Heisenberg uncertainty principle calculator
Enter how precisely a particle's position is known and pick the particle; the calculator gives the least possible spread in its momentum and velocity. Or switch to energy and time, enter a lifetime or pulse length, and see the minimum spread in energy and frequency.
Runs in your browser: nothing you enter is sent anywhere.
How to use it
- Choose position–momentum or energy–time.
- For position, pick the particle (or enter a mass in u) and the position uncertainty Δx with its unit.
- For energy, enter the time interval Δt.
- Momentum: Δp ≥ ħ ÷ (2Δx); velocity: Δv = Δp ÷ m.
- Energy: ΔE ≥ ħ ÷ (2Δt); in eV, ΔE ÷ 1.602176634 × 10⁻¹⁹; frequency: Δf = ΔE ÷ h.
- ħ = h ÷ 2π with h = 6.62607015 × 10⁻³⁴ J·s (exact since 2019).
Worked example
An electron confined to Δx = 0.1 nm: Δp = 1.054572 × 10⁻³⁴ ÷ (2 × 10⁻¹⁰) = 5.27286E-25 kg·m/s, and Δv = 5.27286 × 10⁻²⁵ ÷ 9.10938 × 10⁻³¹ = 578,838.18 m/s.
A proton in a nucleus, Δx = 1 fm: Δp = 5.27286E-20 kg·m/s and Δv = 31,524,512.58 m/s, about a tenth of the speed of light.
An excited state living Δt = 1 ns: ΔE = 5.27286E-26 J = 3.29106E-7 eV, a frequency spread of 79,577,472 Hz.
Tips
- It is a lower bound. Real spreads are usually larger; the equality holds only for Gaussian wave packets.
- Fast particles. If Δv comes out near the speed of light, the non-relativistic Δv = Δp ÷ m is no longer accurate; Δp is still right.
- Not about clumsy measurement. The limit comes from the wave nature of matter, not from disturbing the particle with the measuring device.
Frequently asked questions
What does the uncertainty principle say?
Position and momentum cannot both be known exactly. Their uncertainties, as standard deviations, satisfy Δx × Δp ≥ ħ ÷ 2, where ħ = h ÷ 2π = 1.054571817 × 10⁻³⁴ J·s. Squeeze the position and the momentum must spread out.
Why is it unnoticeable for everyday objects?
ħ is tiny. For a 1 g bead located to 1 µm, Δv is about 5 × 10⁻²⁶ m/s. For an electron confined to an atom, 0.1 nm, the same rule gives at least 578,838 m/s, which is why electrons in atoms cannot sit still.
What is the energy–time relation?
ΔE × Δt ≥ ħ ÷ 2. A state that lives for Δt, or a light pulse lasting Δt, has an energy spread of at least ħ ÷ (2Δt). In frequency that is Δf = 1 ÷ (4π Δt); a 1 ns pulse cannot be narrower than about 80 MHz.
Why do some books use h or ħ instead of ħ/2?
Older and rougher versions write Δx Δp ≳ h or ≳ ħ for order-of-magnitude estimates. The sharp bound ħ/2 holds for standard deviations, and that is what this calculator uses. With ħ or h your answer is 2 or 4π times larger.