Heisenberg Uncertainty

Heisenberg's uncertainty principle (1927): you cannot simultaneously know a particle's position and momentum to arbitrary precision.

Quantitatively: Δx · Δp_x ≥ ℏ/2, where ℏ = h/(2π).

Similarly for energy and time: ΔE · Δt ≥ ℏ/2 — has implications for line widths and short-lived excited states.

NOT a statement about measurement disturbance — it's a fundamental property of wave packets.

Smaller Δx requires a sharper wave packet, which has a broader range of k (= 2π/λ) — and therefore a broader range of momentum.

Macroscopic objects: ℏ is so small that Δx·Δp can be far below any measurable scale ⇒ classical mechanics is recovered.

Implications: there is NO classical 'orbit' for electrons in atoms — only probability clouds.