What does the Heisenberg uncertainty principle really say?

The uncertainty principle says that a quantum particle cannot have exactly defined values of both position and momentum at the same time. The more tightly one is pinned down, the more spread out the other must be. In its standard form, the uncertainty in position multiplied by the uncertainty in momentum can never be less than half the reduced Planck constant. Similar relations exist for other pairs of quantities, such as energy and time. [1] [2]

A free particle with a perfectly definite momentum is described by a wave spread uniformly along all of space, so its position is completely uncertain. A localized particle is modeled as a wave packet, a combination of many wavelengths and therefore many momenta. Squeezing the packet into a smaller region requires mixing in a wider range of momenta, so the momentum uncertainty grows. A bell-curve (Gaussian) shape gives the smallest possible product, exactly ħ/2. [1]

Mathematically, a system can have a definite value of momentum or a definite value of position, but never be in a state that has both, because the operators for the two quantities do not share such states. [2]

The principle is often explained as clumsy measurement: probing a particle disturbs it. Heisenberg's 1927 argument did involve such disturbance, using effects like Compton scattering, and the principle was first framed as limiting what we can know about the electron. But modern textbooks stress that it has nothing to do with the precision of the apparatus. Even perfect instruments would face the same uncertainties, because they come from the wave-like nature of matter. [3] [1]

It also does not mean there are particular things that can never be known. One explainer puts it as: you can never be sure of everything, but that does not mean there are some things you can never be sure of. You can set up an experiment that predicts position well, or one that predicts speed well, but not one that does both. Repeated measurements on many identically prepared particles show a spread of results that reflects the probabilities given by the wave function. [4] [2]

The energy-time version says a quantum state that exists only briefly cannot have a sharply defined energy. The reasoning is that energy is linked to frequency, so measuring energy precisely requires observing the state for many cycles. This relation is not derived in the same way as the position-momentum one. [1]

The meaning of the relations is not fully settled. The Stanford Encyclopedia of Philosophy notes that there is no agreement on the broader interpretive issues of quantum theory, so one cannot expect agreement on what the uncertainty relations mean. Heisenberg himself read them as limits on what can be meaningfully said about position and velocity. The mathematical statement itself is standard and not in doubt. [3] [1]