Heisenberg's Uncertainty Principle Calculator
Calculate the minimum uncertainty in quantum mechanical measurements based on the fundamental limits of nature
Position-Momentum Uncertainty
Energy-Time Uncertainty
Uncertainty Principle Results
Enter your values to see the physical interpretation of these uncertainties.
Quantum Analysis
Detailed quantum mechanical analysis will appear here.
📊 Fundamental Physical Constants
| Constant | Symbol | Value | Unit |
|---|---|---|---|
| Reduced Planck constant | ħ (h-bar) | 1.054571817×10⁻³⁴ | J·s |
| Planck constant | h | 6.62607015×10⁻³⁴ | J·s |
| Speed of light | c | 299792458 | m/s |
| Electron mass | me | 9.1093837015×10⁻³¹ | kg |
| Proton mass | mp | 1.67262192369×10⁻²⁷ | kg |
| Electron volt | eV | 1.602176634×10⁻¹⁹ | J |
| Atomic mass unit | u | 1.66053906660×10⁻²⁷ | kg |
Note: Values from 2019 redefinition of SI base units (CODATA 2018).
📚 Quantum Physics Insights
Wave-Particle Duality
The uncertainty principle arises from the wave-like nature of quantum particles - precise position and momentum cannot both be known.
Measurement Limits
This isn't about measurement technology - it's a fundamental limit to what can be known about a quantum system.
Virtual Particles
The energy-time uncertainty allows short-lived "virtual particles" to pop in and out of existence in vacuums.
Microscopic vs Macroscopic
Uncertainty is negligible for everyday objects but dominates at atomic scales (ħ ≈ 1.05×10⁻³⁴ J·s).
Quantum Fluctuations
Even in perfect vacuums, fields have tiny fluctuations due to the uncertainty principle.
Mathematical Foundation
The principle comes from non-commuting operators in quantum mechanics: [x̂,p̂] = iħ.
🧪 Example Quantum Systems
Hydrogen Atom Electron
Δx ≈ 5.3×10⁻¹¹ m (Bohr radius)
Δp ≈ 2×10⁻²⁴ kg·m/s
Product ≈ 1.06×10⁻³⁴ J·s ≈ ħ
Quantum Dot
Δx ≈ 10 nm
Δv ≈ 10⁵ m/s
Shows clear quantum confinement
Atomic Nucleus
Δx ≈ 1 fm (10⁻¹⁵ m)
Δp ≈ 200 MeV/c
Explains nuclear force range
Laser Pulse
Δt ≈ 1 fs pulse
ΔE ≈ 0.66 eV bandwidth
Critical for ultrafast spectroscopy
Note: This calculator implements Heisenberg's Uncertainty Principle (Δx·Δp ≥ ħ/2 and ΔE·Δt ≥ ħ/2) where ħ (h-bar) is the reduced Planck constant (≈1.05×10⁻³⁴ J·s). The results show the fundamental quantum limits on measurement precision.