3. A spin-1/2 particle subjected to a time-dependant magnetic field is one of the few systems that is both exactly solvable and exhibits a non-trivial Berry phase. Consider an electron (charge -e, mass m) at rest at the origin, in the presence of a magnetic field whose magnitude Bo is constant, but whose direction sweeps out a cone, of opening angle a, at constant angular velocity w. (See Griffiths example 11.4 for a picture). Explicitly, B(t) = Bo (sin a cos(wt), sin a sin(wt), cos a) (1) where I've picked the z direction as the center of the cone. The Hamiltonian is H (t ) =- B . S eh m Bo 2m [sin a cos(wt)or + sin a sin(wt)y + cos Qoz ] (2) hw1 cos a e-int sin a 2 eiwt sin a (3) - cos a where w1 = me. At any instant t, the instantaneous energy eigenstates are just the spinors [X4(n)) which are aligned (+) or anti-aligned (-) with the magnetic field direction n at that instant. They are cos(a/2) X+ (t) = (eiwt sin(a/2)/ X-(t) = e -int sin(a/2) - cos(a/2) with respect to the (fixed) z-direction basis. (a) Show that the exact solution to the time-dependant Schrodinger equation with initial condition x(t = 0) = X+(t = 0) is: x (t ) = cos(At/2) - iw1-w sin(At/2)] cos(a/2) e-int/2 [cos(At/2) - i witw sin(At/2) ] sin(a/2) etiwt/2 (4) where *= VW2 + wi - 2ww1 cos a (5) (b) For w to first order in w/wil