Please do 4, 5, and 6 Exercise # 8In class we introduced Rabi's model, a two level system with a time-dependent perturbationHa = H, = 0 and H = (H)* = aheutwhere a and w are real parameters. This leads to the dynamical equationaħe'(w-ww)t]ih-(1)=where wo(Es – Ea)/h is a positive constant provided Ea < Eg. Then the state vector iswritten in terms of the solutions, Ca(t) and c(t), as|V(t)) = ca(t) e¯iEat/h|Ea) + c»(t) e¯-iEst/h|EL).In what follows, you solve Eq. (1) for the initial state|V(0)) = |Ea).1. Let 3 = w - wo and write down Eq.(1) in components. You should getCa = -iaest(2a)CbC = -iae-ißt(2b)Ca2. Take the time derivative of the above. You get two coupled second-order equations,č, = aet (Bc, – ić,)č, = -ae-ist (Bc, + ic,)(За)(3b)3. Eliminate ca and ca from Eq.(3b) by using Eq.(2). You should get the following:č, + ißc, + a²c, = 0(4)4. Solve this ODE (Hint: assume c, = et). Then you should get the following generalsolution,Cb = e-ißt/2(Aest + Be¬it)(5)where A and B are integration constant, and 2 = Va? + (B/2)2 .5. Based on the initial state, what is c(0)? Show that this value leads to B = -A inЕq (5). 6. Show that Eq.(5) can now be written asC, = 2iAe-ist/2 sin t(6)7. Differentiate Eq.(6) and then plug that into Eq.-(2) to find ca. You should get2ACa =- sin Qt + N cos Nt(7)8. Based on the initial state, what is c,(0)? Use this value to find A in Eq.(7).9. Now put all the pieces of information together to show that the solution of Rabi'smodel isw - woCa(t) = e"(w-ww)t/2sin Nt(8a)cos Nt - i2ΩC(t) = e-i(w-wo)t/2(8b)sin Ntwherew - woN = 1a2 +(9)

Name: Please do 4, 5, and 6 Exercise # 8In class we introduced Rabi's model,
Uploaded: 2024-03-20T06:46:03.000Z
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Description: Please do 4, 5, and 6 Exercise # 8In class we introduced Rabi's model, a two level system with a time-dependent perturbationHa = H, = 0 and H = (H)* = aheutwhere a and w are real parameters. This leads to the dynamical equationaħe'(w-ww)t]ih-(1)=wher