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 if A and B are observables for which the expectations and exist, and such that αA + βB is also an observable for some real numbers α and β, 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. (author)