Uncertainty Principle

The Heisenberg uncertainty principle is not a limitation of measurement technology — it is a fundamental property of quantum states. Position and momentum cannot both be precisely defined for any quantum object.

Key Concepts

Position-momentum: ΔxΔp ≥ ℏ/2
Energy-time: ΔEΔt ≥ ℏ/2
Commutator: [x̂, p̂] = iℏ
Uncertainty = standard deviation in repeated measurements
Minimum uncertainty: Gaussian wave packets

Key Equations

Position-momentum uncertainty

\Delta x\,\Delta p \geq \frac{\hbar}{2}

Energy-time uncertainty

\Delta E\,\Delta t \geq \frac{\hbar}{2}

Commutator

[\hat{x},\hat{p}] = i\hbar

General uncertainty relation

\Delta A\,\Delta B \geq \frac{1}{2}|\langle[\hat{A},\hat{B}]\rangle|

Worked Example

Example Problem

Problem

An electron is localized in Δx = 0.1 nm. Find the minimum uncertainty in momentum.

Solution

Δp ≥ ℏ/(2Δx) = 1.055×10⁻³⁴/(2×10⁻¹⁰) = 5.28×10⁻²⁵ kg·m/s.

Practice

Exercises

7 problems

1 of 7

Drag the slider to set Δx = 0.5 nm and observe how the momentum-space spread changes. What is the minimum Δp in units of 10⁻²⁵ kg·m/s?

Drag the slider to set Δx = 0.50 nm. Watch the momentum-space width (Δp) widen as position narrows — the uncertainty principle live! Enter Δp in units of 10⁻²⁵ kg·m/s.

Δx = 1.00 nm

Δp = 0.5275 × 10⁻²⁵ kg·m/s

Δp = × 10⁻²⁵ kg·m/s

2 of 7

For an electron confined to Δx = 0.5 nm, use the visualization to find the minimum kinetic energy K_min in eV.

Set Δx = 0.50 nm with the slider. The panel below the wave shows K_min = (Δp)²/(2m) live. Enter K_min in eV.

Δx = 1.00 nm

K_min = eV

3 of 7

An atomic state has lifetime τ = 1.0×10⁻⁸ s. Find the minimum energy width ΔE in eV.

Unlock Exercise 3