Study On The Physical Property Of Berry's Phase And The Strict Production Of Berry's Phase In Any Cyclic Evolution Of A Quantum System

In 1984, Berry predicted: A quantal system in an eigenstate, slowly transported round a circuit C by varying parameters R in its Hamiltonian , will acquire a geometrical phase factor in addition to the familiar dynamical phase factor.In this paper, we introduce the adiabatic theorem in Berry's original paper, It means: a quantum system governed by Hamiltonian which depends on time keeps stationary state in slowly varying environment. Therefore the evolution of wave function of the system is the same with the evolution of stationary state. Then according to the adabatic theorem, Berry analyzed non-degenerate periodic quantum system exists geometric phase, and gave general formulation of geometric phase.Because of randomcity of phase factor, if we do a gauge transformation to the eigenstate of as follows:Then ,so it is necessary to standardize the evaluation of eigenstate. In this paper, by means of time-dependent gauge transformation, we do a thorough study on undynamic phase which emerges in the quantum system. The results show that the system will gain an non-integrable phase when a periodic evolution doesn't accomplished; however, only when the system achieves a periodic evolution, an integrable phase will be acquire, namely Berry phase. The circuit C(slow parameters R returns to its original value) is very important, or else the phase will be standardized to zero, in a sense, the phase is invariable under periodic time-dependent gauge transformation, therefore it is observable. Thus is given by a circuit integral in parameter space, and therefore Berry phase is also topological. Accordingly, we do a general interpretation on the topological property of Berry's phase as an example of NMR. In this article, we interpret Aharonov-Bohm effect as a geometrical phase factor,its important meanings lie in two points:(1) A have physics effectï¼›(2) phase factor of wave function is concerned with geometrical factor in the area of(), electromagnetic field is completely descripted by phase factor.To simplify the condition of the appearance of Berry's phase is direction of theory development. Aiming at localization of Berry's work, we introduce two important generalization about Berry's phase: (1) Aharonov and Anandan gavegeometrical phase of periodic evolution of quantum system, and predicted: under the limitation, A-A phase tended to Berry's phase;(2) The system can produce strictly geometric phase from the non-adiabatic to the adiabatic limit evolution in the parameter space.In this paper,we prove any cyclic evolution exist universally quantum system, and find that cyclic evolution is related to the initial state. We can get geometric phase by solving time-dependent Schr?dinger equation, but evolution operator is most effective method to obtain wave function formal solution. As a example of NMR, by calculating evolution operator of satisfied Schr?dinger equation, we get the formal solution of wave function, and find out A-A phase which can be produced under Circle condition and cycle initial state condition in the system . The results show: A-A phases of two conditions are associated with initial state in the system. Under the condition of adiabatic limit, the Berry phase is strictly obtained by computing the two types of A-A phase. The primary productions in this paper are as follows:(1) Berry's phase is the integrable case of undynamic phase, and it has geometric property;(2) By means of evolution operator related to cyclic evolution of quantum system, we solve formal solution of the Schr?dinger equation in the nuclear magnetic resonance (NMR) system, we work out the A-A phase that can be produced under circle condition and cycle initial state condition in the system. Under the condition of adiabatic limit, the Berry phase is strictly obtained by computing the two types of A-A phase...