Uncertainty as a Fourier Trade-off

The two panels are not two measurements of two things. They are the same array of complex numbers, drawn twice: the lower panel is the discrete Fourier transform of the upper one, recomputed every frame. A quantum state does not have a position distribution and, separately, a momentum distribution. It has an amplitude, and those are its two views.

Drag Squeeze in position down and watch. As the violet bump narrows, the teal one widens — and not vaguely: halve σx and σp exactly doubles. That reciprocity is not quantum mechanics. It is a property of Fourier transforms that a radio engineer would recognise instantly: a shorter pulse occupies more bandwidth, which is why a fast data link needs a wide channel and a narrow filter rings.

The bridge is de Broglie’s p = ħk. Momentum is the Fourier variable conjugate to position, so the engineer’s bandwidth theorem becomes a statement about momentum, and the constant that shows up in front is ħ.

That is the whole derivation. σx·σp ≥ ħ/2 is a width theorem multiplied by a conversion factor. Nothing in it mentions a measuring device bumping into a particle, which is why the relation constrains a state that nobody is looking at, and why it holds just as firmly for a photon in a fibre as for an electron in an atom.

With Gaussian selected, drag the width slider from one end to the other and watch the bar at the bottom of the canvas. σx and σp both change by large factors. Their product does not move: 0.5, to four decimal places, at every width.

Now switch to Top hat. It is a perfectly reasonable-looking packet — a particle confined to a region, nothing pathological — and its product is several times the floor. The reason is visible in the momentum panel: hard edges have Fourier tails that decay slowly, and those far tails dominate ⟨p²⟩ even though they carry almost no probability. Sharpness is expensive, and it is expensive far away from where the sharpness is.

The Gaussian is the unique shape that reaches the bound, and that fact quietly runs a lot of physics: it is why the ground state of any harmonic trap is Gaussian, why a laser’s transverse mode is Gaussian, and why “minimum-uncertainty state” and “coherent state” are the same words.

Turn up Quadratic phase and keep your eye on the upper panel. Nothing happens to it. |ψ(x)|² is untouched — the packet is exactly as wide as it was, in exactly the same place. But the momentum panel spreads out and the product climbs off the floor.

The faint grey line inside the position panel is the real part of ψ, and that is where the chirp is hiding: the oscillation gets tighter toward the edges. Momentum is the rate at which phase winds, so a phase that winds at different rates in different places is a packet carrying a spread of momenta. Two states can look identical on a probability plot and still be different states with different physics.

Contrast that with Give it a push. A uniform phase winding e^(ik₀x) slides the whole momentum distribution sideways and changes neither width. Uniform winding is motion; non-uniform winding is spread.

Set a Gaussian with no chirp, confirm the bar is sitting on the floor, and press Release it. The packet evolves as a free particle — each momentum component simply accumulates its own phase, which is exact, not a time-stepping approximation.

Two things happen at once. The momentum panel does not move at all, because a free particle conserves momentum and nothing is there to change the distribution. And the position panel spreads, following σx(t) = σ₀√(1 + (ħt/2mσ₀²)²), because the fast components run ahead and the slow ones fall behind. So the product grows, and a minimum-uncertainty state stops being one the instant you stop holding it.

What is actually building up is the chirp you added by hand a moment ago. Free evolution generates quadratic phase all on its own — which is why the tightest packet spreads fastest, and why an electron microscope’s resolution and an optical fibre’s bandwidth are limited by the same equation.

The oscillator shapes make the connection to energy explicit. Select Oscillator n=1: the product is exactly 1.5, and n=2 gives exactly 2.5. Those are the same numbers as the harmonic oscillator’s energy levels, (n + ½)ħω, because for a harmonic trap the energy simplyis a weighted sum of ⟨x²⟩ and ⟨p²⟩. The ladder of energies and the ladder of uncertainty products are one ladder.

It also explains the ground state that nobody can turn off. A particle in a box cannot sit still at the bottom, because being localised at all forces a momentum spread, and a momentum spread is kinetic energy. That zero-point energy is what the tunnelling page leans on when a particle turns up somewhere it has no business being, and what the double slit is doing when a narrower slit produces a wider pattern. Try Two bumps here and compare the momentum panel with the fringes on that page — they are the same calculation.