The Study On The Berry Phase And Fidelity Of Two Spin 1/2 Particles

In the whole system of quantum mechanics,quantum geometric phase occupy an important place.Since Berry proposed the existence of geometric phase in the periodic adiabatic process of quantum Hamilton systems in 1984,the theory of geometric phase for has attracted enormous interests.There has been many experimental trials and theoretical analysis on geometric phase in fields of nuclear physics,atomic and molecular field,quantum calculation,optics,condensed matter physics and canonical field scopes.Geometric phase has been able to prove that exists in the evolution of any quantum systems,many physicists have done the further research and discussion on the geometric phase.In this thesis,we mainly focused on the geometric phase for the interaction of two spin particles.In the first part,we introduce the development outline,basic principle and properties,application of the Berry geometric phase and fidelity.In the second part,we obtained the geometric phase of two spin particles through the dynamic system of the so(5) group and the evolution operator of coherent state under the magnetic field.We obained the instantaneous eigenstate and the corresponding eigenvalue of the Hamiltonian through the evolution operator,we used the so(6) group to constructe the spin relevant operators that satisfied the certain relations.We used the commute relation to deformate the Hamiltonian,we constructed the unitary evolution operator,used the Taylor theorem to change the unitary evolution operator,we obtained the geometric contaction system,through the calculation of the geometric contaction in the degeneracy space.In the third part,we obained the fidelity,we changed the the unitary evolution operator to the matrix deformation.So we can obained D~+(ρ),d/dt D~+(ρ) and d/dt d D(ρ),then,we obained the fidelity of the system.