Heisenberg's Uncertainty Principle

Chapter 7 ended with a rule: a particle travels as a wave of probability amplitude and arrives as a single dot. Apply it to one slit. Passing through a gap of width a is itself a rough measurement of position, since every particle that gets through was, at that moment, somewhere inside the gap. A classical cannonball would fly on in a beam as narrow as the gap. A wave diffracts, spreading sideways, and the narrower the gap, the wider the spread: the first dark band of the pattern lies at an angle θ whose sine is λ/a, where λ is the wavelength.

Translate that spread into momentum. A particle deflected through angle θ carries a sideways momentum equal to p times the sine of θ, where p = h/λ is its de Broglie momentum from chapter 6. Out to the first dark band, the sideways momentum therefore ranges up to about p·λ/a = h/a. Multiply by the slit width and the wavelength cancels: slit width times sideways momentum spread is about h, Planck's constant, whatever the particle and whatever its speed. Pin the position tighter and the momentum grows looser, in inverse proportion. That bargain is the uncertainty principle in its rawest form.

The bargain has been watched with remarkably heavy particles. In experiments reported in 2002, Olaf Nairz, Markus Arndt and Anton Zeilinger in Vienna sent C₇₀ fullerene molecules, cages of 70 carbon atoms weighing 840 atomic mass units and heated to about 900 kelvin, through a slit they could narrow from about 20 micrometres to 70 nanometres. Down to about 4 micrometres the beam on their detector shrank with the slit, like a stream of tiny cannonballs. Below that, the waves took over: with a 1.4-micrometre slit the beam was 17 micrometres wide, and with a 70-nanometre slit it had broadened to 43 micrometres, just as the bargain predicts.

The modern statement is exact. Prepare a particle in exactly the same way a great many times; physicists call such a collection of identically prepared systems an ensemble. On each run measure the particle's position, and the results scatter. Their average marks the centre of the scatter, and their spread is the standard deviation Δx: the square root of the average squared distance of the results from that average. On a second batch of identical runs, measure momentum instead and call that spread Δp. Kennard's theorem states that for every possible quantum state, Δx·Δp ≥ ħ/2, where ħ, the reduced Planck constant, equals h/2π, about 1.055×10⁻³⁴ J·s. The floor, ħ/2, is about 5.27×10⁻³⁵ J·s.

Notice what this does and does not say. It does not say that a single position measurement must be blurry: a good detector registers one particle's position as finely as its pixels allow. It says that no preparation, however ingenious, can make a state whose position results and momentum results are both tightly bunched. The spreads belong to the state, to the two distributions it predicts, not to the error bars of an apparatus. And the trade is strictly between partners: position along x pairs with momentum along x, while position along x and momentum along y can both be sharp.

Put in numbers. Confine an electron to Δx = 0.1 nm, about the size of an atom, and its velocity spread must be at least 5.8×10⁵ m/s, about 580 kilometres per second: an electron in an atom is necessarily a fast-moving blur. A 0.145-kilogram baseball located to within a micrometre needs a velocity spread of only 3.6×10⁻²⁸ m/s. Baseballs obey the principle too; ħ is simply far too small to notice at their scale.

Why should nature impose such a trade? Because, as chapter 6 showed, momentum is wavelength: p = h/λ, or equivalently p = ħk, where the wavenumber k = 2π/λ counts radians of phase per metre. A particle with one perfectly definite momentum is described by a single endless wave, crest after identical crest across all of space. Such a wave has no 'here': the probability density for finding the particle, |ψ|², is the same everywhere. To describe a particle that is somewhere, you need a wave packet, a localized lump of wave, and the only way to build one is to add together many waves of different wavelengths.

In 1822 Joseph Fourier showed that practically any shape can be built by adding sine waves; the recipe listing how much of each wavelength is needed is now called the Fourier transform. Add waves whose wavenumbers spread over a range Δk and they reinforce one another only within a region about 1/Δk wide; everywhere else their crests and troughs fall out of step and cancel. A narrow packet demands a broad mix of wavelengths, and a narrow mix yields a wide packet. Measured as standard deviations, the trade is exact: Δx·Δk ≥ 1/2. Multiply both sides by ħ, use p = ħk, and Kennard's inequality appears. The uncertainty principle is a theorem about waves, and particles obey it because particles travel as waves.

