Consider a perturbation to a two-level system with matrix elements where and are positive constants with the appropriate units. (a) According to first-order

Consider a perturbation to a two-level system with matrix elements

where τ and α are positive constants with the appropriate units.

(a) According to first-order perturbation theory, if the system starts off in the state C_a = 1, C_b = 0 at t = -∞, what is the probability that it will be found in the state b at t = ∞?
(b) In the limit that τ → 0, H_ab = aδ(t). Compute the τ → 0  limit of your expression from part (a) and compare the result of Problem 11.4.
(c) Now consider the opposite extreme: ω₀τ >> 1. What is the limit of your expression from part (a)? Comment: This is an example of the adiabatic theorem (Section 11.5.2).

H = Ha 20 11/ 2(2/1)-2- H = H = 0. bala UP.