uncertainty relation

An uncertainty relation is the precise mathematical inequality behind Heisenberg's principle. For position and momentum it reads ΔxΔp ≥ ℏ/2, where Δx and Δp are the statistical spreads of the two quantities and ℏ is Planck's constant divided by 2π. The inequality sets a hard floor: the product of the two spreads can shrink only so far, never to zero together.

The relation is not limited to position and momentum. A more general version, due to Robertson, applies to any two observables and ties their minimum joint spread to how much their operators fail to commute. When two quantities can be measured together with no trade-off, their operators commute and the right-hand side vanishes; when they cannot, a nonzero floor appears. The canonical commutator of x and p is exactly what produces the ℏ/2 in the familiar case.

It is worth being careful about what the relation describes. It is a statement about the statistical spread of results you would get over many identical preparations, not about a single measurement of a single particle. You can measure one particle's position as accurately as your apparatus allows; the relation governs how the distributions of repeated outcomes are constrained, which is a sharper and more honest claim than 'you can't measure two things at once'.