Quantal phase factors accompanying adiabatic changes

A quantal system in an eigenstate, slowly transported round a circuit C by varying parameters $\mathbf{R}$ in its Hamiltonian $\hat{H}(\mathrm{R})$, will acquire a geometrical phase factor $\exp {i\gamma(\mathrm{C})}$ in addition to the familiar dynamical phase factor. An explicit general formula for $\gamma$(C) is derived in terms of the spectrum and eigenstates of $\hat{H}(\mathbf{R})$ over a surface spanning C. If C lies near a degeneracy of $\hat{H}, \gamma$(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 $\gamma$(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.