Heisenberg Uncertainty Principle Calculator

The Heisenberg uncertainty principle states that there is a fundamental limit to how precisely two given quantities can be known together. For position and momentum: Δx·Δp ≥ ħ/2, meaning the more precisely a particle’s position is known (small Δx), the more uncertain its momentum (large Δp), and vice versa. This is not a limitation of measuring instruments, but an intrinsic property of nature — quantum particles simply do not have sharply defined position and momentum simultaneously.

Δx is the uncertainty (standard deviation) of position in meters, Δp is the uncertainty of momentum in kg·m/s, and ħ (read “h-bar”) is the reduced Planck constant = 1.054571817×10⁻³⁴ J·s. The ≥ sign means their product is never smaller than ħ/2 ≈ 5.27×10⁻³⁵ J·s. This calculator computes the lower bound: enter Δx to get the minimum Δp, or vice versa.

ħ is Planck’s constant divided by 2π: ħ = h/(2π), with h = 6.62607015×10⁻³⁴ J·s so that ħ = 1.054571817×10⁻³⁴ J·s. They are the same quantity, just a convention: h goes with ordinary frequency f (E = hf), while ħ goes with angular frequency ω (E = ħω). Since ω = 2πf, the two formulas are identical. The uncertainty principle is written with ħ because it arises naturally from the commutator [x, p] = iħ in quantum mechanics.

Because ħ is extremely small (10⁻³⁴). Example: a dust grain of mass 10⁻⁹ kg whose position is known to 1 micrometer has momentum uncertainty only Δp ≥ 5.27×10⁻²⁹ kg·m/s, i.e. velocity uncertainty Δv ≥ 5.27×10⁻²⁰ m/s — far below anything measurable. The effect only becomes significant at atomic scales: an electron confined to 1 angstrom (10⁻¹⁰ m) has Δp ≥ 5.27×10⁻²⁵ kg·m/s, equivalent to a velocity uncertainty of about 579,000 m/s.

The second form of the uncertainty principle links energy uncertainty ΔE to a time interval Δt: ΔE·Δt ≥ ħ/2. A classic application is the natural linewidth of spectral lines: an excited state living only Δt = 10⁻⁸ s has energy uncertainty ΔE ≥ 5.27×10⁻²⁷ J, so the frequency of the light it emits is not perfectly sharp. The shorter a state’s lifetime, the broader its energy spread.

No. Energy uncertainty does not mean energy appears from nothing. The relation is fully consistent with energy conservation: it limits how sharply the energy of a short-lived state can be defined. Virtual particles in Feynman diagrams are a perturbation-theory bookkeeping device, not real particles violating energy conservation — the total energy of every observed physical process is still conserved.

Take an electron confined to atomic size, Δx = 10⁻¹⁰ m. Then Δp ≥ ħ/(2Δx) = 1.054571817×10⁻³⁴/(2×10⁻¹⁰) = 5.27286×10⁻²⁵ kg·m/s. Dividing by the electron mass 9.1093837015×10⁻³ⁱ kg gives Δv ≥ 5.79×10⁵ m/s. This is why electrons cannot sit still in the atomic nucleus: localizing them gives them large momentum (and kinetic energy) — the basis of matter’s stability.

They can be known, just not both with unlimited precision at the same time. You may measure position as accurately as you like — at the price of momentum becoming highly uncertain, and vice versa. The limit is quantitative (ħ/2), not a total ban. For macroscopic bodies this bound is so tiny that in everyday practice position and velocity can be known together precisely enough.

Use h when working with ordinary frequency f in hertz (E = hf, p = h/λ), and ħ when working with angular frequency ω = 2πf or angular wavenumber k = 2π/λ (E = ħω, p = ħk). The two are always interchangeable via h = 2πħ. This calculator uses ħ because the standard form of the uncertainty principle is Δx·Δp ≥ ħ/2; if you have the h version of a formula, remember its equivalent form Δx·Δp ≥ h/(4π).

No — it applies universally to all particles and quantum systems, from electrons and protons to photons. For photons, the position–momentum version becomes an uncertainty between position and wavelength. Even macroscopic systems obey the same relation; only the ħ/2 bound is tiny compared to everyday scales so the effect is unobservable. The principle is a mathematical consequence of the wave nature of matter, not a property of one special particle.