position operator

The position operator is the observable that represents where a particle is located. In the most common, position-based way of writing quantum mechanics, its action is delightfully simple: it just multiplies the wavefunction by the coordinate. To find the average position you weight each location by how much probability the wavefunction places there.

Although the operation looks like plain multiplication, the position operator is still a genuine operator with eigenstates and eigenvalues. Its eigenvalues are all the possible positions along the line, and its eigenstates are perfectly sharp spikes located at single points. A particle truly in such an eigenstate would have an exactly known position — but such a state is an idealization, infinitely spread out in momentum and not normalizable on its own.

The position operator is one half of the most famous pairing in quantum mechanics. It does not commute with the momentum operator, and that single fact — captured in the canonical commutation relation — is the mathematical source of the uncertainty principle. The more sharply a state is peaked in position, the more widely it must be spread in momentum, and vice versa.