canonical commutation relation
The canonical commutation relation is the cornerstone rule that the position and momentum operators do not commute: their commutator equals iℏ. It says that measuring position and then momentum is genuinely different from measuring momentum and then position, by an amount fixed by Planck's constant. This one short equation is arguably the single most important formula in quantum mechanics.
Almost everything distinctively quantum can be traced back to this relation. The uncertainty principle follows from it directly, with the size of iℏ setting the smallest possible product of the spreads in position and momentum. The structure of the harmonic oscillator, the quantization of energy, and the very fact that ℏ sets the scale of all quantum effects are all consequences of this single commutator.
The word canonical signals that this is the basic, defining relation from which others are constructed, mirroring the role that position and momentum play as a fundamental pair in classical mechanics. When physicists 'quantize' a classical system, the central step is precisely to demand that its position and momentum obey this commutation relation, replacing classical variables with operators that no longer commute.
Position and momentum fail to commute by exactly iℏ — the seed of quantum behavior.
Here ℏ = h/2π is the reduced Planck constant. In the classical limit ℏ becomes negligible compared with the quantities involved, the right-hand side effectively vanishes, and position and momentum recover their familiar simultaneous sharpness.