Operators and observables

In classical mechanics, a quantity like energy or momentum is simply a number attached to a particle's state. Quantum mechanics represents each measurable quantity — called an observable — not as a number but as an operator, a mathematical action that transforms one wavefunction into another. Position is represented by an operator, momentum by a different operator (built from spatial derivatives), energy by yet another. The wavefunction itself is where all the physical information lives; the operator is the tool that extracts a particular piece of it.

The link between an operator and what a measurement actually produces runs through eigenvalues. A wavefunction that is an eigenstate of a given operator is a special one that the operator leaves unchanged in form, merely scaled by a number — that number is the eigenvalue, and it is exactly the value a measurement of that observable returns for that state. This is not a minor mathematical detail: it is the complete answer to "what values can a measurement give?" A system in an eigenstate of energy has a perfectly sharp, well-defined energy — no spread, no uncertainty in that quantity — even though its wavefunction may be spread out in position. Measurement outcomes are always drawn from an operator's eigenvalues, never anything in between.

Different observables generally have different operators with different sets of eigenstates, and a wavefunction that is a clean eigenstate of one observable is typically not an eigenstate of another — a state with sharp, definite momentum, for instance, is necessarily spread out over all of space in position, with no sharp position value at all. This mismatch between which states are "sharp" for which observables is the operator-based version of the uncertainty principle: it is not a statement about measurement disturbance, but about the mathematical structure of which quantities can and cannot be simultaneously sharp for the same wavefunction.