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 nontrivial factor in the quantum evolution operator. For a charged particle, the infinite degeneracy 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 quantum evolution operator. This result agrees with the fact that the instantaneous hamiltonian is invariant under magnetic translation as well as rotation. Simple experimental verification is proposed.