A general law for quantum mechanical joint probabilities: generalisation of the Wigner formula and the collapse postulate for successive measurements of discrete as well as continuous observables

Abstract

The fundamental prescriptions of quantum theory have so far remained incomplete in that there is no satisfactory prescription for the joint probabilities of successive observations of arbitrary sequence of observables. The joint probability formula derived by Wigner is based on the collapse postulate due to Von Neuman and Lüders and is applicable only to observables with purely discrete spectra. Earlier attempts to generalize the collapse postulate to observables with continuous spectra have been unsatisfactory as they lead to only finitely additive (and notσ-additive) joint probabilities in general. In this paper a suitable generalisation of the Wigner joint probability formula is proposed, which is completely satisfactory in the sense that it leads toσ-additive joint probabilities for successive observations of arbitrary sequence of observables, consistent with all the other basic prescriptions of quantum theory. This general law for quantum mechanical joint probabilities is arrived at by a reformulation of earlier results on expectation values in successive measurements. The generalized Wigner joint probability formula is also shown to be a consequence of a general collapse postulate, which allows for changes in state due to measurement from normal states to non-normal states also. As an illustration of our results, the probability distribution of the outcomes of a momentum measurement which immediately succeeds a position measurement is computed, and this seems to shed an entirely new light on the uncertainty principle.