ones at the bottom are not exempt from quantum mechanics — they are objects whose quantum description happens to be indistinguishable from the classical one at every resolution that exists. A picture with a threshold in it invites the reading that something different is going on either side of it, and nothing different is going on.

The claim to carry forward is that λ = h/p has no scale in it. It is not a rule about small things; it is a rule about everything, whose consequences vanish for large things because h is 6.6 × 10⁻³⁴ joule seconds and a tennis ball’s momentum is not.