Heisenberg's Uncertainty Principle

Once matter is accepted to have wave character, a serious conceptual problem follows: a wave, by its very nature, is spread out over some region of space, so it does not make sense to ask exactly where a wave is located the way one asks where a classical particle is. Werner Heisenberg (1927) turned this observation into a precise, quantitative principle.

Statement. It is impossible to determine, simultaneously and with unlimited precision, both the exact position and the exact momentum (or velocity) of a microscopic particle such as an electron. If $\Delta x$ is the uncertainty in position and $\Delta p$ is the uncertainty in momentum, then

Since $p = mv$, and the mass $m$ of the particle is taken as fixed and known exactly, this can be rewritten in terms of the uncertainty in velocity, $\Delta v$:

The product of the two uncertainties can never be smaller than $h/4\pi$ -- so making one uncertainty very small (measuring position very precisely) necessarily makes the other uncertainty very large (the momentum becomes correspondingly ill-defined), and vice versa. This is not a statement about the limits of our measuring instruments; it is a fundamental feature of nature at the quantum scale, built into the wave character of matter itself. …