Heisenberg's Uncertainty Principle Calculator

Enter a position uncertainty (Δx) and pick a particle to find the minimum possible uncertainty in momentum and velocity, using Δx·Δp ≥ ħ/2.

Quick Facts

Uncertainty relation
Δx · Δp ≥ ħ/2
Position and momentum cannot both be known to arbitrary precision at once.
Reduced Planck constant
ħ ≈ 1.054571817 × 10⁻³⁴ J·s
ħ = h / 2π, where h = 6.62607015 × 10⁻³⁴ J·s is the exact SI-defined Planck constant.
Minimum-uncertainty case
Δp_min = ħ / (2Δx)
This calculator uses the theoretical best case (equality); real measurements give Δp ≥ this value.
Energy-time form
ΔE · Δt ≥ ħ/2
A related but physically distinct uncertainty relation, not computed here.

Your Results

Calculated
Minimum Momentum Uncertainty
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Δp ≥ ħ / (2Δx)
Minimum Velocity Uncertainty
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Δv = Δp / m
Δv as % of Light Speed
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(Δv / c) × 100%
Regime
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Classical vs. relativistic check

Ready

Enter a position uncertainty, choose a particle, then press Calculate.

How the Heisenberg Uncertainty Calculator Works

The Heisenberg Uncertainty Principle is one of the foundational results of quantum mechanics. It states that a particle's position and momentum cannot both be known with arbitrary precision at the same time: Δx · Δp ≥ ħ/2, where Δx is the uncertainty (standard deviation) in position, Δp is the uncertainty in momentum, and ħ = h/2π ≈ 1.054571817 × 10⁻³⁴ J·s is the reduced Planck constant. This is not a statement about clumsy instruments — it is a built-in limit on how sharply position and momentum can be defined simultaneously, derived from the wave nature of matter.

Deriving the minimum velocity uncertainty

This calculator takes the equality case — the smallest uncertainty the principle allows — and solves for momentum: Δp_min = ħ / (2Δx). Since momentum is mass times velocity (p = mv), dividing by the particle's mass converts that into a velocity uncertainty: Δv_min = Δp_min / m. For example, an electron (m ≈ 9.109 × 10⁻³¹ kg) confined to a position uncertainty of 0.1 nm has a minimum velocity uncertainty of roughly 5.8 × 10⁵ m/s — a classic textbook result. The calculator also reports that velocity as a percentage of the speed of light (c ≈ 2.998 × 10⁸ m/s); percentages approaching or exceeding 1% signal that the non-relativistic formula p = mv is losing accuracy and a relativistic treatment would be more appropriate.

Choosing a particle and units

Pick Electron, Proton, or Neutron for standard rest masses, or select Custom to enter any mass in kilograms (useful for atoms, ions, or composite particles). Enter Δx in whichever length unit suits the scale of the problem — picometers or angstroms for atomic-scale confinement, nanometers for typical textbook problems, or micrometers and millimeters for larger, more classical objects. Smaller Δx and smaller mass both push Δv higher, since Δv = ħ / (2mΔx) is inversely proportional to both quantities.

Frequently Asked Questions

What does the Heisenberg Uncertainty Principle actually say?
It states that the uncertainty in a particle's position (Δx) and the uncertainty in its momentum (Δp) cannot both be made arbitrarily small at the same time: Δx · Δp ≥ ħ/2, where ħ = h/2π ≈ 1.054571817 × 10⁻³⁴ J·s. This is a fundamental limit built into the structure of quantum mechanics, not a flaw in measuring equipment.
What formula does this calculator use?
It takes the minimum-uncertainty case Δp = ħ / (2Δx), then converts that momentum uncertainty into a velocity uncertainty with Δv = Δp / m, where m is the mass of the selected particle. This gives the smallest possible spread in velocity consistent with the given position uncertainty.
Why is the velocity uncertainty so much larger for an electron than for a baseball?
Because Δv = ħ / (2mΔx) is inversely proportional to mass. An electron's mass (9.109 × 10⁻³¹ kg) is about 30 orders of magnitude smaller than a baseball's, so the same position uncertainty produces a velocity uncertainty that is utterly negligible for the baseball but enormous — hundreds of kilometers per second — for the electron.
Is there also an energy-time uncertainty principle?
Yes. A related relation, ΔE · Δt ≥ ħ/2, links the uncertainty in a system's energy to the time interval over which it is observed. It has a different physical origin than the position-momentum relation and is not computed by this calculator.

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