Uncertainty Principle
The Tighter the Squeeze, the Wider the Spread
Squeeze It and It Spreads
Widen a green laser’s beam to about a centimeter, and shine it through a slit whose width you can set, onto a wall two meters away. Then close the slit slowly. At first the bright patch on the wall narrows, as a shadow should. When the slit is about a millimeter wide, the patch stops narrowing. Close the slit further and the patch grows taller again. At a tenth of a millimeter, its middle band is about two centimeters tall, with fainter bands above and below.
Closing the slit pins down more closely where the light went through. What the light loses is direction. It leaves the slit headed a little up or a little down, and the narrower the slit, the wider the range of headings. Send single photons through, one at a time, and a camera in place of the wall records each as a single dot, but the dots fill the same widened band. So each photon’s heading is spread out, and a photon’s heading is the direction of its momentum, the quantity that collisions never lose.
This is the uncertainty principle, found by Werner Heisenberg: the more sharply a particle’s place is fixed, the more widely its momentum must be spread, and the other way round. That is a floor: a particle can be spread wider in both. In its exact form, each spread is measured as the typical distance of the results from their average. Take the spread in a particle’s position and multiply it by the spread in its momentum. The product never falls below a floor set by Planck’s constant, the number that ties the energy of each lump of light to how fast its wave swings.
The floor is tiny, which is why no one meets it in daily life. Pin a grain of sand weighing a milligram to within a thousandth of a millimeter, and its speed is spread by so little that it would take tens of thousands of years to drift the width of an atom. Pin an electron to the width of an atom, and its speed is spread by more than a thousand kilometers a second.
A Short Wave Is Many Waves
Why should pinning down a place spread the momentum? Because light travels as a wave, and so, it turns out, does every particle: a wave whose height gives the chances of finding it, though each one is found at a single point. A wave that runs on and on can have a single wavelength. A wave squeezed into a short stretch cannot. The only way to build a short burst is to add many waves of different wavelengths, lined up so that their crests reinforce inside the stretch and cancel everywhere else. The shorter the burst, the wider the range of wavelengths it needs.
Sound shows the same rule. A long note on a flute has a clear pitch. A hand clap, over in an instant, has none: it is a jumble of pitches. The rule is older than quantum physics: it is the mathematics of waves, and radio engineers design by it. A signal that switches quickly needs a wide band of frequencies, which is why a fast data link takes up more of the radio spectrum than a slow one. None of this is quantum. It holds for every wave, in water, air or light.
Quantum physics adds one rule. Every particle has a wavelength, and its momentum is Planck’s constant divided by that wavelength: the shorter the wave, the more momentum. So a particle whose wave is squeezed into a small space is a blend of many wavelengths, and so of many momenta. At the slit the light keeps its one wavelength, but its wave is squeezed in height. A wave running at a slant crosses the slit with its crests farther apart up and down, so waves of the one length at a range of slants are what give it many wavelengths up and down. Each slant is a direction the light leaves in.
The spread does not wait for anyone to look. The parts of a squeezed wave head off in their different directions, and the wave grows. Pin an electron to the width of an atom and let it go, and its wave spreads over more than a meter within a millionth of a second. Inside an atom, the pull of the nucleus holds it in.
Not a Clumsy Instrument
Heisenberg first explained his principle with a thought experiment. To see where an electron is, you bounce light off it into a microscope. Light of short wavelength shows the electron sharply but strikes it hard. The light could have bounced off at any angle that still enters the lens, so the direction of the kick is unknown, and a wider lens, which shows the electron more sharply, leaves it less certain. Light of long wavelength strikes gently but shows only a blur.
The story invites a picture in which the spread is made by our clumsy probing. The floor says something else. Prepare a thousand particles in the same way. Measure the position of half of them and the momentum of the other half, so that no particle is measured twice. The positions still scatter, the momenta still scatter, and the two spreads still obey the floor. The spread belongs to the state the particles were prepared in, not to anything a measurement did to them.
Heisenberg’s own version, the error of one measurement times the kick it gives, is a different claim. With the error and the kick defined state by state, in one widely used way, it can be beaten, and experiments have beaten it, with the spins of neutrons and the polarization of photons, pairs that clash as position and momentum do. The floor held every time: a gentle measurement adds little kick to a quantity that was spread widely already. Physicists still disagree on how best to define the error and the kick. Take the worst case over every state a particle could be in, and a version of Heisenberg’s rule can be proved to hold.
Not every pair of quantities clashes. A particle can have a sharp position along one line and along another at the same time, and an atom can have a sharp energy and a sharp amount of turning, its angular momentum, about one axis. Its turning about two different axes is a pair that clashes. The pairs that clash are those for which the order of two measurements matters, and the more it matters, the higher the floor. Whether a photon passes an upright filter and whether it passes a slanted one is such a pair. Photon by photon, that is how a slanted filter between two crossed ones lets light through: asking the slanted question unsettles the upright answer.
Could sharp values hide underneath after all? Some theories give every particle a definite path at every moment. Even there, no one can prepare particles sharper than the floor: the paths are steered by the particle’s wave, and learning a position more finely changes the wave, and so the speeds. Particles prepared alike scatter just as widely, and no experiment has ever made a state that beats the floor. Tests of entanglement add a price: any hidden values must be tied together across any distance at once, as they are in those theories.
