ℏ is the reduced Planck constant: h/(2π) ≈ 1.0546×10⁻³⁴ J·s. It appears in the uncertainty principle because quantum states are described by wave functions with angular frequency, not ordinary frequency.

Why is the result a minimum uncertainty, not the actual uncertainty?

The uncertainty principle gives a lower bound: Δx·Δp ≥ ℏ/2. Real measurements can have larger uncertainties due to experimental imperfections. Only an ideal minimum-uncertainty (coherent or squeezed) state saturates the bound exactly.

How does particle mass affect the calculation?

Mass converts between momentum uncertainty (Δp = m·Δv) and velocity uncertainty. A heavier particle has a smaller minimum velocity spread for the same position uncertainty.

What units should I use?

Use SI units for best results: meters (m) for position, kg·m/s for momentum, kg for mass, joules (J) for energy, and seconds (s) for time. The calculator also accepts common sub-units like nm, eV, and ms.

Does this apply to macroscopic objects?

Technically yes, but the minimum uncertainties are so small (due to the tiny value of ℏ) that they are completely unmeasurable for everyday objects. The principle is only practically relevant at atomic and subatomic scales.

Heisenberg Uncertainty Calculator

Name: Heisenberg Uncertainty Calculator
Availability: InStock
Author: Nham Vu

Calculate minimum position or momentum uncertainty using the Heisenberg uncertainty principle (Δx·Δp ≥ ℏ/2). Also covers energy-time uncertainty.

Heisenberg: Δx · Δp ≥ ℏ/2 where ℏ = 1.0546 × 10⁻³⁴ J·s

What do you know?

Position uncertainty (Δx)

Unit

Enter a value on the left and click Calculate

Constants & Reference

ℏ (h-bar) 1.0546 × 10⁻³⁴ J·s

h (Planck) 6.6261 × 10⁻³⁴ J·s

mₑ (electron) 9.1094 × 10⁻³¹ kg

mₚ (proton) 1.6726 × 10⁻²⁷ kg

Summary

Calculate minimum position or momentum uncertainty using the Heisenberg uncertainty principle (Δx·Δp ≥ ℏ/2). Also covers energy-time uncertainty.

How it works

Select a calculation mode: position-momentum or energy-time uncertainty.
Enter the known uncertainty value and, for velocity mode, the particle mass.
The calculator applies Δx·Δp ≥ ℏ/2 (or ΔE·Δt ≥ ℏ/2) to find the minimum uncertainty.
Results are shown in SI units with scientific notation for small values.

Use cases

Verify textbook quantum mechanics problems involving position and momentum bounds.
Estimate the minimum speed spread of electrons confined to a nanoscale region.
Calculate minimum energy uncertainty for a particle with a known lifetime.
Explore how confinement size affects momentum uncertainty in nanostructures.
Check units and order-of-magnitude estimates in quantum physics coursework.

Frequently Asked Questions

Last updated: 2026-07-01 · Reviewed by Nham Vu