05 2 v 2 9 A pr 2 00 2 Magnetic translation and Berry ’ s phase factor through adiabatically rotating a magnetic field ∗

For a spin subjected to an adiabatically changing magnetic field, the " solid angle result " as embodied by a rotation operator is the only non-trivial factor in the evolution operator. For a charged particle, the infinite degeneracy of the energy levels calls for a rigorous investigation. We find that in this case, it is the product of the rotation operator and a magnetic translation operator that enters into the evolution operator and determines the geometric phase. This result agrees with the fact that the instantaneous hamiltonian is invariant under magnetic translation as well as rotation. Experimental verification of the result is proposed.