The uncertainty principle is one of the deepest truths of quantum mechanics: you cannot know both the exact position and the exact momentum of a particle at the same time. Here a Gaussian wave packet is shown simultaneously in position space and momentum space — squeeze it narrow in one and it inevitably spreads in the other. This same trade-off underpins electron microscopes, MRI signal limits and the quantum noise floor of gravitational-wave detectors like LIGO.
Δx · Δp ≥ ℏ/2 — Δx is the position spread, Δp the
momentum spread, and ℏ the reduced Planck constant. For a Gaussian,
Δx = σ and Δp = 1/(2σ), giving exactly
Δx·Δp = 1/2 (in units where ℏ = 1).
Heisenberg derived the principle in 1927 while still in his mid-twenties; it is not about clumsy measuring instruments but a fundamental property of waves themselves — even a perfect apparatus could never beat the ℏ/2 limit.
This simulation displays a single Gaussian wave packet simultaneously in position space, as the probability density |ψ(x)|², and in momentum space, as |φ(p)|². The two pictures are connected by a Fourier transform, so they are not independent. As you make the packet narrow in position the momentum distribution must broaden, and vice versa, illustrating the fundamental quantum limit Δx·Δp ≥ ℏ/2 in real time.
The Position width σ_x slider squeezes or stretches the left-hand bell curve, while the Wave number k₀ slider shifts the average momentum, sliding the right-hand peak left or right. The readout bar reports Δx, Δp and their product live. This same trade-off sets practical limits in electron microscopy, in nuclear magnetic resonance and in the quantum noise floor of gravitational-wave detectors such as LIGO.
What does the Heisenberg uncertainty principle actually say?
It states that the position and momentum of a quantum particle cannot both be known to arbitrary precision at the same time. The product of the two uncertainties has a hard lower bound, written as Δx·Δp ≥ ℏ/2, where ℏ is the reduced Planck constant. This is a property of nature, not a fault of the measuring device.
What do the two panels in this simulation show?
The left panel plots the position-space probability density |ψ(x)|², telling you where the particle is likely to be found. The right panel plots the momentum-space density |φ(p)|², telling you which momenta are likely. They are two views of the same quantum state, linked by a Fourier transform.
What do the two sliders do?
The Position width σ_x slider, ranging from 0.02 to 2.0, narrows or widens the wave packet in position space. The Wave number k₀ slider, ranging from −10 to 10, sets the central momentum, which slides the momentum-space peak along its axis without changing its width. Watch the readout bar respond as you drag either one.
Because position and momentum wave functions are Fourier transforms of each other. A sharply localised function in one domain is mathematically forced to be spread out in the other. Squeezing the position packet to a width σ gives a momentum width of 1/(2σ), so as σ shrinks, 1/(2σ) grows. The trade-off is unavoidable.
For a Gaussian packet the position spread is Δx = σ and the momentum spread is Δp = 1/(2σ), in units where ℏ = 1. Their product is therefore exactly Δx·Δp = 1/2, which equals ℏ/2. That is why the readout sits at the lower bound no matter how you set the width.
A Gaussian wave packet is the unique minimum-uncertainty state. For any other shape the product Δx·Δp would be strictly larger than ℏ/2. Since this simulation only ever shows a Gaussian, the product stays pinned at its theoretical minimum of 0.500, which is exactly the equality case of the inequality.
The relationships it draws are exact for an ideal Gaussian state. It uses analytic formulae rather than numerically transforming the data, so the displayed Δx = σ and Δp = 1/(2σ) are precise. It is a stationary snapshot, however, and does not evolve the packet in time or include wave-packet spreading under a Hamiltonian.
No. Shifting k₀ moves the centre of the momentum distribution but leaves its width unchanged, so Δp stays at 1/(2σ). Likewise Δx is set only by σ. The product depends solely on the width slider, which is why k₀ lets you explore momentum direction without breaking the minimum-uncertainty relation.
Not fundamentally. While measurement can disturb a system, the uncertainty principle is deeper than that. It arises because a quantum state simply does not possess a sharp position and a sharp momentum at once. Even a perfect, non-disturbing apparatus could never beat the ℏ/2 limit.
It sets resolution limits in electron microscopy, constrains signal-to-noise in nuclear magnetic resonance and MRI, and defines the standard quantum limit for precision interferometers. Gravitational-wave detectors such as LIGO use squeezed light specifically to redistribute this unavoidable uncertainty and improve sensitivity.
Werner Heisenberg formulated it in 1927, while in his mid-twenties, as part of the development of matrix mechanics. It was soon recast in terms of the standard deviations of conjugate variables by Earle Kennard, giving the familiar Δx·Δp ≥ ℏ/2 form used throughout quantum mechanics today.