The Geometrical Structure Of Berry Phase In Quantum Adiabatic Approximation

The concept of phase factor of state vector is a subtleaspect in quantum mechanics, about which thetime-dependent solution of Schr?dinger equation is stronglyconcerned. If the potential in the equation evolves withdefinite time-period and the evolution is slowly sufficiently,then a nontrivial problem appears, i.e. whether the statevector can evolve back to its original form? The answer isknown as 'no' for the appearance of time-dependent phasefactor. Then whether one can redefine the state vector andsimultaneously absorb the phase factor? The answer is againnegative;the reason is presented in my context. The development of quantum phase factor is a mainstream of physics in last few decades;the well-knownparadigm in the region is Aharonov-Bohm effect and Berryphase factor. And moreover, it has been the most importantsignificance of development of quantum mechanics recently. In this thesis we have considered the geometrical structureof Berry phase systemically, in which the fiber bundletheory is thoroughly applied. Before doing this, the quantumadiabatic approximation and some relative basic conceptsare introduced. The thesis is arranged as following: in thefirst chapter, the Berry phase is introduced in detail, and themost relevant AA phase is also presented;the condition forapplying the quantum adiabatic approximation is derived too.In the second chapter, the quantum adiabatic approximationand some other approximation methods, and the relationshipbetween these methods are systemically introduced. It isshowed that when the condition for adiabatic approximationis not satisfied, one has to consider the higher order quantumadiabatic approximation method. Basing upon aboveintroduction, in chapter III the geometrical structure of Berryphase is concisely discussed. The discussion consists of twodifferent cases: one is degenerate and the other isnon-degenerate case. While studying the topic, we find thatthe fiber bundle is a perfect method to investigate thesubject.In summary, we have obtained the following results inthis paper:1. If the curve line that evolves as a function of timeparameter is closed, then the Berry phase is an element ofHoler group Un (d)on the fiber bundle ξ [ξ is a fiberbundle formed by basis manifold M and fiber ofHamiltonian eigenvector] .2. The induced gauge potential is the connection matrix ofξ .3. The induced gauge field if the curvature tensor of fiberbundle ξ .