expectation value

The expectation value of a quantity is the average result you would get if you measured it on a great many systems all prepared in the very same state. Written with angle brackets, like ⟨x⟩ for average position or ⟨E⟩ for average energy, it is computed by weighting each possible outcome by its probability and summing. It distils a whole probability distribution down to a single representative number.

It is essential to read the word 'expectation' carefully: it does not mean the result you expect on any single measurement. The expectation value can even be a number that no individual measurement could ever return — the average of a die's faces is three and a half, a value never actually rolled. In quantum mechanics ⟨x⟩ might sit at the centre of a symmetric cloud where the particle is, in fact, almost never found.

Expectation values are how the smooth predictions of quantum mechanics connect to the laboratory and to classical physics. Ehrenfest's theorem shows that the averages ⟨x⟩ and ⟨p⟩ evolve much as a classical particle's position and momentum would, which is part of why the everyday world looks Newtonian. Paired with the spread, or standard deviation, around the mean, the expectation value gives a compact and faithful summary of what a quantum measurement will typically yield.