Plain-English Meaning
You cannot know exactly where a particle is and how fast it's moving at the same time. The more precisely you pin down its location (small Δx), the more uncertain its momentum (large Δp). This isn't about bad instruments — it's built into nature itself.
When a quantum question feels ambiguous, translating it into state, observable, probability, and evolution language usually clarifies the answer.
Deeper Explanation
Δx·Δp ≥ ℏ/2 (with ℏ ≈ 1.055×10⁻³⁴ J·s). Rigorously, ΔA·ΔB ≥ |⟨[Â,B̂]⟩|/2 for any pair of observables. Since [x̂, p̂] = iℏ, this gives Δx·Δp ≥ ℏ/2. Similarly, ΔE·Δt ≥ ℏ/2 — short-lived states have wide energy widths (natural linewidth).
Worked Example
Problem: An electron is confined to 1 nm. What is the minimum uncertainty in its momentum and speed?
- Δx = 10⁻⁹ m
- Δp ≥ ℏ/(2Δx) = 1.055×10⁻³⁴/(2×10⁻⁹) = 5.28×10⁻²⁶ kg·m/s
- Δv = Δp/m = 5.28×10⁻²⁶/9.11×10⁻³¹ ≈ 5.8×10⁴ m/s
Result: Δp ≥ 5.28×10⁻²⁶ kg·m/s; Δv ≥ ~58 km/s
At A Glance
Category: Quantum
Levels covered: High School, College, Masters, PhD
Best use: Start with the formula meaning, then move to the worked example and quiz so the equation turns into a tool instead of a memorised line.