5. Consider a two-level perturbed system with (t) = ca(t)vae where the coefficients ca and c, evolve according to -Habe

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5 Consider A Two Level Perturbed System With T Ca T Vae Where The Coefficients Ca And C Evolve According To Habe 1 (75.02 KiB) Viewed 10 times

5. Consider a two-level perturbed system with (t) = ca(t)vae where the coefficients ca and c, evolve according to -Habe -iEat/h -iwot "Cb : cb (t) =- + co(t)e-Ent/h i 2hw, Cb h Here H' is the time-dependent perturbation introduced to the system and the matrix elements H, = (v₁\H'\v₁) are such that Haa = 0 = Hbb Now suppose that for some time-independent potential V, the perturbed Hamiltonian and its matrix elements are given by H' = e-it, Ha Vba -iwt e 2 2 = Hab (V). Assume the system starts off in state a. We have used the shorthand Vij = (a) Show that c, evolves according to the second-order equation: Vab|2 4h2 b=0. (b) The general solution for the last equation above is given by Hewat ca ba cb + i(w-wo) co + 3 co(t) = e-i-t [Aert + Be-iwrt] W = 3 7 Vab 2 Vab/2 ħ² Use the information on the initial conditions to show that ca and c, are given by -Vbaei(wo-w)t/2 sin(wrt), Ca(t) = e(w-wo)1/2 cos(wrt) - i ||| ezwt (w - wo)² + w_wo 2w, (c) What is the transition probability, Pab(t), that the system will be in state , at some later time t? (d) Prove that, at any given time t, the probability of finding the system in either state is always equal to 1. [10] sin(wrt [6] [4] or [5] [25]