The hamiltonian for a spin 1/2 particle in a magnetic field b is h = -yb-s where y is the gyromagnetic ratio and s is the spin operator. at t = 0 the particle is prepared in the state x(0) the spin up and spin down states in the s2 basis, and travels (in the y-direction) through an apparatus where it experiences a magnetic field in the z-direction of magnitude b = bo +az (bo is a constant and a is the field gradient coefficient). assume the magnetic field outside the apparatus is zero. the particle emerges from the (z) = ax (z) bx£, where x are apparatus at time t = t. (a) write down the eigen-states and eigen-energies of the particle during the time it travels through the appa- ratus. (b) in terms of the parameters of the system, write down the expression for the particle's state x(t) while travelling through the apparatus. (c) show that when the particle emerges from the apparatus at time t = t, the spin components have acquired momentum along the z-direction in opposite directions. find the magnitude of the momentum and comment (d) consider the situation where the direction of the magnetic field in the apparatus is now changed to the x- direction with magnitude b = bo+ax. if the particle was prepared at t = 0 in the same state x(0) = ax bx and emerges at t = t, follow similar steps to those above to arrive at an expression for the state x(t = t) and compare with your answer to (c). ay(z) data 1 1 0 1 ,x+ , x 1. v2 v2 () (;