### 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.