Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S3 / 3 / 2

The wave function is a probability amplitude: by the Born rule, $|\psi (x){|}^{2}dx$ is the probability of a position measurement in an interval of width $dx$. For a normalized bound state $\int |\psi {|}^{2}dx=1$, the expected values areThe integrals run over the allowed domain. If the given state is not normalized, divide both by $\int |\psi {|}^{2}dx$. Its variance is $⟨x^2⟩-⟨x{⟩}^2$; neither $\psi$ itself nor its overall phase is a directly measured position probability.