# symmetry-xk-fourier-complex.mp4

Chapter section: Position / Wave Number Uncertainty

Caption (verbatim):

Trading off spread in position for spread in wave number

Image description (verbatim):

A complex wave function and its Fourier transform sweep between the localization extremes

## Before the animation

Chapter lines 1025–1025:

As we know, a single-mode wave function has a single wave number. But what position does it have? There is no way to answer this as the wave is uniform over all position space. The same statement holds in reverse. A wave packet ideally localized at one position is uniform over all $k$-space. Anywhere in between these extremes, as a wave packet is more localized in one space, it is more spread out in the dual space. 

## After the animation

Chapter lines 1033–1033:

We can easily see this relationship drawn on paper, but we also hear it in music. The precise pitch of a tuning fork requires long sustain, while the percussive clap of a clave has no clear pitch. This tradeoff is the root of the Heisenberg uncertainty principle, or what is known in popular science as "quantum fuzziness."

Chapter lines 1035–1035:

With this visual understanding of the tradeoff in the spread of the magnitude envelopes in $x$ and $k$ space, we can define the corresponding statistical standard deviation, or uncertainty, in position and wave number.