Quantal Phase Factors Accompanying Adiabatic Changes

A quantal system in an eigenstate, slowly transported round a circuit C by varying parameters R in its Hamiltonian hat{H}(R), will acquire a geometrical phase factor exp {iγ(C)} in addition to the familiar dynamical phase factor. An explicit general formula for γ(C) is derived in terms of the spectrum and eigenstates of hat{H}(R) over a surface spanning C. If C lies near a degeneracy of hat{H}, γ(C) takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy. As an illustration γ(C) is calculated for spinning particles in slowly-changing magnetic fields; although the sign reversal of spinors on rotation is a special case, the effect is predicted to occur for bosons as well as fermions, and a method for observing it is proposed. It is shown that the Aharonov-Bohm effect can be interpreted as a geometrical phase factor.