Uncertainty
y = ∫₋₆⁶ exp(−(2 + 1.5 · sin(t)) · u²) · cos(x · u) du
Open in the app The dials and keys named below are the app's.
What it draws
The curve is an integral, computed at every point: y = ∫ e^(−a·u²) · cos(x·u) du from −6 to 6, with a = 2 + 1.5·sin t. e^(−a·u²) is a Gaussian packet in u, and multiplying it by cos(x·u) and integrating asks how much of the frequency x it contains. So y is the Fourier transform of the packet, drawn against frequency, while the packet itself is never on screen. The limits ±6 stand in for ±∞: the integrand there is below 10⁻⁷ even for the widest packet.
The integral has a closed form, √(π/a) · e^(−x²/(4a)): the transform of a Gaussian is a Gaussian. As a breathes between 0.5 and 3.5 every 2π ≈ 6.28 seconds at Speed 1, the peak at x = 0 moves between 2.51 and 0.95. The width goes the other way. The curve is down to 1/e of its peak at x = 2√a, which runs from 1.41 to 3.74.
Why it breathes backwards
The packet e^(−a·u²) falls to 1/e at u = 1/√a, so a large a is a narrow packet. Its transform falls to 1/e at x = 2√a, and the product of the two widths is 2 whatever a is. Squeeze the packet and its spectrum spreads; let it spread and the spectrum sharpens. When the unseen packet is at its narrowest, the curve on screen is at its widest and lowest.
In quantum mechanics u is position and x is momentum, and this trade is the uncertainty principle. In standard deviations of the squared magnitudes, the packet has σ = 1/(2√a) and its transform σ = √a, a product of exactly 1/2. That is the smallest value Heisenberg's inequality σₓ·σₚ ≥ ħ/2 allows with ħ = 1, and the Gaussian is the only shape that reaches it.
History
Werner Heisenberg stated the principle in 1927, with a thought experiment about a gamma-ray microscope, and Earle Kennard proved the inequality σₓ·σₚ ≥ ħ/2 the same year. Its mathematical core is older: the tighter a function, the wider its Fourier transform, a fact of harmonic analysis that holds for sound and radar pulses just as well.
Try
- Draw the packet instead, exp(−(2 + 1.5·sin(t))·x²): the same breathing the other way round, narrowest when the transform was widest.
- Shift the packet, integral(exp(−(2 + 1.5·sin(t))·(u − 2)²)·cos(x·u), u, −6, 6): the same envelope now multiplied by cos 2x, with its first zero at x = π/4 ≈ 0.79. A shift in position is a ripple in frequency.
- Change cos to sin: the curve is flat zero, because the packet is even and a sine picks up only the odd part.
- Freeze the width, integral(exp(−2·u²)·cos(x·u), u, −6, 6): a still Gaussian of height √(π/2) ≈ 1.25.