Strike a drum once, sharply, and you hear a thud with no clear pitch. Hold a long note on a flute and the pitch is pure. Musicians know this without thinking about it: a sound that is very short in time cannot have a single, precise pitch. That everyday fact about waves is the key to one of the most famous ideas in physics.

A tradeoff in every wave

To build a short, sharp burst of sound, you have to add together many waves of different pitches, arranged so that they reinforce one another for an instant and cancel everywhere else. A pure tone, with a single frequency, must go on for a long time. So short bursts need a wide spread of frequencies. More generally, the more tightly a wave is bunched up in time or space, the wider the spread of frequencies or wavelengths it contains. This is a mathematical property of all waves, from sound to ripples on a pond, worked out in the tools that the French mathematician Joseph Fourier published in 1822.

Since matter behaves as waves, and a particle's wavelength is tied to its momentum, the same tradeoff applies to particles. A wavefunction squeezed into a small region must contain a wide range of wavelengths, and so a wide range of momenta. This is the uncertainty principle that Werner Heisenberg announced in 1927. Here we look at what it really means and what it does.

Not just clumsy measuring

Heisenberg first explained his principle with a thought experiment: to see an electron, you must bounce light off it, and the light kicks it. That story suggests the electron has an exact position and momentum that we merely disturb. The modern understanding goes further. A particle does not possess a sharp position and a sharp momentum at the same time; the wavefunction simply has no such state. The limit is in the nature of the thing, not just in our tools, and no future instrument will remove it.

The same rule means that confining a particle more tightly raises its energy of motion, because its wavefunction must then contain shorter wavelengths. As the example below shows, this fact sets the size of every atom.

Order matters

One consequence is that the order of measurements matters. In everyday life, checking a car's location and then its speed gives the same information as doing it the other way round. In quantum mechanics, some pairs of measurements interfere with each other, so changing their order can change the results. The three filters of the last level were an example: the middle measurement changed what the last one found.

Energy and time

A similar tradeoff links energy and time, though it works a little differently because time is not something a particle has. A state that lasts only a short time does not have a sharply defined energy. Atoms show this directly. An excited state that quickly drops to a lower level gives off light whose color is slightly spread out, while a state that lives longer produces a sharper line. Atomic clocks rely on long lived states, whose extremely sharp lines make precise tick marks.

Never perfectly still

The most striking result is that nothing confined can be perfectly at rest. A particle sitting motionless at the bottom of a bowl would have both an exact position and an exact momentum of zero, which the uncertainty principle forbids. So even at its lowest possible energy, a confined particle keeps a minimum jiggle, called zero point energy. At absolute zero, the coldest possible temperature, atoms still move.

This is not a tiny curiosity. Helium atoms are light and attract each other only weakly, so their zero point jiggle is enough to keep them from locking into a solid. At ordinary pressure, helium stays liquid and never freezes, all the way down to absolute zero. It becomes solid only when squeezed to about 25 times atmospheric pressure.

Worked example: why atoms do not collapse

Classical physics predicted a disaster: an electron circling a nucleus should radiate its energy away and spiral inward in a tiny fraction of a second. Bohr's allowed levels said that it doesn't, but not why. The uncertainty principle gives a clear reason.

The electron is attracted to the nucleus, so it would lower its energy by moving closer. But squeezing the electron into a smaller region forces its wavefunction to contain shorter wavelengths, and shorter wavelengths mean greater momentum and more energy of motion. Pull in too close and the cost in motion energy outweighs the gain from attraction. The atom settles at the size where the two effects balance. Work through the numbers for hydrogen and that size comes out at about a twentieth of a billionth of a meter, which is just what the hydrogen atom measures.

So the uncertainty principle is not only a limit on knowledge. It holds up the world. The solid floor under your feet owes its size and firmness partly to the refusal of electrons to be confined more tightly than nature allows. The next level adds the other half of the story.