Nonadiabatic noncyclic geometric phase of a spin-1/2 particle subject to an arbitrary magnetic field

We derive a formula of the nonadiabatic noncyclic Pancharatnam phase for a quantum spin-1/2 particle subject to an arbitrary magnetic field. The formu la is applied to three specific kinds of magneic fields. (i) For an orienta ted magnetic field, the Pancharatnam phase is derived exactly. (ii) For a r otating magnetic field, the evolution equation is solved analytically. The Aharonov-Anandan phase is obtained exactly and the Pancharatnam phase is co mputed numerically. (iii) We propose a kind of topological transition in on e-dimensional mesoscopic ring subject to an in-plane magnetic field, and th en address the nonadiabatic noncyclic effect on this phenomenon.