BERRY'S PHASE IN QUANTUM MECHANICS AND ITS APPLICATIONS

Abstract

The aim of this project is to explain Aharonov-Bohm effect and to study the spin-half particle using Berry phase. If we consider Quantum Hamiltonian with external magnetic field that varies in time, we can solve the time dependent shcr̈dinger equation for arbitrary nth stationary state of the Hamiltonian. If we assume that the final change of parameters will be done very slowly (adiabatically), in the adiabatic theorem we say forever in the nth stationary state if we started in it. This state is given up to a phase factor. If we choose this factor and then change external parameter cyclically return after a long time to the initial value the state gets an extra phase consisting of the well known dynamical phase and a new geometrical phase that is Berry's phase. Berry's phase often appears in quantum two-level system such as spin half particle in a rotating magnetic field, as the magnetic field is slowly evolved through a closed path the particle has been shown to acquire an additional phase called Berry's phase. Finally one of the manifestations of Berry's phase is Aharonov Bohm effect. The Aharonov Bohm effect demonstrates that the wave function of a quantum particle traveling close to a solenoid experiences a phase shift even if the magnetic field is zero outside the solenoid. This is due to the non–vanishing vector potential that exists ever where the magnetic field is zero.