Reciprocity between moduli and phases in time-dependent wave functions

For time (t)-dependent wave functions, we derive rigorous conjugate relations between analytic decompositions (in the complex t plane) of phases and log moduli. We then show that reciprocity, taking the form of Kramers-Kronig integral relations (but in the time domain), holds between observable phases and moduli in several physically important instances. These include the nearly adiabatic (slowly varying) case, a class of cyclic wave functions, wave packets, and noncyclic states in an “expanding potential”. The results define a unique phase through its analyticity properties, and exhibit the interdependence of geometric phases and related decay probabilities. Several known quantum-mechanical applications possess the reciprocity property obtained in the paper.