The same bargain rules sound and every other wave, with no quantum mechanics involved. A sharp handclap is a jumble of frequencies with no definite pitch, while a pure tone must last many cycles. One shape strikes the best possible bargain: the Gaussian, or bell curve, whose Fourier transform is another bell curve. A Gaussian wave packet, at the moment it is narrowest, reaches Δx·Δp = ħ/2 exactly; every other shape does worse.

Heisenberg illustrated his relation with a thought experiment that has misled students ever since. Imagine locating an electron with a microscope. No microscope resolves details much finer than the wavelength of its light, so to see the electron sharply you need very short waves, ultimately gamma rays. But short-wavelength photons carry large momenta, h/λ. A gamma-ray photon that scatters off the electron into the lens gives it a recoil kick, and because the photon could have entered anywhere across the lens, the sideways kick is unknown by about h/λ times the sine of the lens's half-angle. Sharpening the position blurs the momentum, and the product is again about h.

The picture is vivid, but taken alone it suggests something false: that the electron had an exact position and momentum all along and the light merely jostled it. Bohr objected as soon as he read the paper, pointing out that what matters is not that the momentum changes but that the change cannot be known within the same experiment, and Heisenberg acknowledged the criticism in a note added in proof. The modern reading goes further. Kennard's inequality is a theorem about the state alone, and it limits every state that can be prepared, whether or not anything ever measures it. Chapter 7's which-path lamp shows both sides. Feynman's own analysis is a kick story: to tell the slits apart, the light needs a wavelength shorter than the slit spacing, and photons that short knock each electron sideways far enough to smear the fringes away. Yet the deeper rule concerns information, not force: any record of the path erases the fringes, kick or no kick.

Disturbance by measurement is real, but it is a separate, subtler question. In 2003 Masanao Ozawa derived a general relation between the error of one measurement and the disturbance it inflicts on a second quantity, and showed that the naive product of the two, which the microscope story says can never fall below the bound, sometimes can. In 2012 Yuji Hasegawa's group in Vienna, measuring neutron spins, and Aephraim Steinberg's group in Toronto, measuring photon polarization, observed such sub-bound products. How best to define error and disturbance is still debated. The standard-deviation bound on states has never been in doubt.

A second relation, ΔE·Δt ≳ ħ/2, carries Heisenberg's name but has a different pedigree. Time in quantum mechanics is not an observable with a spread across an ensemble; it is the parameter that clocks run on, and Wolfgang Pauli argued in 1933 that it cannot be an operator like position. Here Δt means the time a state takes to change appreciably. Leonid Mandelstam and Igor Tamm made that precise in 1945: a state whose energy spread is ΔE needs at least about ħ/(2ΔE) to evolve into something measurably different. A state of perfectly sharp energy never changes at all, which is exactly what a stationary state is.

The most visible consequence is the width of spectral lines. An excited atom decays with a characteristic lifetime τ, and because the excited state exists only for a time of order τ, its energy is spread over a width Γ = ħ/τ, a result Victor Weisskopf and Eugene Wigner derived in 1930. The emitted photons therefore span a small range of frequencies, the natural line width. Sodium's yellow D₂ line comes from a state that lives 16.25 nanoseconds, so it is about 9.8 MHz wide, roughly two parts in a hundred million of its 509-terahertz frequency. Doppler motion and collisions usually broaden real lines further, but no line emitted by decaying atoms is ever narrower than its natural width, which is why atomic clocks favour transitions whose upper states live a long time.

Particle physicists run the rule backwards to measure lifetimes far too short for any clock. The Z boson, a carrier of the weak nuclear force discovered at CERN in 1983, appears as a peak in collision rates centred on 91.19 GeV with a width of 2.4955 GeV. Its lifetime is therefore ħ/Γ, about 2.6×10⁻²⁵ seconds, during which even light crosses less than a tenth of a proton's radius. Every way the Z can decay shortens its life and widens its peak, including decays into neutrinos that no detector sees, so the width counts them. Data from CERN's Large Electron-Positron collider, reanalysed in 2020, give 2.996 ± 0.007 kinds of light neutrino: three, counted with the uncertainty principle.

