What this calculator does
The uncertainty principle sets a floor on how precisely position and momentum can both be known. Pin down one more tightly and the other necessarily becomes less definite, and no improvement in instruments changes that.
It is not a statement about clumsy measurement. It reflects the fact that quantum objects do not possess simultaneously definite values of both quantities, so the limit is a property of nature rather than of our apparatus.
The formula
The product of the two uncertainties is at least the reduced Planck constant divided by two. Dividing that figure by the known uncertainty gives the minimum possible uncertainty in the other quantity.
| Term | Meaning |
|---|---|
| Reduced Planck constant (ħ) | Planck's constant divided by 2π, equal to 1.054571817 × 10⁻³⁴ J·s. |
| Uncertainty | The spread in possible measured values, not an error in the instrument. |
| Conjugate variables | Pairs like position and momentum, or energy and time, that are bound by an uncertainty relation. |
The inputs explained
| Field | What to enter |
|---|---|
| Known uncertainty | Choose whether you are supplying a position uncertainty or a momentum uncertainty. |
| Value | The known uncertainty: metres for position, kilogram metres per second for momentum. |
When to use it
Estimating an electron's momentum spread
Confining an electron to atomic dimensions forces a large momentum uncertainty, which is why electrons in atoms have such high kinetic energies.
Understanding why atoms do not collapse
Confining an electron closer to the nucleus raises its momentum uncertainty and therefore its energy, which sets a limit on how tightly it can be bound.
Checking a quantum mechanics problem
Uncertainty estimates are a standard way of getting an order-of-magnitude answer without solving the full equations.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does confining a particle affect its momentum?
Position uncertainties from atomic to nuclear scales.
| Position uncertainty | Minimum momentum uncertainty |
|---|---|
| 1e-9 m | 5.2729e-26 kg·m/s |
| 1e-10 m | 5.2729e-25 kg·m/s |
| 1e-12 m | 5.2729e-23 kg·m/s |
| 1e-15 m | 5.2729e-20 kg·m/s |
Questions
Is this just a limitation of our instruments?
No. It is a property of quantum systems themselves. A particle does not have simultaneously definite position and momentum waiting to be discovered, so no improvement in measurement technique can beat the limit.
Why does it not affect everyday objects?
Because ħ is so tiny. For anything of ordinary mass the implied uncertainties are far below what could ever be measured, so classical behaviour is recovered completely.
Does this explain why atoms are stable?
Substantially, yes. An electron falling into the nucleus would be confined to a tiny region, forcing an enormous momentum and therefore kinetic energy. The balance between that and electrostatic attraction sets the size of the atom.
Are there other uncertainty relations?
Yes, energy and time obey a similar one, which is why short-lived particles have poorly defined masses and why spectral lines from brief transitions are broadened.
For the matter-wave picture behind it, see the de Broglie wavelength calculator. For photon energies at these scales, see the photon energy calculator.