The Heisenberg Uncertainty Seesaw

A quantum state shown in both conjugate spaces at once: the position wavefunction $|\psi(x)|^2$ and its Fourier transform $|\phi(k)|^2$, with the standard deviations $\sigma_x$ and $\sigma_p$ drawn as extent bars. A slow breathing modulates the squeeze so the seesaw is live; as the position packet narrows the momentum packet must broaden. Because the momentum-space amplitude is the Fourier transform of the position one, narrowing a packet in $x$ necessarily widens it in $p$. A gauge tracks the product $\sigma_x \sigma_p$ against the $\hbar/2$ floor it can never cross: a Gaussian sits exactly on the bound (the minimum-uncertainty state), while a box, triangle or double-bump state sits strictly above it.