Uncertainty principle

Level AdvancedDifficulty ★★★★★Application⌖ Open in the map

What is it?

Position and momentum wave functions are Fourier transforms of each other, and a function and its transform cannot both be narrow: $\sigma_x\sigma_p \ge \hbar/2$ . The same inequality limits time–frequency resolution in signal processing (spectrograms, radar).

Formulas

\sigma_x\,\sigma_p \ge \frac{\hbar}{2}, \qquad \sigma_t\,\sigma_f \ge \frac{1}{4\pi}

The mathematics behind it

Fourier transform★★★★★fundamental
Position and momentum wave functions are Fourier pairs; the uncertainty principle is a Fourier inequality.

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.