exp series Ilts of the preceding questions. 2. Consider two spin 1/2's, S, and S2, coupled by an interaction of the form a(t)S, S2; a(t) is a function of time which approaches zero when approaches infinity, and takes on non-negligible values (on the order of a) only inside an interval, whose width is of the order of t, about 1 = 0. a. At t = - co, the system is in the state | +,-> (an eigenstate of S₁. and S2 with the eigenvalues +h/2 and h/2). Calculate, without approximations, the - +) of state of the system at t = +co. Show that the probability P(+ - finding, at t=+co, the system in the state | -, +> depends only on the integral +8 8 a(t) dt. b. Calculate P(+ - +) by using first-order time-dependent pertur- bation theory. Discuss the validity conditions for such an approximation by comparing the results obtained with those of the preceding question. c. Now assume that the two spins are also interacting with a static magnetic field Bo parallel to Oz. The corresponding Zeeman Hamiltonian can be written : - Bol1S1zY2S22) Ho = where y, and y₂ are the gyromagnetic ratios of the two spins, assumed to be different. 1357 COMPLEMENT Ex Assume that a(t) = a, et Calculate P(++) by first-order time-dependent perturbation theory. With fixed ao and t, discuss the variation of +) with respect to Bo <--+)6