Classical Model for Quantum Berry's Phase Factors

Abstract：

Having considered the similarity between the classical evolution equation and the Schrodinger equation, in this paper we show that a geometric phase factor with topological property,which we call the classical Berry's phase factor, appears properly in the solutions of the classical evolution equations with slowly-changing parameters. The general solutions of the approximate equations in any order in the non-degenerate case are given when the adiabatic condition is violated. As an example, an electrically charged particle moves in an adiabatically varying magnetic field is studied in detail, and a classical model for the Berry's phase factor is obtained explicitly. The phase factor can be explained in geometry as a holonomy of the hermitian linear fiber bundle on a unit sphere S² of the parameter space.