Abstract
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 <A 2> and <B 2> 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.
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Helmberg, G. (1969).Introduction to Spectral Theory in Hilbert Space, p. 300. North-Holland, Amsterdam.
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Farina, J.E.G. Variance and covariance in quantum mechanics and the spreading of position probability. Int J Theor Phys 21, 83–103 (1982). https://doi.org/10.1007/BF01857848
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DOI: https://doi.org/10.1007/BF01857848