Phase Evolution in a Multi-Component System

We derive a general expression for the expectation value of the phase acquired by a time dependent wave function in a multi component system, as excursions are made in its coordinate space. We then obtain the mean phase for the (linear dynamic $E \otimes \epsilon$) Jahn-Teller situation in an electronically degenerate system. We interpret the phase-change as an observable measure of the {\it effective} nodal structure of the wave function.