Δ-XThe Heisenberg Runner / Uncertainty Principle

Δ-X is never in one place long enough to have one speed. That is not a character flaw. It is a theorem about waves.

Any wave confined to a short stretch of space has to be built by adding many wavelengths together. A single pure wavelength goes on forever, the same everywhere. To make a short packet you add neighbouring wavelengths so that they cancel away from the centre, and the shorter the packet, the wider the range you need. The first relation on this page is the exact form, for spreads measured as standard deviations: the width in position times the width in wavenumber is at least one half. That is mathematics, not physics. It holds for sound, radio, water waves, anything.

You can hear it. A tone burst with a spread in time of one millisecond cannot have its frequency pinned down to better than about 80 hertz. Make a click shorter and it loses its pitch altogether.

Quantum mechanics adds one sentence. Louis de Broglie's relation says a particle's momentum is ħ times its wavenumber. Multiply through, and the wave fact becomes the Heisenberg relation: position spread times momentum spread is at least ħ/2. Earle Kennard proved it in this exact form in 1927, and showed that packets with a Gaussian, bell-curve shape reach the bound exactly. Every other shape does worse.

The consequence is not small. Confine an electron to the width of an atom, about a tenth of a nanometre, and its momentum must spread so much that its velocity spread is at least roughly 600 kilometres per second. Nothing pushed it. Squeezing the wave did that. Electrons in atoms move fast because atoms are small.

Δ-X keeps running, because standing still in a small space was never an option.