Electron localized to 0.10 nm
Inputs: Δx = 0.10 nm, mass = electron.
Substitution: Δp = ħ/(2Δx).
Result: Δp ≥ 5.2729×10^-25 kg·m/s and Δv ≥ 5.7884×10^5 m/s.
Heisenberg Uncertainty Principle Calculator
Calculate minimum uncertainty in position, momentum, velocity, energy, or time using ΔxΔp ≥ ħ/2 and ΔEΔt ≥ ħ/2. Enter one value to solve the minimum paired uncertainty, or enter both to check whether the pair satisfies the bound.
Formula summary
- ΔxΔp ≥ ħ/2
- Δpmin = ħ/(2Δx)
- Δxmin = ħ/(2Δp)
- ΔEΔt ≥ ħ/2
- ħ = h/(2π), so ħ/2 = 5.2729×10^-35 J·s
Variables and units
| Symbol | Meaning | SI unit |
|---|---|---|
| Δx | Position uncertainty, as a standard deviation | m |
| Δp | Momentum uncertainty, as a standard deviation | kg·m/s |
| Δv | Velocity uncertainty for a massive non-relativistic particle, using Δp = mΔv | m/s |
| ΔE | Energy uncertainty or energy width | J |
| Δt | Time scale, lifetime, or pulse duration | s |
| h | Planck constant | J·s |
| ħ | Reduced Planck constant, h/(2π) | J·s |
Position–Momentum
Electron mass is used for Δv = Δp/m.
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Product Δx·Δp
—
Compare with ħ/2 = 5.2729×10⁻³⁵ J·s
Minimum Δp
—
If Δx is given
Minimum Δx
—
If Δp is given
Bound ratio
—
1.00 = at the limit
Velocity uncertainty Δv
—
Uses Δp = mΔv for massive non-relativistic particles
Enter Δx, Δp, or both to see the minimum result and a substituted inequality.
Visual: product vs ħ/2
Log-scale boundary: Δp = ħ/(2Δx)
Energy–Time
Product ΔE·Δt
—
Compare with ħ/2 = 5.2729×10⁻³⁵ J·s
Minimum ΔE
—
If Δt is given
Minimum Δt
—
If ΔE is given
Bound ratio
—
1.00 = at the limit
Enter ΔE, Δt, or both to see the minimum result and a substituted inequality.
Visual: product vs ħ/2
Log-scale boundary: ΔE = ħ/(2Δt)
Worked examples
Proton localized to 1 fm
Inputs: Δx = 1 fm, mass = proton.
Substitution: Δp = ħ/(2×10^-15 m).
Result: Δp ≥ 5.2729×10^-20 kg·m/s and Δv ≥ 3.1523×10^7 m/s.
Femtosecond laser pulse
Inputs: Δt = 10 fs.
Substitution: ΔE = ħ/(2Δt).
Result: ΔE ≥ 0.03291 eV.
Short-lived excited state
Inputs: Δt = 1 ns lifetime.
Substitution: ΔE = ħ/(2×10^-9 s).
Result: ΔE ≥ 3.2911×10^-7 eV.
Macroscopic 0.145 kg object
Inputs: Δx = 1 μm, mass = 0.145 kg.
Substitution: Δv = ħ/(2mΔx).
Result: Δv ≥ 3.6365×10^-28 m/s, far below everyday velocity scales.
How it works
- Position–momentum: Heisenberg says
Δx · Δp ≥ ħ/2withħ = h/(2π) = 1.054571817×10⁻³⁴ J·s. If you provide Δx, we compute minimum Δp = ħ/(2Δx), and vice versa. - Velocity uncertainty: For a massive, non-relativistic particle, Δp = mΔv. Choose a mass preset to compute Δv from the momentum uncertainty.
- Energy–time: The common form
ΔE · Δt ≥ ħ/2relates energy spread to a lifetime, pulse width, or evolution timescale. Same algebra: 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 shows the product ratio, while the log-scale graph shows the boundary curve and your entered point when both values are provided.
All calculations convert inputs to SI internally. Displayed results can be converted back to the selected output units, and each Copy SI button copies the unrounded SI value.
Physics caveats and constants
- Δx, Δp, ΔE, and Δt are uncertainty widths. For position and momentum, the textbook inequality uses standard deviations of a quantum state, not ordinary measurement error.
- The equality case is a Gaussian minimum-uncertainty wave packet. Other state shapes normally produce products larger than ħ/2.
- Energy-time uncertainty is not identical to position-momentum uncertainty because time is not an operator in standard non-relativistic quantum mechanics. Treat it as a lifetime, linewidth, bandwidth, or evolution-time estimate.
- Velocity uncertainty uses Δp = mΔv, so it is for massive non-relativistic particles. The photon preset intentionally disables mass-based velocity output.
Sources: NIST CODATA fundamental constants; Encyclopaedia Britannica on the uncertainty principle; Stanford Encyclopedia of Philosophy on uncertainty relations.
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.
FAQ
What is the Heisenberg uncertainty principle formula?
The common textbook forms are ΔxΔp ≥ ħ/2 and ΔEΔt ≥ ħ/2. Here ħ is the reduced Planck constant, h/(2π).
How do I calculate minimum momentum uncertainty?
Enter Δx and leave Δp blank. The tool converts Δx to meters and calculates Δp_min = ħ/(2Δx).
How do I calculate uncertainty in velocity?
Choose a particle mass or enter a custom mass. The calculator first finds Δp, then uses Δv = Δp/m for a massive non-relativistic particle.
What is ħ/2?
ħ/2 is half the reduced Planck constant. Numerically it is about 5.2729×10^-35 J·s.
What units should I use?
You can use the dropdowns for common classroom and physics units, including nm, Å, eV, MeV, fs, and eV/c. Internally the calculator uses SI values.
Why is my product below the limit?
If the ratio is below 1, the entered pair gives a product smaller than ħ/2. That usually means a unit conversion error or a pair that cannot represent a valid quantum state in this model.
Does uncertainty mean measurement error?
No. The uncertainty here is the spread of a quantum state, often represented by a standard deviation. It is not simply imprecision in the measuring device.
Why is energy-time uncertainty different?
Position and momentum are operator observables in standard quantum mechanics, while time is normally a parameter. Energy-time uncertainty is still useful for lifetimes, linewidths, and pulse bandwidths, but its interpretation is more context-dependent.