Chapter 5 left a time bomb under every atom: a classical electron circling a nucleus should radiate, lose energy and spiral inward in about 1.6×10⁻¹¹ seconds. Bohr forbade the crash by decree; the uncertainty principle explains it. Falling inward means squeezing the electron into a smaller region, and confinement has a price. An electron confined within a distance r has a momentum spread of order ħ/r, so even with zero average momentum its kinetic energy is at least of order ħ²/(2·m·r²), where m is its mass. Halve the radius and this confinement energy quadruples.

The electrical attraction pulls the other way. An electron at distance r from a proton has potential energy −e²/(4πε₀r), where e is the elementary charge and ε₀ the vacuum permittivity; halve the radius and it merely doubles in depth. Add the two and the total rises steeply at small r and slowly at large r, with a valley between. The bottom of the valley sits at r = 4πε₀ħ²/(m·e²) = 5.29×10⁻¹¹ metres, the Bohr radius a₀, with a total energy of −13.6 eV, hydrogen's measured ground-state energy. Because the estimate fudges factors of order one, landing exactly on target is partly luck, but the scale is no accident. Squeeze the electron to half that radius and the total energy rises to zero; to a tenth, it soars above 1,000 eV.

So atoms have a size, a few tenths of a nanometre, fixed by a tug-of-war between attraction and confinement. Bulk matter, with countless electrons, needs one more ingredient: the Pauli exclusion principle of chapter 18, which forbids electrons from all crowding into the same state. Freeman Dyson and Andrew Lenard proved in 1967 that together these keep ordinary matter stable, and Elliott Lieb and Walter Thirring found a far shorter proof in 1975, built on a sharpened uncertainty inequality.

The atom's argument generalizes. The lowest-energy state of a classical ball in a bowl is the ball at rest at the bottom: exact position, zero momentum. Quantum mechanics forbids that state, since it needs Δx and Δp to vanish together. Any confined particle therefore keeps a minimum energy even at absolute zero, its zero-point energy. Nothing can be perfectly still. Chapters 11 and 13 compute it for a box and a spring; in helium it is large enough to stop the liquid freezing, even at absolute zero, unless squeezed to about 25 atmospheres. It is a floor, not a fuel tank: there is no lower state to fall into.

Light has zero-point fluctuations too, and engineers now sculpt them. Two complementary properties of a light wave, closely related to its amplitude and its phase, obey an uncertainty relation of the same form, so the noise in one can be pushed below the zero-point level at the cost of extra noise in the other, without beating the bound on the product. Such squeezed light has been injected into the LIGO and Virgo gravitational-wave detectors since 2019, cutting quantum noise where it matters. You cannot abolish uncertainty, but you can move it to where it does no harm.

Position and momentum are the first example of incompatible observables: measurable quantities that no state can make definite together. In the language of chapter 14 each observable is an operator, a rule that acts on the wavefunction, and for incompatible pairs the order of operations matters. Position then momentum differs from momentum then position, and the difference, called the commutator, is always iħ, written [x̂, p̂] = iħ. Max Born and Pascual Jordan made it the cornerstone of matrix mechanics in 1925, and in 1929 Howard Percy Robertson proved that any two observables with a non-zero commutator obey an uncertainty relation of this kind. Spin along x and spin along z are another incompatible pair (chapter 17).

None of this makes the quantum world vague. The uncertainty principle is a precise inequality inside a theory whose predictions are among the sharpest in science: in 2023 the electron's magnetic moment was measured to 0.13 parts per trillion, and it agrees with theory to about one part in a trillion. Charges and masses are perfectly definite, probability distributions are predicted exactly, and the principle forbids far more than it permits, from collapsing atoms to motionless springs. Uncertainty is not a licence for anything goes. It is one of the strictest laws we have.