Heisenberg's uncertainty principle

Position and momentum cannot both be known to arbitrary precision at once — narrowing the uncertainty in one necessarily widens the uncertainty in the other, a limit built into the wave nature of matter, not a flaw in measuring instruments.

Practise this concept

The uncertainty principle is often told as a story about clumsiness: to see an electron you must bounce a photon off it, and the photon kicks it, so you spoil the momentum while learning the position. The story is memorable and it is wrong — or at least it describes a real effect that is not what the principle is about. Heisenberg's limit survives no matter how gentle the probe, because it is not about probing at all.

It is about waves. A wave with one sharp wavelength has to extend over all space; the moment you confine it to a small region, you can only build that localised shape by adding together many different wavelengths. This is ordinary wave mathematics, true of sound and water waves as much as of electrons. What quantum mechanics adds is de Broglie's link between wavelength and momentum. Position spread and momentum spread are then two faces of the same wave, and Δx·Δp ≥ ħ/2 is a theorem about it.

So the electron does not secretly possess an exact position and an exact momentum that we are prevented from reading. It has a wavefunction, and a wavefunction narrow in position is made of a wide range of momenta. Note also what the principle does not say: any single quantity can be measured as precisely as you like. The trade-off applies only to particular pairs — position with momentum, energy with time — and it is why an electron confined to an atom cannot simply sit still at the nucleus, since squeezing its position would cost more kinetic energy than the binding provides.

Key ideas

Requires: Wavefunctions and probability

Formulas

\Delta x \cdot \Delta p = \frac{\hbar}{2}

Symbol	Name	Unit
$dx$	position uncertainty	m (metre)
$dp$	momentum uncertainty	kg*m/s (kilogram metre per second)

Common misconceptions

The uncertainty exists because measuring a particle disturbs it, so better instruments would remove the limit.
The particle has an exact position and momentum at every instant; we simply cannot learn both.
The principle applies to every pair of quantities, so nothing at all can be measured precisely.

Resources

Articles

Part of these UT courses

This concept is mapped as covering part of the following course(s) in the University of Tartu curriculum.

Physics of the microworldLOFY.01.009
Quantum mechanicsLOFY.04.073