| Since the title of this paper is "Geometric Phase in Quantum Mechanics", it is necessary to explain the meaning of "phase" in quantum mechanics before the investigation for "Geometric Phase". After inspecting the "Geometric Phase", we should better explain it mathematically when the phase has it root deeply in topology.The title of this paper's first chapter is "Basic Knowledge", it can be classified into quantum mechanics for undergraduate course and Higher quantum mechanics for postgraduate course. The frame of this chapter is base on professor Liang's < new evolution for modern physics > course for graduate in institution of theoretical physics in Shanxi university.At the beginning of the first chapter the derivation from the Lagrange and Hamilton to the motion equation of a charged particle in magnetic fields in classic mechanics is presented. Then the momentum P in the acquired motion equation is replaced by quantum operator, thus we obtain Schrodinger equation for charged particle moving in magnetic fields in Quantum Mechanics. Then we discuss the Gauge transformation and gauge invariance of Schrodinger equation, following which the Dirac phase is introduced. The AB problem and Dirac magnetic monopole is also introduced. Adiabatic approximation is pulled in for explaining time-dependent Hamilton system and Berry phase. The AA phase in non-adiabatic approximation is introduced after Berry phase. Then time-dependent invariant is introduced and a kind of method to solve the Schrodinger equation explicitly called Gauge transformation method. Once again we discussed the Berry Phase andAharonov-Bohm effect along with adiabatic approximation. At the end of this chapter a concrete calculation example of Gauge transformation is given.In chapter 2 of this paper, two example applying gauge transformation for Quantum Geometric Phase in adiabatic condition are given, with explicit calculation process .The two examples is similar in objects to calculate, they are two state quantum system expressed in sphere coordination.In chapter 3 as the end of this paper, conclusions and expectation based on forenamed analysis is given, especially for Geometric Phase's features.This paper gives a full and explicit analysis for Geometric Phase, from basic theory to familiar calculation modules in articles. We focus on the application of gauge transformation method to Geometric Phase Calculation. |
