The uncertainty principle says a particle cannot have a sharp position and a sharp momentum at the same time: Δx · Δp >= h-bar/2. Drag the position spread slider below and watch the two curves fight — squeeze one and the other widens by exactly the same factor, while the product readout holds at 1.00.
The Uncertainty Principle: Two Spreads That Fight
A minimum-uncertainty wave packet, drawn twice — once in position and once in momentum. Squeeze one curve and the other widens by exactly the same factor, because their product is pinned at h-bar/2. Nothing here measures anything; the trade-off is built into the packet itself.
me = 9.1093837139e-31 kg · mp = 1.67262192595e-27 kg
Minimum-uncertainty (Gaussian) packet: Δx · Δp = h-bar/2 exactly.
What Is the Uncertainty Principle Simulator?
The uncertainty principle simulator is a free interactive physics lab that runs in your browser — nothing to install and no sign-up. Slide the position spread and watch the momentum spread widen in step, with the product pinned at h-bar over 2. It reports product divided by h-bar/2 δx · δp ÷, position spread δx, momentum spread δp, product δx · δp and Speed spread Δv as you drag the sliders.
| Control | Range | Step |
|---|---|---|
| Position spread as a power of ten | -14 – -6 pm | 0.05 |
What the Slider Is Really Changing
The single slider on this page sets Δx, the width of the wave packet itself — not the error bar on a ruler. It moves an exponent rather than a plain number, so one sweep of the thumb carries you from ten femtometres, roughly a nuclear diameter, out to a micrometre, and the readout switches units as it goes. Every position you stop at describes a different, perfectly legitimate particle. The momentum curve drawn underneath is not a second thing you chose; it is the one momentum spread that packet is obliged to have.
That obligation is why the product ÷ h-bar/2 readout never moves off 1.00. The textbook statement is an inequality, Δx · Δp >= h-bar/2, and most wave packets sit some way above the floor. A Gaussian is the exception: it is the one shape that achieves the bound exactly, turning the inequality into the equality Δx · Δp = h-bar/2. So a frozen 1.00 is not a rounding artefact or a hard-coded constant — it is the simulator reporting that this packet is as economical as quantum mechanics permits. Narrowing it in position is really narrowing it into a broader mixture of wavelengths, which is the same trade seen from the wave side; the De Broglie wavelength calculator converts any single one of those momenta into the wavelength that goes with it.
The three particle buttons isolate the part of the physics that students most often merge together. Hold Δx still and switch from electron to proton to dust grain: Δp does not move at all, because Δp = h-bar/(2Δx) contains no mass. Mass only appears one step later, when that momentum spread is turned into a speed spread, Δv = Δp/m. So the speed uncertainty falls by a factor of about 1836 from electron to proton, and by a further twelve orders of magnitude for the dust grain — which is precisely why quantum indeterminacy is invisible in everyday life. The particle never stops being uncertain; its speed uncertainty simply becomes too small to notice.
One last thing this lab is built to disprove: nothing here measures anything. There is no detector, no photon bouncing off the particle, no disturbance. The two curves are one packet described in two languages, related exactly as a shape and its Fourier transform are related, and the trade-off between them is a fact about what a particle is rather than about how clumsily we probe it. For the full argument and where Heisenberg's own microscope thought-experiment misleads, read the Heisenberg uncertainty principle explained, or step back to quantum mechanics: a beginner guide for the wider picture.
Frequently asked questions
What does the position slider actually change?
It sets the width of the wave packet itself — the standard deviation of the position distribution, written Δx. The slider moves an exponent rather than a number, so one sweep covers everything from 10 femtometres to a micrometre. Nothing about the particle is being measured or disturbed; you are simply choosing a different packet, and the momentum curve underneath is the one that packet is forced to have.
Why does the product readout always show 1.00?
Because the simulator draws a Gaussian packet, and a Gaussian is the one shape that sits exactly on the floor of the uncertainty relation. The general statement is Δx · Δp >= h-bar/2, an inequality; a Gaussian turns it into an equality, Δx · Δp = h-bar/2. So 1.00 is not a coincidence or a rounding artefact — it is the readout telling you this packet is as efficient as quantum mechanics allows. Any other shape would push the number above 1.00, never below it.
Why does changing the particle not change the momentum spread?
Δp = h-bar/(2Δx) contains no mass at all. Confinement alone fixes the momentum spread, so an electron, a proton and a dust grain squeezed into the same width all carry exactly the same Δp. Mass only enters one step later, when momentum is converted into speed: Δv = Δp/m. That is why the speed spread collapses by a factor of about 1836 when you switch from the electron to the proton, while the Δp readout does not move.
Does this simulate a measurement disturbing the particle?
No, and that is the misconception the lab exists to correct. Nothing in the simulator observes, detects or disturbs anything. The two curves are properties of one wave packet, related to each other the way a shape and its Fourier transform are related. The trade-off is built into what a particle is, not into the clumsiness of an instrument. The Measure position button is named after the situation it models — a particle confined to a small region — but it changes the packet, not the reading.
What does the confinement energy readout mean?
It is the kinetic energy that the momentum spread implies, E = Δp²/2m, converted to electronvolts. Squeezing a particle into a small box is not free: the momentum spread it forces on the particle carries energy. For an electron held inside roughly an atom's width the figure lands near an electronvolt, which is why atoms are the size they are — squeeze harder and the energy cost climbs faster than the electrostatic attraction can pay for it.
References & formula source
- Eisberg & Resnick — Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, chapter on the uncertainty principle.
- Griffiths — Introduction to Quantum Mechanics, section on the generalised uncertainty principle and the minimum-uncertainty (Gaussian) wave packet.
- NIST — Fundamental Physical Constants (CODATA), reduced Planck constant h-bar = 1.054571817 × 10^-34 J·s.
- Further reading: Uncertainty principle — Wikipedia