Expectation values and measurement

When a wavefunction is not an eigenstate of the quantity being measured — the usual situation, since most physically realistic states are not perfectly sharp in any one observable — quantum mechanics cannot predict what a single measurement will return with certainty. What it predicts instead is the expectation value: the average result that would emerge from measuring that same quantity on a large number of identically prepared systems, each outcome weighted by its quantum-mechanical probability. This is calculated directly from the wavefunction and the relevant operator, and it is a genuine, well-defined prediction — just not a prediction about any one measurement.

Confusing the expectation value with the outcome of a single measurement is the most common misreading of this idea. Any individual measurement returns one specific eigenvalue of the operator, chosen probabilistically, not the expectation value itself — which typically is not even one of the possible eigenvalues at all, in the same way that an average die roll of 3.5 is not a value any single roll can produce. Measure many identically prepared systems and the average of all those individual, eigenvalue-only outcomes converges on the expectation value; no single measurement needs to land anywhere near it.

Measurement in quantum mechanics is also not a passive read-out of a pre-existing value; performing it changes the system. Before measurement, the wavefunction may be spread across many possible outcomes; immediately after a measurement returns a particular eigenvalue, the wavefunction is found to have collapsed onto the corresponding eigenstate — a genuinely different state from whatever it was before. This is why identically prepared systems, sharing the same pre-measurement wavefunction, can still yield different individual results: the measurement is not revealing a hidden pre-existing answer, it is part of what determines the outcome.