How does Planck’s constant appear in Heisenberg’s uncertainty principle?

In quantum mechanics, Planck’s constant ($h$) is fundamental to the uncertainty principle, which was articulated by the German physicist Werner Heisenberg.

The principle states that it is impossible to measure certain pairs of conjugate observables—such as position and momentum—with absolute precision simultaneously. The product of the uncertainties in these measurements is constrained by a value derived from Planck’s constant. Specifically, the product of the uncertainty in position and the uncertainty in momentum is equal to or greater than:

$$\frac{h}{4\pi }$$

This relationship also applies to other conjugate pairs, such as energy and time, where the product of their respective uncertainties is similarly bounded by $h/(4\pi )$ or more. Because $h$ is an exceedingly small value, these uncertainties are negligible for macroscopic objects but become significant at the scale of atoms and subatomic particles.1