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Uncertainty Principle Calculator

Calculate Heisenberg uncertainty limits for position-momentum and energy-time conjugate pairs.

Inputs

m
kg

Results

Min Momentum Uncertainty (Δp)

0 kg·m/s

Min Velocity Uncertainty (Δv)

578,878.03 m/s

Min Energy Uncertainty (ΔE)

0 J

Min Time Uncertainty (Δt)0 s
Δx·Δp / (ℏ/2)1
De Broglie Wavelength0 m
How to Use This Calculator
  1. Enter Position Uncertainty (Δx) in meters — how precisely you know the particle's location. 1e-10 m is atomic scale.
  2. Enter Particle Mass (kg): the default is an electron (9.109e-31 kg); a proton is 1.67e-27 kg.
  3. Read Min Momentum Uncertainty (Δp) — the fundamental limit imposed by Heisenberg's relation Δx·Δp ≥ ℏ/2.
  4. Min Velocity Uncertainty (Δv = Δp/m) shows how precisely speed can be known given the position measurement.
  5. The Ratio (Δx·Δp)/(ℏ/2) shows how close your measurement scenario is to the quantum limit — a value of 1.0 saturates the bound.

How the result changes with Position Uncertainty (Δx)

Position Uncertainty (Δx)Min Momentum Uncertainty (Δp)Min Velocity Uncertainty (Δv)Min Energy Uncertainty (ΔE)
00 kg·m/s57.89 m/s0 J
00 kg·m/s16.54 m/s0 J
00 kg·m/s8.91 m/s0 J
00 kg·m/s6.43 m/s0 J

What each input means

Position Uncertainty (Δx)
Uncertainty in position measurement
Particle Mass
Mass of the particle (default: electron)

How this is calculated

Formula

Δx × Δp ≥ ℏ/2

Worked example, using the default values

  1. Identify Input Parameters
    Position Uncertainty (Δx) = 1e-10, Particle Mass = 9.109e-31 = 2 input(s) provided
  2. Calculate Min Momentum Uncertainty
    Min Momentum Uncertainty = Number(minMomentumUncertainty.toExponential(3))
    5.273e-25 = 5.273e-25
  3. Calculate Min Velocity Uncertainty
    Min Velocity Uncertainty
    578878.03 = 578878.03
  4. Calculate Min Energy Uncertainty
    Min Energy Uncertainty = Number(minEnergyUncertainty.toExponential(3))
    1.526e-19 = 1.526e-19
  5. Calculate Min Time Uncertainty
    Min Time Uncertainty = Number(minTimeUncertainty.toExponential(3))
    3.455e-16 = 3.455e-16
  6. Calculate Δx·Δp /
    Δx·Δp /
    1 = 1

Engine last updated . Checked against 3 independently-derived tests — how we verify calculators.

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