Heisenberg's uncertainty principle

If matter genuinely behaves as a wave, there is an unavoidable price to pay: a wave, by its very nature, is spread out rather than located at one exact point, so a particle described by a wave cannot simultaneously have a perfectly sharp position and a perfectly sharp momentum. Werner Heisenberg turned this observation into a precise, quantitative statement now called Heisenberg's uncertainty principle: it is impossible to determine, simultaneously and with unlimited accuracy, both the position and the momentum of a microscopic particle. Formally, if $\Delta x$ is the uncertainty in position and $\Delta p$ is the uncertainty in momentum, their product can never be reduced below a fixed minimum:

This is not a statement about the limits of our measuring instruments - it is a fundamental limit built into the wave nature of matter itself. Making $\Delta x$ smaller (pinning down the position more precisely) necessarily forces $\Delta p$ to grow larger, and vice versa; the product of the two can shrink no further than $h/4\pi$.

Why you never notice this in daily life. For any macroscopic object, $h/4\pi$ is such an astronomically tiny number compared to the object's own momentum that the uncertainty is completely undetectable - which is exactly why classical mechanics works perfectly well for footballs, cars and planets. The principle only becomes significant for particles as light as an electron.

Worked illustration: the electron in the first Bohr orbit. Take the electron in hydrogen's first orbit, whose Bohr radius is $0.529$ Å. Suppose its position within that orbit is known to an accuracy of 0.5% of the radius:

Substituting this into Heisenberg's relation (using it as an equality to get the minimum possible uncertainty) and converting $\Delta p=m\Delta v$:

$\Delta v\geq\frac{6.626\times10^{-34}}{4\times3.1416\times(9.11\times10^{-31})\times(2.645\times10^{-13})}\approx2.19\times10^{8}\ \text{m s}^{-1}$ …