The geometric phase of the quantum systems with slow but finiterate of the external time-dependent field

With the help of the time-dependent gauge transformation technique, we havestudied the geometric phase of a spin-half particle in a rotating magneticfield. We have found that the slow but finite frequency of the rotatingmagnetic field will make the difference between the adiabatic geometric phase andthe exact geometric phase. When the frequency is much smaller than theenergy space and the adiabatic condition is perfectly guaranteed, theadiabatic approximation geometric phase is exactly consistent with theadiabatic geometric phase. A simple relation for the accuracy of theadiabatic approximation is given in terms of the changing rate of the frequency ofthe rotating magnetic field and the energy level space.