A New Form for the Statistical Postulate of Quantum Mechanics

A new type of representation of the wave function of quantum mechanics, and a new form of the statistical interpretation of the wave function, are presented. The variable in the new representation is a mapping of points of differential space, closely related to Hilbert space, but having an associated measure for suitably defined sets of points in it. In the new statistical interpretation, an eigenvalue of every observable defined in the system is associated with each point in differential space. This can be done in such a way that, for each observable, the measure (=probability) of the set of all differential-space points belonging to a given set of eigenvalues is equal to the quantum-mechanical probability, calculated in the usual way (Born statistical postulate), that an experiment will yield an eigenvalue in this set. Thus we obtain an interpretation of quantum mechanics in terms of probability densities or probabilities instead of probability amplitudes.