• Tags: uncertainty, heisenberg, principle
• Difficulty: Moderate

The uncertainty principle states that certain pairs of physical quantities (conjugate variables) cannot be simultaneously measured with arbitrary precision:

\Delta x$$\Delta p ≥ ℏ/2 (position-momentum) \Delta E$$\Delta t ≥ ℏ/2 (energy-time)

where ℏ = h/(2π) = $1.055 \times 10^{-34}$ J·s.

This is not a limitation of measurement instruments but a fundamental property of nature arising from the wave nature of matter. Practical consequences: an electron confined to a box of width $\Delta x$ has a minimum momentum uncertainty $\Delta p$ ≈ ℏ/(2$\Delta x$), giving minimum KE ≈ ($\Delta p$)^{2}/(2m) ≈ ℏ^{2}/(8m\Delta$$x^{2}). This explains why electrons in atoms have non-zero kinetic energy even in the ground state.

For macroscopic objects, the uncertainty is negligibly small. For a 1 kg object with $\Delta x$ = 10^{-10} m: $\Delta p$ ≈ $5 \times 10^{-25}$ kg·m/s, $\Delta v$ ≈ $5 \times 10^{-25}$ m/s (undetectable).