Heisenberg Uncertainty Principle
The Heisenberg uncertainty principle sets fundamental limits on measuring position and momentum simultaneously.
Learn with examples.
The Formula
The Heisenberg uncertainty principle states that the more precisely you know a particle's position, the less precisely you can know its momentum, and vice versa. This is not a limitation of measurement technology but a fundamental property of nature.
German physicist Werner Heisenberg formulated this principle in 1927. It arises from the wave nature of matter described by quantum mechanics. A wave that is sharply localized in space must contain many different momentum components, and vice versa.
The reduced Planck constant ℏ (h-bar) equals h/(2π) ≈ 1.055 × 10⁻³⁴ J·s. There is also an energy-time form of the principle: ΔE × Δt ≥ ℏ/2. This means energy can fluctuate for very short time intervals, which allows phenomena like quantum tunneling and virtual particles.
Variables
| Symbol | Meaning |
|---|---|
| Δx | Uncertainty in position (meters, m) |
| Δp | Uncertainty in momentum (kg·m/s) |
| ℏ | Reduced Planck constant (1.055 × 10⁻³⁴ J·s) |
Example 1
An electron is confined to a region 1 × 10⁻¹⁰ m wide (about the size of an atom). What is the minimum uncertainty in its momentum?
Given: Δx = 1 × 10⁻¹⁰ m
Apply the formula: Δp ≥ ℏ/(2Δx) = 1.055 × 10⁻³⁴ / (2 × 10⁻¹⁰)
Δp ≥ 5.28 × 10⁻²⁵ kg·m/s
Example 2
A proton's momentum is known to within Δp = 1 × 10⁻²⁰ kg·m/s. What is the minimum uncertainty in its position?
Apply the formula: Δx ≥ ℏ/(2Δp) = 1.055 × 10⁻³⁴ / (2 × 10⁻²⁰)
Δx ≥ 1.055 × 10⁻³⁴ / 2 × 10⁻²⁰
Δx ≥ 5.28 × 10⁻¹⁵ m (about the size of a nucleus)
When to Use It
Use the Heisenberg uncertainty principle to determine the fundamental limits of simultaneous measurements in quantum systems.
- Estimating the minimum kinetic energy of confined particles
- Understanding why electrons don't collapse into the nucleus
- Calculating the natural linewidth of spectral emissions
- Analyzing quantum tunneling probabilities