Engineering & Science
Heisenberg Uncertainty Principle: Momentum Spread & Natural Linewidth
Calculate minimum quantum momentum spread delta_p*c >= hbar*c / (2 * delta_x) (using hbar*c = 197.327 eV*nm) and excited-state / ultrafast-laser energy linewidth delta_E >= hbar / (2 * tau) (using hbar = 658.212 meV*fs).
Skip 10^-34 Joule-second exponent headaches with exact CODATA eV*nm and meV*fs constants!
Werner Heisenberg (1927) and Earle Kennard proved that non-commuting quantum observables satisfy strict wavepacket Fourier lower bounds: position-momentum uncertainty sigma_x * sigma_p >= hbar / 2, and time-energy lifetime-linewidth uncertainty delta_t * delta_E >= hbar / 2! Typing raw SI hbar = 1.05457e-34 J*s and dividing by e = 1.60218e-19 J invites scientific-notation errors in spreadsheets—whereas two exact CODATA conversion constants make atomic, nuclear, and ultrafast laser calculations effortless: hbar * c = 197.32698 eV*nm (identical to 197.32698 MeV*fm!) and hbar = 658.21195 meV*fs!
Confine an electron inside a delta_x = 0.10 nm atomic wavepacket: its minimum relativistic momentum spread is delta_p * c = 197.32698 / (2 * 0.10) = 986.63 eV! Likewise, a femtosecond laser pulse or short-lived core-hole state lasting tau = 15.0 fs carries a minimum transform-limited energy spread delta_E = 658.21195 / (2 * 15.0) = 21.940 meV (and 329.106 meV at attosecond-scale tau = 1.0 fs)!
| Quantum Confinement / Lifetime Scenario | Input Spread (dx or dt) | Minimum Conjugate Uncertainty | Physical Quantum Consequence |
|---|---|---|---|
| Atomic Electron Localization | dx = 0.10 nm | dp * c >= 986.63 eV | keV-scale zero-point kinetic energy |
| Ultrafast Laser / Core-Hole Decay | dt = 15.0 fs | dE >= 21.940 meV | Broadened spectral bandwidth |
| Sub-Femtosecond Attosecond Pulse | dt = 1.0 fs | dE >= 329.106 meV | Spans > 0.3 eV photon spectrum |
Minimum Momentum Uncertainty delta_p * c (eV) = 197.32698 / (2 * delta_x_nm)
Divides CODATA hbar * c = 197.32698 eV*nm by 2 * spatial standard deviation B2 (nm).
=IF(B2 <= 0, "Invalid Quantum Uncertainty", ROUND(197.32698 / (2 * B2), 2))Minimum Transform-Limited Energy Spread delta_E (meV) = 658.21195 / (2 * delta_t_fs)
Divides CODATA hbar = 658.21195 meV*fs by 2 * temporal pulse/lifetime duration B3 (femtoseconds).
=IF(B3 <= 0, "Invalid Quantum Uncertainty", ROUND(658.21195 / (2 * B3), 3))Try the example
Use the original values or change them to test your understanding.