On a ring threaded by a flux line moves a charged quantum particle whose rotation is hindered by an angle-dependent potential. When the potential is rigidly rotated through , the particle (in the nth eigenstate of the potential on the ring) acquires a geometric phase . A general formula is , where is the dimensionless quantum flux and is a Schrödinger current associated with the state. Properties of are obtained in terms of the transmission coefficient round the ring. vanishes when the box is impenetrable or when is integer or half-integer. Energy levels form bands, with playing the role of Bloch pseudomomentum. For unhindered semiclassical states above the barrier, a WKB theory gives the geometric phase and hence the (previously calculated) classical Hannay angle . This does not vanish when is integer or half-integer, but these cases correspond to band edges where the semiclassical states are degenerate and differ greatly from the true asymptotic states. The theory is illustrated by the exact calculation of for a model where the potential is a delta function.