Heisenberg Uncertainty Principle
Fundamental limit on measurement precision
Formula
Δx · Δp ≥ h / (4π)
or Δx · Δp ≥ ℏ / 2
- Δx = uncertainty in position
- Δp = uncertainty in momentum
- h = Planck constant (6.626 × 10⁻³⁴ J·s)
- ℏ = h/(2π) (reduced Planck)
Example
Given: Δx = 1.0 × 10⁻¹⁰ m (atomic scale).
Δp ≥ (6.626 × 10⁻³⁴) / (4π × 1.0 × 10⁻¹⁰) kg·m/s
Δp ≥ 5.27 × 10⁻²⁵ kg·m/s
For electron (m = 9.109 × 10⁻³¹ kg), Δv ≥ Δp/m ≈ 5.8 × 10⁵ m/s.
Answer: Δv ≥ 5.8 × 10⁵ m/s
Notes
- Not due to measurement imperfections; fundamental property of nature.
- Also applies to energy-time: ΔE Δt ≥ ℏ/2.
- Macroscopic objects: h very small, so uncertainties negligible.
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