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1. Consider the time evolution of a spin 21 degree of freedom S=(Sx,Sy,Sz) situated in a time-dependent magnetic field, B(t)=B0ez+B1(costex+sintey) where e is the unit vector along the axis. For the magnetic moment =S let us define the angular frequencies 0=B0 and and 1=B1. (i) Derive an effective static Hamiltonian for its time evolution, by considering a reference frame rotating along with B1(t). (ii) For the initial condition (0)=S=21,Sz=21, at time t=0, calculate the probability that the spin flips to S=21,Sz=21 after time period T, by solving the Schrdinger equation for the effective Hamiltonian obtained in (i). 2. Calculate the same transition probability as Problem 4.1(ii) in terms of the first order time-dependent perturbation. Identify the condition when the calculation is a good approximation to the exact result in 4.1 (ii). The first-order formula may be used without its derivation