  For time (t) dependent wave functions we derive rigorous conjugate relations
between analytic decompositions (in the complex t-plane) of the phases and of
the 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 non-cyclic states in an "expanding potential".
The results exhibit the interdependence of geometric-phases and related decay
probabilities. Several known quantum mechanical theories possess the
reciprocity property obtained in the paper.
