Incerteza de Heisenberg

Heisenberg uncertainty: Δx·Δp ≥ ℏ/2

The uncertainty principle says you can't pin down both the position and the momentum of a quantum particle to arbitrary precision at once: Δx·Δp ≥ ℏ/2, where ℏ = h/2π ≈ 1.055·10⁻³⁴ J·s is the reduced Planck constant. The catch is that this is a fundamental limit built into nature, not some shortcoming of your equipment. Squeeze Δx down and Δp has to grow to compensate. To make it concrete: localizing an electron within Δx = 10⁻¹⁰ m (roughly an atomic radius) forces Δp ≥ 5.27·10⁻²⁵ kg·m/s. The same idea shows up in energy and time, ΔE·Δt ≥ ℏ/2. Werner Heisenberg worked the principle out in 1927, and it sits at the heart of the Copenhagen interpretation of quantum mechanics.

Applications

Quantum measurement theory. Scanning tunneling microscopy (STM), where tunneling happens precisely because position stays fuzzy. Zero-point energy in harmonic oscillators. Quantum cryptography, where the BB84 protocol leans on the fact that measuring a state disturbs it. And the natural linewidth of atomic spectral lines, since ΔE·Δt sets how much the line broadens.

FAQ

Is this a limit of measuring devices? No. Hand someone a flawless instrument and the product Δx·Δp still can't drop below ℏ/2. The constraint belongs to the quantum state itself.

Why doesn't this apply to baseballs? Because ℏ is so tiny (10⁻³⁴) that the uncertainty vanishes into nothing for anything you can hold. It only becomes noticeable at atomic and subatomic scales.

What is the energy–time form used for? It accounts for why short-lived excited states (small Δt) end up with broader spectral lines (larger ΔE), and it's what lets virtual particles exist in quantum field theory.