Uncertainty. Heisenberg, 1927

A perfectly pure note would have to last for ever, so it is an ideal. Real notes start and stop, and that always adds a little spread of pitch. A tuning fork comes close: it rings for many seconds on almost a single pitch. The rule is general. The longer a note lasts, the purer its pitch can be, and a very short blip has no clear pitch at all. The wavelength is the speed of sound, about 343 metres a second in air, divided by the number of vibrations a second.

A pure wave has one wavenumber k, and its height is spread evenly over every x:

With v ≈ 343 m/s in air, 220 Hz gives λ ≈ 1.56 m. Its spreads are Δk = 0 and Δx = ∞.

Joseph Fourier showed, in work on heat begun in 1807, that almost any shape can be built by adding plain waves. The trade you see here holds for every kind of wave: sound, light, water and radio. Engineers meet it all the time, because a very short radio pulse needs a very wide band of frequencies. This mix uses a bell-shaped spread of wavelengths, which gives the smallest product possible. Any other recipe gives a bigger product, never a smaller one.

The pulse on screen is a real sum of 51 plain waves, each with wavenumber kⱼ = 2π/λⱼ and a bell-shaped weight:

Measured as standard deviations, the two spreads obey a fixed rule. The page works them out from the sum itself:

Louis de Broglie proposed in 1924 that every particle has a wavelength set by its momentum, and in 1927 electron waves were seen bouncing off crystals. Heisenberg sent off his paper in March 1927. The spreads describe a crowd of electrons prepared exactly alike. Find each one and they turn up in different places. Time each one and they move at different speeds. Pin the crowd's place and their speeds scatter. Pin it hard enough and some even move backwards.

De Broglie's rule ties momentum to wavelength. Multiply Δx · Δk ≥ ½ by ħ:

The electron here moves at v₀ = 1,000 km/s on average, so λ₀ = h/mv₀ = 0.727 nm. Pinned to Δx = 0.10 nm, it has Δv ≥ 579 km/s.

Heisenberg's paper explained the limit with a microscope: to see an electron you must bounce light off it, and the light's kick disturbs it. Niels Bohr objected that this missed the real reason, which lies in the wave nature of matter and light. Heisenberg added a note before the paper appeared, admitting Bohr's point. Today the limit is stated as a property of the waves themselves. It holds even when each electron is measured only once, for its place or for its speed, so no kick is to blame. The kick picture still survives in popular accounts.

Earle Kennard proved the exact form later in 1927. Here each Δ is a standard deviation over many particles prepared alike:

Heisenberg's own 1927 estimate was rougher, δq δp ∼ h. The gold line on the chart is Δv = ħ / (2mΔx) for an electron, about 58 km/s at 1 nm.

In everyday units h is a decimal point followed by 33 zeros, then 66. Against the 160 grams of a cricket ball that is nothing. Against an electron, which weighs about a billion billion billion times less than a gram, it is everything. That is why the quantum blur rules inside atoms and hides from daily life. Big things do not obey different laws. The same rule applies to them, but its blur is far too small to notice.

Electron (9.11 × 10⁻³¹ kg): 5.8 × 10⁵ m/s. Hydrogen atom (1.67 × 10⁻²⁷ kg): 315 m/s. Dust (10⁻¹² kg): 5.3 × 10⁻¹³ m/s. Cricket ball (0.16 kg): 3.3 × 10⁻²⁴ m/s.

Before quantum theory this was a real crisis. In the old picture of an electron circling like a planet, it should give off light and spiral into the centre in less than a billionth of a second, so atoms should not exist. Niels Bohr's model of 1913 simply declared that the electron could stand only on certain steps. Uncertainty gives a deeper reason: the cloud cannot shrink without its motion growing. It sets the size of atoms, and so helps set the size of everything made of them.

Take the push the squeeze demands, p ≈ ħ/r, and add the electric pull:

The lowest point is at

This rough estimate lands exactly on hydrogen's measured energy. With p ≈ ħ/2r instead it would be off by a factor of four, so treat it as a sketch. The full calculation gives the same −13.6 eV.

Earle Kennard proved the exact inequality later in 1927, and Howard Percy Robertson extended it to other pairs of quantities in 1929. Einstein never accepted that the limit was final and tried for years to beat it with thought experiments. On page 22 of the book he attacks it with a box. The brass microscope on this page is Heisenberg's own first picture, the one the card warns about. Its kick is real, but it is not the reason for the limit.