A Short Flash Holds Many Colors
The same rule works in time. A flash of light that lasts a long time can be a single, pure color. A short flash cannot. It must hold a band of colors, and the shorter the flash, the wider the band. A green flash lasting a trillionth of a second needs colors spread over only a few tenths of a nanometer: still pure green. A flash only one wave long holds every color the eye can see.
Such flashes are tools of today. The lasers that cut the cornea in eye surgery fire flashes a few hundred femtoseconds long, millionths of a billionth of a second. Physicists have made flashes of visible light shorter than one swing of its wave. They hold everything from the near infrared to the ultraviolet.
For light, a color is an energy: each photon carries its frequency times Planck’s constant. So the photons of a short flash have energies spread over a wide range. This is the energy-and-time form of the uncertainty principle, and it differs in kind from the first. Time is not a quantity a particle carries, as its position is; it is the clock on the wall. Put exactly, the relation says how fast a state can change. A state with a perfectly sharp energy never changes at all, and a state that lasts only a short time cannot have a sharp energy.
So a particle that lives briefly has no sharp mass. The Z boson, one of the carriers of the weak force, lasts about 3 × 10⁻²⁵ seconds. Its mass, 91.2 billion electron volts (GeV), is smeared over a band 2.5 GeV wide. The other way round, an atomic clock counts the swings of a state that lasts. In strontium clocks the excited state lasts about two minutes, so its color is sharp to about three parts in a billion billion. A clock needs a line that sharp, though how well it keeps time also depends on the laser that reads the line.
The relation is often told as energy borrowed: empty space may break the conservation of energy for a moment, so long as it pays the energy back quickly. It does not say that. Energy is conserved at every moment. The virtual particles of that story are steps in a calculation, not things that briefly exist.
Why Atoms Have a Size
The squeeze is why atoms do not collapse. The electron in a hydrogen atom is drawn toward the proton. Squeezed closer, its momentum would have to spread more, and its energy of motion would rise. Coming closer lowers its energy of position, but the energy of motion rises faster as the squeeze tightens. The two balance when the electron is most likely to be found about a twentieth of a nanometer from the proton, which makes a hydrogen atom about a tenth of a nanometer across. Strictly, the floor on the two spreads is not enough to prove this by itself. The proof uses a sharper form of the same squeeze: crowding any part of the wave close to the proton costs energy of motion.
The same squeeze sets the scale of nuclear energy. A proton held inside a nucleus, tens of thousands of times smaller than an atom, has millions of electron volts of energy of motion, where an electron in an atom has a few. To keep it there, the strong force must set it in a pit of energy deeper than that, so when nuclei rearrange, millions of electron volts change hands. That is why a nuclear reaction releases about a million times more energy than a chemical one.
For a whole lump of matter, even the sharper squeeze is not enough on its own. The proof that ordinary matter is stable needs one more rule: electrons also obey the exclusion principle, so no two of them can share the same state. Without it, matter would fall in on itself. Every electron would crowd into an atom’s lowest state, so atoms heavier than helium would shrink, and packing atoms together would release ever more energy for each atom added. The same pair, the squeeze and the exclusion rule, holds up white dwarf stars against their own weight.
The squeeze also means that nothing held in place can lose all its energy of motion. Even at its lowest energy, a confined particle keeps a spread of momentum, and with it some energy of motion that cannot be taken away. Helium shows it best. Its atoms are light and pull on each other only weakly, and that leftover energy keeps helium liquid all the way down to absolute zero. It freezes only when pressed to about 25 times the pressure of the air.
Putting the Noise Where No One Looks
The floor is on the product of two spreads, not on either one. Nothing forbids squeezing one spread far below its usual size, as long as the other grows to make up for it. Light prepared this way is called squeezed light.
The detectors that sense gravitational waves use it. LIGO, the Laser Interferometer Gravitational-Wave Observatory, measures the stretching of space by sending laser light along two arms 4 kilometers long and comparing the light that comes back. At its best it is limited by the light itself: photons arrive at random times, and their pushes jiggle the mirrors. Squeezed light moves that randomness from one to the other: from when the crests arrive to how high they are, or back. LIGO makes it in a crystal kept at a steady warmth. There, green laser light splits into pairs of infrared photons, with no steady infrared wave among them, and these are fed in where no laser light enters. So what is squeezed is the scatter that even empty space’s field shows.
There is a catch. At high frequencies the trouble is when the photons arrive, and at low frequencies it is how hard they push, and squeezing one makes the other worse. Each LIGO detector now uses a cavity 300 meters long that turns the squeeze from one to the other across the range of frequencies. In its fourth observing run, squeezed light turned this way let LIGO see 15 to 18 percent farther than it could without squeezed light, enough to expect up to 65 percent more events.
So the uncertainty principle is not the fault of clumsy instruments. It is what traveling as a wave means: a particle cannot sit in one place and also have one momentum, just as a sound cannot be short and also have one pitch. Knowing that, engineers can choose where the fuzz goes.
