Last updated: 2026-06-16

Heisenberg Uncertainty Principle

The Heisenberg uncertainty principle is a cornerstone of quantum mechanics. It states that the position and momentum of a particle cannot both be known with unlimited precision: the better the position is pinned down, the more uncertain the momentum becomes. The two uncertainties obey $\Delta x\,\Delta p \ge \hbar/2$, where $\hbar$ is the reduced Planck constant.

This calculator takes a position uncertainty $\Delta x$ and a particle mass $m$, and returns the minimum momentum uncertainty $\Delta p$ and the corresponding velocity uncertainty $\Delta v$.

How the principle works

The limit is not caused by clumsy instruments. It is built into the wave nature of matter: a particle localized to a small region must be built from a wide range of wavelengths, and wavelength corresponds to momentum. Squeezing the position therefore unavoidably broadens the spread of momenta. The principle sets the smallest possible product of the two uncertainties.

Formula

The relation $\Delta x\,\Delta p \ge \hbar/2$ is an inequality; this calculator reports the minimum-uncertainty case where equality holds.

Worked example

An electron ($m = 9.109 \times 10^{-31}\ \text{kg}$) is localized to within $\Delta x = 1 \times 10^{-10}\ \text{m}$, roughly the size of an atom.

Step 1 — momentum uncertainty:

Step 2 — velocity uncertainty:

The velocity is uncertain by hundreds of kilometres per second — a clear sign that a bound electron cannot follow a definite orbit.

Uncertainty at different scales

As the confinement region grows, the velocity uncertainty falls in proportion. For everyday macroscopic masses the effect becomes immeasurably small.

Real-world relevance

The uncertainty principle underlies the stability of atoms: an electron cannot collapse into the nucleus because confining it ever more tightly would demand an ever-larger momentum. It also sets the resolution limits of electron microscopes and explains zero-point energy, the residual motion that persists even at absolute zero.

Limitation

This calculator evaluates the minimum-uncertainty product for the position–momentum pair only. Real measurements often have larger uncertainties, and other conjugate pairs (such as energy and time) follow their own uncertainty relations not covered here.