Compute the lower bound of the position-momentum uncertainty product Δx·Δp ≥ ħ/2, for quantum-mechanics principle demos and measurement-limit discussions.
| Item | Formula / Note | Source / Standard |
|---|---|---|
| Uncertainty principle | Δx·Δp ≥ ħ/2 | Heisenberg 1927 |
| Energy-time | ΔE·Δt ≥ ħ/2 | Quantum mechanics |
| Derivation | From commutator [x,p] = iħ | Dirac 1930 |
| Standard deviations | Δ are standard deviations | Quantum measurement |
| Non-zero floor | Zero-point energy unavoidable | Quantum oscillator |
| Value of ħ | ħ = 1.054571817×10⁻³⁴ J·s | CODATA 2018 |
Constants used on this page and their authoritative sources, for verification and citation
| Constant | Value | Unit | Source |
|---|---|---|---|
| Reduced Planck constant ħ = h/(2π) | 1.054571817e-34 | J·s | CODATA 2018 (derived from exact h and π) |
| Planck constant h | 6.62607015e-34 | J·s | SI defining constant (exact) |
| Elementary charge e | 1.602176634e-19 | C | SI defining constant (exact) |
Note: the Heisenberg relation holds for any conjugate pair (Δx·Δp ≥ ħ/2, ΔE·Δt ≥ ħ/2); it is a fundamental quantum limit, not a limitation of measurement precision.