With b 1 and b 2 as time independent find the matrix

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with b 1 and b 2 as time- independent, find the matrix equation with being the eigenvalue of the matrix. (5 pts) (c) Find the eigenvalues . (10 pts) (d) If c 1 (0) = 1 and c 2 (0) =0, find the expression for c 1 (t) and c 2 (t) . (10 pts) Prob. 3 An infinite potential well of length L is shown in the figure. At t = 0 , the electron is in its ground state. Assume that the ground state and the first excited state are sufficient to form a complete set of eigenfunctions for all questions below. At t = 0 , an constant field F is added to the potential V(x) = F(x+L/2) for L/2 < x < L/2 to perturb the system. (a) Use the finite-base approximation to find the trial energy after perturbation. Is this consistent with the energy input at t = 0 ? (10 pts) (b) Model the system as a two-level system with static coupling for t > 0 , estimate H 1 , H 2 and V 12 . (10 pts) (c) Find the estimated eigenfunctions ( + and ) and eigenvalues ( E + and E ) for t > 0 . (10 pts) (d) Find c 1 (t) and c 2 (t) and the Rabi oscillation frequency in the two-level system. (10 pts) (e) If L = 1nm and F = 0.1eV/nm, compute the Rabi frequency. (5 pts)
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