The assignment of wave functions to energy densities and probability densities

In many problems involving wave propagation, the squares of the absolute values of the wave functions rather than the wave functions themselves are the physically observable quantities. These quantities are interpreted as energy densities in the case of electromagnetic radiation, for example, and as probability densities in quantum mechanics. The objective of the present paper is to give a theorem involving Fourier integrals which indicates a set of physical measurements of energy densities and probability densities in one-dimensional problems which lead to an essentially unique determination of the wave functions. For use in quantum mechanics, the measurements which are required can be expressed as the mean values of certain operators constructed from the position and momentum operators. These will also be given.