The Calculation And Generalization Of Geometric Phase Of Quantum System

The research of Mesoscopic system has become a compelling field in physicalkingdom. Lots of new physical phenomena have been found. Among them, thediscovery of Aharonov-Bohm effect is very important for the defining of geometricphase at experimental aspect.It was discovered by Berry that the total phase factor contains acircuit-dependent component in addition to the familiar dynamical componentdefined by the energy of quantum system, which accompanies the periodic evolutionof the system and the Hamiltonian returning to original condition, provided that theHamiltonian is cyclic and adiabatic in an non-degeneracy quantum system. Peopleattach importance to the concept of geometric phase since it is brought forward. As toall sides' efforts, a series of limitation is cancelled gradually. One of the mostimportant generalizations is given by Aharonov and Anandan, who give the notion ofnonadiabatic geometric phase (Aharonov-Anandan phase). we get the final generalconclusion that geometric phase exists in any evolution of quantum system. largenumbers of experiments also prove the existing of geometric phase. Because of its importance in theory and experiment, many scientists dedicatethemselves to the calculation of geometric phase. However, there is not a generalmethod for all the quantum system by now. In this paper, operator decompositionapproach is used to calculate the non-adiabatic geometric phase of anharmonicoscillator. As an example we focus on the isotonic oscillator, a type of anharmonicoscillator. The Aharonov-Anandan phase is derived when we choose the groud stateand the first excited state as cyclic initial states. Then we generalize our result bychoosing three states or more states as cyclic initial states. furthermore, we givegeneral formula of Aharonov-Anandan phase for time-independent systems anddiscuss its applicability. In the end, we show the outlook for the development andapplication of geometric phase.