- **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).$
Part of JPH-01 — Modern Physics: Photoelectric Effect & Matter Waves
Heisenberg's Uncertainty Principle
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