The extention of Berry's theory on geometric phase

To the cyclic Hamiltonian system, where we have done the parameter transiti on t-->R(t), we study the problem of the acquirement of Berry geometric pha se gamma(n) (C) by the "strict" evolution from the non-adiabatic to the adi abatic-limit. Our results-show that there exist four types of evolution sta tes, all of which can satisfy the above "strict" evolution along the same c losed curve C in the space formed by the parameter R and can obtain the sam e Berry geometric phase gamma(n)(C), When Berry first found the geometric p hase gamma(n) (C), he only considered one evolution state, which is just th e adiabatic approximation case of one of the four "strict" evolution states mentioned above. So Berry's theory on geometric phase can be extended into the four types of strict evolution shown in this paper.