Problem 1: Exponential Switching Consider the following perturbation to a two-level system: Vfi (t)=x(1-e-at) e[t] Here x is the strength of the perturbation and a is the rate at which this

strength is applied. In the plot below, the steepest curve is for the largest value of a. "Exponential Switching Figure 1 [a] Show that, in the limit of large time and infinitely slow exponential turn on, the transition probability as the following limit: x² w₂2 Use this expression is a useful asymptote that you will apply in parts [b] and [c] below. [b] Consider an exponential switching perturbation for which x = 0.1 w₁ and a = 0.1 w₁i. Plot the transition probability, using time steps of 1/ wf, and show the asymptote derived in [a]. Plt → arbitrarily large, a +0 = (1) (2) [c] Repeat the analysis of [b] but now with a much slower rate of turn-on: a = 0.001 wf. Plot the transition probability, using time steps of 1/ w, and show the asymptote derived in [a]. What is the difference between your results the two cases? Be sure o discuss the overall probability as well as the undulat in the transition probability with time. [d] Assume that the time-evolving state is a linear combination of the initial and final states. Substitute this into the Schrodinger equation, in the interaction picture, to obtain an ODE for M₁. Numerically solve this ODE to obtain p[t], and compare the result with your perturbation approximations of [b] and [c]. Comment on the accuracy of your perturbation approximation.