Fourier reciprocity
Fourier reciprocity is the mathematical fact, true of all waves, that a signal which is narrow in one domain is necessarily broad in the reciprocal one. A short, sharp pulse in time is built from a wide range of frequencies; a pure single tone, narrow in frequency, must ring on for a long time. This is a theorem about functions and their Fourier transforms, and it long predates quantum mechanics — it governs music, radio, and image processing alike.
Quantum mechanics inherits this reciprocity wholesale. The wavefunction in position and the wavefunction in momentum are exactly a Fourier transform pair, with Planck's constant ℏ setting the exchange rate between the two. Because a function and its Fourier transform cannot both be sharply peaked, a wavefunction localized in space must contain a wide span of momenta, and one with sharp momentum must spread across space.
Seen this way, the Heisenberg uncertainty principle is not a strange new postulate added by hand but the same old bandwidth theorem applied to matter waves. The factor of ℏ/2 in ΔxΔp ≥ ℏ/2 is what you get when you translate the general Fourier inequality into physical units using the de Broglie relation between momentum and wavelength. The mystery, in a sense, is just wave mathematics in disguise.
Position and momentum wavefunctions are Fourier transforms, so neither can be sharp when the other is.
This reframing is illuminating but not a complete deflation of the principle. The physical content lies in identifying momentum with the conjugate Fourier variable scaled by ℏ — that de Broglie link is what makes the abstract theorem a statement about real particles.