Heisenberg Uncertainty Principle Calculator
Enter a position uncertainty in picometres to find the minimum momentum spread (in eV/c), or enter an energy uncertainty in eV to find the minimum time uncertainty (in attoseconds), using Heisenberg's uncertainty principle Δx·Δp ≥ ℏ/2.
Uncertainty pair
pm
Δp = ℏ / (2·Δx) — smallest allowed momentum spread
- 1
HUP constant (ℏ/2)
ℏ/2 = 98,663.49 pm·eV/cDerived from ℏ = 1.0546 × 10⁻³⁴ J·s converted to picometre·eV/c units. - 2
Minimum momentum uncertainty
Δp = (ℏ/2) ÷ Δx = 98,663.49 ÷ 100 = 986.63
How does this calculator work?
Heisenberg's principle: Δx·Δp ≥ ℏ/2 and ΔE·Δt ≥ ℏ/2 (ℏ ≈ 1.0546 × 10⁻³⁴ J·s). Minimum momentum spread Δp = ℏ/(2Δx); minimum time uncertainty Δt = ℏ/(2ΔE). These are hard quantum limits — not engineering limitations — reflecting the wave-like nature of particles.
Formula
How this is calculated
The Heisenberg uncertainty principle is a fundamental theorem of quantum mechanics — not a limitation of measuring instruments. It states that certain pairs of physical quantities, called conjugate variables, cannot simultaneously have arbitrarily small uncertainties. The more precisely one is known, the less precisely the other can be.
For position and momentum: Δx·Δp ≥ ℏ/2, where Δx is the standard deviation in position, Δp is the standard deviation in momentum, and ℏ = h/(2π) ≈ 1.0546 × 10⁻³⁴ J·s is the reduced Planck constant. The calculator accepts Δx in picometres (1 pm = 10⁻¹² m, useful for atomic scales; 100 pm ≈ 1 ångström) and returns the minimum Δp in eV/c (a natural momentum unit in particle physics). The conversion used is 1 eV/c = eV/c = 5.344 × 10⁻²⁸ kg·m/s.
For energy and time: ΔE·Δt ≥ ℏ/2. The calculator accepts ΔE in eV and returns the minimum Δt in attoseconds (1 as = 10⁻¹⁸ s). For example, an excited atom with a 1 eV energy width has a lifetime of at least 329 attoseconds. This form is used to estimate the natural linewidth of spectral transitions: narrower spectral lines correspond to longer-lived states. Results are at the quantum minimum (Gaussian wave packet); real systems have larger products.
Frequently asked questions
No. It is intrinsic to quantum mechanics, not a consequence of clumsy measurements. Even in principle, a quantum state cannot simultaneously have a precise position and a precise momentum. The principle reflects Fourier duality: a wave packet localized in position must contain a broad spread of wavelengths — and hence momenta.
eV/c (electronvolt per speed of light) is a convenient momentum unit in particle physics: 1 eV/c ≈ 5.344 × 10⁻²⁸ kg·m/s. An attosecond (as) equals 10⁻¹⁸ seconds — the timescale of electron motion in atoms. The calculator shows SI equivalents (kg·m/s and seconds) in the results grid.
ℏ (h-bar) = h/(2π) ≈ 1.0546 × 10⁻³⁴ J·s, where h ≈ 6.626 × 10⁻³⁴ J·s is Planck's constant. The 2π factor arises because the uncertainty relations are most naturally stated using angular momentum and angular frequency (radians per second) rather than ordinary frequency.
Also known as
TG we-Calculate Editorial Team. (2026). Heisenberg Uncertainty Principle Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/heisenberg-uncertainty-calculator
TG we-Calculate Editorial Team. "Heisenberg Uncertainty Principle Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/heisenberg-uncertainty-calculator.
TG we-Calculate Editorial Team, "Heisenberg Uncertainty Principle Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/heisenberg-uncertainty-calculator
@misc{wecalculate_heisenberg_uncertainty_calculator, title = {Heisenberg Uncertainty Principle Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/heisenberg-uncertainty-calculator}}, year = {2026}, note = {TG we-Calculate} }
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