ΔxΔp = ħ .
On the right side is ħ, the ubiquitous Planck constant (here divided by 2π) which turns up in every equation of quantum mechanics. Momentum is represented by p, and position (distance) by x. The Δ symbols are used here to mean “uncertainty of” (not “change of”): Δx is the uncertainty of position, Δp is the uncertainty of momentum. The product of these two uncertainties is equal to the constant ħ. Since ħ is, on the human scale, a very small quantity, Δx and Δp can be so close to zero in the macroscopic world that there is for all practical purposes no uncertainty whatever in the position and momentum of large objects. If we wish to specify a man’s position to within the size of a single atom, his speed could, in principle, be determined to an accuracy of about 10–26 m/s. Needless to say, the inaccuracies of measurement take over long before the inherent fundamental limitation of accuracy implied by the uncertainty relation can play any role. But in the world of particles, this is not so. Masses and distances are so small that the uncertainty principle is of vital importance. An electron, in order to be localized within a distance 10–10 m (which is equivalent to assigning it this great an uncertainty of position), has an inherent uncertainty of speed of about 106 m/s.
The uncertainty principle, when discussed by itself outside the framework of quantum mechanics, is often assigned a profundity and depth that are not really justified. It has obvious philosophic implications and it is especially popular with those who wish to attack science, for it shows that even the “exact” scientist is prohibited by nature from measuring things as exactly as she might like. One might also argue that nature is shielding its innermost secrets by allowing scientists to proceed only so far and no farther in their downward quest. In truth, the uncertainty principle is fundamental and presents in capsule form an important part of the physical content of quantum mechanics. Nevertheless, it may be viewed as just one more aspect of the wave nature of matter, in which case it seems considerably less mysterious. (See Essay Q8.)