eigenstate

An eigenstate of an operator is a special state that the operator leaves pointing in the same direction, merely rescaling it by a number. While a generic state gets reshaped when the operator acts, an eigenstate emerges unchanged in form, multiplied only by its eigenvalue. In that sense an eigenstate is perfectly matched to the operator — it is one of the operator's natural, characteristic states.

Eigenstates have a vivid physical meaning. If a system happens to be in an eigenstate of some observable, then that observable has a single, definite value — its eigenvalue — and a measurement returns that value with certainty, no randomness at all. This is the only situation in which a quantum measurement is fully predictable; for any other state the outcome is genuinely probabilistic.

Because the eigenstates of a Hermitian operator form a complete, orthogonal set, any state whatsoever can be written as a superposition of them. Measuring the observable then projects the state onto one eigenstate, and the probability of each outcome is set by how much of that eigenstate the original state contained. This is the machinery behind the Born rule and the apparent collapse on measurement.