7. Differentiate Eq.(6) and then plug that into Eq.(2) to find ca. You should get 24 94/2 ( sin t +N cos Nt (7) 8. Based on the initial state, what is c,a(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's model is w- wo Ca(t) = e(w~ww)}/2 sin t (8a) ca(t) sin t (8b) where wo N = a2 + (9)
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Please do 7, 8, and 9
![Exercise # 8
In class we introduced Rabi's model, a two level system with a time-dependent perturbation
Ha = H, = 0 and H = (H)* = aheut
where a and w are real parameters. This leads to the dynamical equation
aħe'(w-ww)t]
ih-
(1)
=
where wo
(Es – Ea)/h is a positive constant provided Ea < Eg. Then the state vector is
written 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 get
Ca = -iaest
(2a)
Cb
C = -iae-ißt
(2b)
Ca
2. 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 general
solution,
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 in
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