💡 Uncertainty principle
Heisenberg's uncertainty principle: Δx · Δp ≥ ℏ/2 (position-momentum) and ΔE · Δt ≥ ℏ/2 (energy-time), where ℏ = h/(2π). So minimum Δp = ℏ/(2Δx), minimum Δx = ℏ/(2Δp); minimum ΔE = ℏ/(2Δt), minimum Δt = ℏ/(2ΔE). The energy-time form applies to the lifetime of excited states.
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🧮Step-by-Step Solution▼ Show
| Form | Formula | Notes |
|---|---|---|
| Position-momentum | Δx · Δp ≥ h/(4π) = ℏ/2 | ℏ = h/(2π) |
| Energy-time | ΔE · Δt ≥ h/(4π) = ℏ/2 | Lifetime of excited state |
About Heisenberg's uncertainty principle
Position-momentum: Δx · Δp ≥ ℏ/2. You cannot simultaneously know position and momentum to arbitrary precision; the product of their uncertainties has a minimum. So Δp_min = ℏ/(2Δx) and Δx_min = ℏ/(2Δp).
Energy-time: ΔE · Δt ≥ ℏ/2. For an excited state with lifetime Δt, the energy spread is at least ΔE ≈ ℏ/(2Δt). Shorter-lived states have broader energy (e.g. natural line width).