Assignment:
A system is in an eigenstate | psi_i > with energy E_i. The perturbation
V(t) = H'exp(-((alpha)^2)(t^2))
is turned on at t_i = -infinity and left on until t_f = infinity. Here H' is independent of time, and alpha is a constant. Show that at t_f = infinity, the probability that the system has evolved into the eigenstate | psi_f > with energy E_f is
P(i -> f) = (pi)/((hbar^2)(alpha^2)|< psi_f | H' |psi_i >|^2 exp[-((E_f - E_i)^2)/(2(hbar^2)(alpha^2)].