Variance and covariance in quantum mechanics and the spreading of position probability

A rigorous discussion of the concept of expectation value of an unbounded observable is given, and of its variance. It is shown that ifA andB are observables for which the expectations <A²> and <B²> exist, and such that A+B is also an observable for some real numbers and , neither of which vanishes, then a quantum mechanical analog of covariance and correlation coefficient can be defined. The quadratic variation with time of the variance of position of a particle moving freely in one dimension is deduced rigorously, assuming only that there is a time at which the variances of position and momentum exist.