Heisenberg Uncertainty
Find the minimum uncertainty in momentum (and velocity) for a particle confined to a region Δx — or the reverse — and the energy–time limit ΔE·Δt ≥ ℏ/2.
What is the Uncertainty Principle?
In quantum mechanics a particle is described by a wave. A wave squeezed into a small region (small Δx) must be built from many wavelengths, which means a large spread in momentum Δp. You can never pin down both at once — not because of clumsy instruments, but because of what a particle is.
The limit only matters at atomic scales: ℏ ≈ 1.05×10−34 J·s is tiny. It explains why electrons don’t fall into the nucleus and why atoms have a size at all.
Live Graph
log–logReal-World Scales
| System | Δx | Minimum Δv |
|---|---|---|
| Electron in an atom | 10−10 m | 5.8×105 m/s |
| Electron in a nanowire | 10−8 m | 5.8×103 m/s |
| Proton in a nucleus | 10−15 m | 3.2×107 m/s |
| Dust grain (10−15 kg) | 10−6 m | 5.3×10−14 m/s |
| Tennis ball (57 g) | 10−3 m | 9.3×10−31 m/s |
| ℏ = 1.054571817×10−34 J·s · h = 6.62607015×10−34 J·s | ||
Forms of the Principle
Divide by the mass to get the velocity spread. Light particles in small spaces move fast — heavy ones barely notice.
A state that lives only Δt has an energy spread ΔE. Short-lived excited atoms and particles emit broadened spectral lines.
A confined particle can never be completely at rest: it always keeps a minimum kinetic energy. This is why helium stays liquid even at absolute zero.
Sources & Verification
- D. J. Griffiths & D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018 — Ch. 1 (The Wave Function) and Ch. 3 (Formalism).
- E. Tiesinga, P. J. Mohr, D. B. Newell & B. N. Taylor, “CODATA recommended values of the fundamental physical constants: 2018”, Reviews of Modern Physics 93, 025010 (2021) — values of the constants used.