Heisenberg's Uncertainty Principle Calculator
Calculate minimum uncertainties in position and momentum for quantum particles
Calculate Quantum Uncertainty
Standard deviation in position measurement
For velocity uncertainty calculation
Uncertainty Results
Formula: Δx · Δp ≥ ℏ/2 = 5.273 × 10^-35 J⋅s
Where: ℏ = h/2π = 1.055 × 10^-34 J⋅s (reduced Planck constant)
Physical Context
Example: Electron in Motion
Problem Setup
Particle: Electron (mass = 9.109 × 10⁻³¹ kg)
Speed: 2.00 × 10⁶ m/s (well below speed of light)
Speed precision: 0.5% uncertainty
Momentum: p = mv = 1.82 × 10⁻²⁴ kg⋅m/s
Momentum uncertainty: Δp = 9.10 × 10⁻²⁷ kg⋅m/s
Calculation
Using Δx ≥ ℏ/(2Δp):
Δx ≥ (1.055 × 10⁻³⁴ J⋅s) / (2 × 9.10 × 10⁻²⁷ kg⋅m/s)
Δx ≥ 5.80 × 10⁻⁹ m = 5.8 nm
Result Interpretation
The minimum position uncertainty (5.8 nm) is much larger than the size of an atom (~0.1 nm), demonstrating that quantum uncertainty becomes significant for particles at the atomic scale.
Common Particle Masses
Physical Constants
Quantum Tips
The uncertainty principle is fundamental to quantum mechanics
It's not due to measurement limitations but quantum nature
Smaller particles have larger relative uncertainties
Effects become negligible for macroscopic objects
Understanding Heisenberg's Uncertainty Principle
What is the Uncertainty Principle?
Werner Heisenberg's uncertainty principle states that you cannot simultaneously measure certain pairs of quantum properties (like position and momentum) with perfect accuracy. The more precisely you know one property, the less precisely you can know its complementary property.
Why Does This Happen?
- •It's a fundamental property of quantum mechanics, not a measurement limitation
- •Particles exhibit wave-particle duality at quantum scales
- •Position is a particle-like property, momentum is wave-like
- •The act of measurement affects the quantum system
Mathematical Formulation
Δx · Δp ≥ ℏ/2
where ℏ = h/(2π)
- Δx: Standard deviation in position measurement
- Δp: Standard deviation in momentum measurement
- ℏ: Reduced Planck constant = 1.055 × 10^-34 J⋅s
- h: Planck constant = 6.626 × 10^-34 J⋅s
Note: This inequality represents the minimum possible uncertainty product. Real measurements will typically have larger uncertainties.
Applications and Implications
Atomic Structure
Explains why electrons don't spiral into the nucleus and determines the size of atoms and energy levels.
Nuclear Fusion
Position uncertainty allows particles to overcome energy barriers in stellar fusion processes.
Quantum Technology
Foundation for quantum computing, cryptography, and precision measurement techniques.