projective measurement

A projective measurement is the clean, idealized kind of measurement that textbooks use to introduce the subject. You pick an observable — say, the energy or the spin direction — and the possible results are its eigenvalues. When you measure, the state is 'projected' onto the eigenstate belonging to whichever value you read off, and the probability of each value follows the Born rule. The name comes from the mathematical step of projecting the state onto a particular direction in its space of possibilities.

A hallmark of this idealized measurement is repeatability: if you measure the same observable again immediately afterward, you are guaranteed to get the same answer, because the first measurement has already left the system sitting squarely in that eigenstate. This is why projective measurement is sometimes called a sharp or strong measurement — it extracts the full answer at once and leaves the system in a definite, corresponding state.

Real laboratory measurements are usually messier than this clean picture. Detectors are imperfect, they disturb the system in extra ways, and many useful measurements deliberately extract only partial information. The fully general framework physicists use to handle these cases is broader, but the projective measurement remains the simple, exact backbone of the theory — the case everything else is compared against.