Probability density of quantum expectation values

We consider the quantum expectation value \mathcal{A}=\ of an observable A over the state |\psi\> . We derive the exact probability distribution of \mathcal{A} seen as a random variable when |\psi\> varies over the set of all pure states equipped with the Haar-induced measure. The probability density is obtained with elementary means by computing its characteristic function, both for non-degenerate and degenerate observables. To illustrate our results we compare the exact predictions for few concrete examples with the concentration bounds obtained using Levy's lemma. Finally we comment on the relevance of the central limit theorem and draw some results on an alternative statistical mechanics based on the uniform measure on the energy shell.Comment: Substantial revision. References adde

Quantum probability distribution of arrival times and probability current density

This paper compares the proposal made in previous papers for a quantum probability distribution of the time of arrival at a certain point with the corresponding proposal based on the probability current density. Quantitative differences between the two formulations are examined analytically and numerically with the aim of establishing conditions under which the proposals might be tested by experiment. It is found that quantum regime conditions produce the biggest differences between the formulations which are otherwise near indistinguishable. These results indicate that in order to discriminate conclusively among the different alternatives, the corresponding experimental test should be performed in the quantum regime and with sufficiently high resolution so as to resolve small quantum efects.Comment: 21 pages, 7 figures, LaTeX; Revised version to appear in Phys. Rev. A (many small changes

The most probable wave function of a single free moving particle

We develop the most probable wave functions for a single free quantum particle given its momentum and energy by imposing its quantum probability density to maximize Shannon information entropy. We show that there is a class of solutions in which the quantum probability density is self-trapped with finite-size spatial support, uniformly moving hence keeping its form unchanged.Comment: revtex, 4 page