Heisenberg Uncertainty
Δx · Δp ≥ ℏ/2 — tune position spread; momentum spread responds inversely. Live product.
Key Notes
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.
Formulas
Position-momentum (1D)
ℏ = h/(2π) ≈ 1.054 × 10⁻³⁴ J·s.
Energy-time
Short-lived states have broader natural line widths.
Approximate (with h)
Loose form often used in JEE — factor differs by 4π.
Equality case (Gaussian)
Achieved by a coherent state (Gaussian wave packet).
Important Points
Uncertainty is a property of QUANTUM STATES, not an artifact of measurement — even an ideal apparatus cannot beat it.
Sharper position localisation requires a wave packet built from a wider range of wavelengths ⇒ wider spread in p.
Why electrons don't crash into nuclei: confined to a tiny Δx, they must have huge Δp ⇒ huge KE ⇒ would escape. Atoms are stable BECAUSE of uncertainty.
ΔE·Δt: a state living only for Δt has energy uncertain by ~ℏ/Δt — explains natural line broadening in spectroscopy.
Uncertainty does NOT mean 'we just don't know' — quantum states genuinely lack precise simultaneous values for incompatible observables.
Macroscopic: a bullet (m = 0.01 kg) localised to 0.1 mm has Δp ≥ ℏ/(2·10⁻⁴) ~ 5×10⁻³¹ kg·m/s — completely negligible compared to classical p.
Heisenberg Uncertainty notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 12 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.