Heisenberg Uncertainty Calculator

How it works

• Position–momentum: Heisenberg says Δx · Δp ≥ ħ/2 with ħ = h/(2π) = 1.054571817×10⁻³⁴ J·s. If you provide Δx, we compute minimum Δp = ħ/(2Δx), and vice versa.
• Energy–time: The looser form ΔE · Δt ≥ ħ/2 relates energy spread to the timescale of change/measurement. Same math: minimum partner = ħ/(2·known).
• Products & ratios: We show your product and the ratio to ħ/2. A ratio ≥ 1 satisfies the bound. A ratio < 1 means the pair is impossible in simple quantum mechanics.
• Visualization: The bar indicates how your product compares to the limit. Caps at 300% to avoid runaway widths; values still show numerically.

Non-relativistic context, Gaussian wave packets assumed for the “minimum” case. Real systems can have larger products depending on state shape.

Why Heisenberg uncertainty matters (and how to use these numbers)

The Heisenberg uncertainty principle sets a floor on how well pairs of conjugate quantities can be specified. For position and momentum, Δx · Δp ≥ ħ/2 tells you that squeezing a particle’s position wavefunction forces its momentum spread to widen. This isn’t about clumsy measurement; it’s about how wavefunctions are built from Fourier components. A narrow pulse in space requires many momentum components, creating a broad spread in momentum space. The calculator treats Δx and Δp as standard deviations—the common textbook form. If you enter one, it returns the minimum possible value of the other, assuming a Gaussian wave packet (the minimum-uncertainty shape).

Energy–time is trickier because time isn’t an operator in non-relativistic quantum mechanics. Still, ΔE · Δt ≥ ħ/2 gives a useful rule of thumb: rapid processes (small Δt) require broad energy spreads (large ΔE). You see this in ultrafast lasers: a 10 femtosecond pulse naturally spans a wide spectrum because its coherence time is so short. In nuclear and particle physics, short-lived excited states have wide energy widths (Γ), reflecting the same relation. Our calculator uses the same algebra: given ΔE, it returns the minimum Δt, and vice versa.

Ratios help you sanity-check homework. If your Δx·Δp ratio is below 1, the numbers can’t describe a physical state in the simple Heisenberg picture—revisit your units or assumptions. If it is exactly 1, you’re at the limit. Anything above 1 is allowed; real systems often sit above the bound because their wavefunctions aren’t perfect Gaussians. The bar visualization caps at 300% to stay readable, but the numeric ratio shows the true value. The product values also carry the same units as ħ (J·s) for both pairs, since 1 J·s = 1 kg·m²/s.

Try the presets: the position example uses an atomic-scale Δx to show that even modest momentum spreads satisfy the bound. The laser-like energy–time example highlights why short pulses carry broad spectra. You can also test macroscopic masses to see why classical intuition works: even “tiny” positional uncertainties translate to negligible momentum lower bounds. If you need relativistic precision, remember that momentum becomes p = γmv; that only raises Δp for a given Δx, so it never violates the bound. Keep units consistent, and let the ratio guide whether your numbers pass the fundamental test.