Question

3.16 For a spin-1/2 system undergoing Rabi oscillations, assume that the resonance condition holds. a) Solve the differential equations for the coefficients α + (r). Use your results to find the transformed state vector |ψ(t)) and the state vector ψ(t), assuming the most general b) Verify that a π-pulse (wit-T) produces a complete spin flip. Calculate both the c) Assume that the interaction time is such that ω/-: π/2. Find the effect on the system d) Discuss the differences between the original reference frame and the rotating reference initial state of the system. transformed state vector |ψ(t)) and the state vector | ψ(t)) if the initial state is+) frame in light of your results.

Answer 1

A spin 1/2 system under going Rabi oscillations Let w = w_0 9a) The coefficients alpha plus or minus (t) the transformed vector is th alpha_+ 9t0 = hw_1/2 alpha - (t) (1) mt alpha - (t) = hw_1/2 alpha + (t) so now alpha_+ are Coefficient of state vector and w_1 is frequency Here substituting first equation alpha_+ (t) = -w^2_1/2 alpha (t) therefore The general solution of a second order different quation is alpha + (t) = A e^t/2 + Be -t/2 and alpha - (t) = -Ae^1/2 + Be^t/2 \r\n Substitute Ae^1/2 + Be^-to n t/2 The psi(t) = [Ae^t/2 + Be^-t/2 -Ae^t/2 + Be^-t/2] and [psi (t)] [psi (t)] = [Ae^i(t/2) + Be^-i(psi) -Ae^i(t/2) + Be^-i(e theta/2) the initial state vectors can be given = [psi(0)] = [A + B -A + B] = [psi(0)] = [A + B -A + B] therefore still 1/2 system the general up state is psi(0) = [cos theta/2 e sin theta/2]