• summary_type: concept
• word_count: 160

The uncertainty principle establishes fundamental limits on simultaneous measurement: \Delta x$$\Delta p ≥ ℏ/2 and \Delta E$$\Delta t ≥ ℏ/2, where ℏ = h/(2π). This is not a measurement limitation but a property of nature arising from the wave nature of matter. Key applications: (1) Electron in a box of width L: minimum KE ≈ ℏ^{2}/(8$mL^{2}$) (zero-point energy). (2) Electrons cannot exist in the nucleus: confinement to ~10^{-15} m gives KE ~ 10 MeV, exceeding nuclear binding energies. (3) Atomic stability: balance between kinetic energy (increases with confinement) and potential energy (decreases with distance) sets the atomic size at the Bohr radius. (4) Natural linewidth: excited states with lifetime τ have energy uncertainty $\Delta E$ ≈ ℏ/τ, broadening spectral lines. For macroscopic objects, the uncertainties are negligibly small (~10^{-30} m or less).