momentum operator

The momentum operator represents a particle's momentum, and in the position-based picture it does something more interesting than mere multiplication: it takes a derivative. Written as −iℏ times the rate of change of the wavefunction with position, it reads how quickly the wave's phase winds along space. A wave that oscillates rapidly carries large momentum; a gently varying wave carries little.

The appearance of the imaginary unit and the derivative is exactly what is needed to make the operator Hermitian, so that its eigenvalues come out as real momenta. Its eigenstates are pure waves of a single wavelength, spread evenly over all of space — a state of perfectly definite momentum is completely delocalized, with no preferred position at all. This is the mirror image of the position operator's sharp spikes.

Momentum and position form the archetypal incompatible pair. Their operators do not commute, and the resulting canonical commutation relation forbids any state from having both a sharp position and a sharp momentum. This is not a limitation of our instruments but a structural feature of the theory, the deepest reason the uncertainty principle holds.