Perturbative solution to the time-dependent two-level problem and the validity of the Rosen-Zener conjecture

We consider the time-dependent two-level problem of quantum mechanics, where the levels are coupled by a radio-frequency pulse with an arbitrary time-dependent envelope $V\left(t\right)\mathrm{}$. We derive an approximate solution for the system's transition amplitude $P\left(\infty \right)$ which is correct to the third order of perturbation theory, and which applies to all pulses $V\left(t\right)\mathrm{}$ with finite first and second moments which obey the following: $\lim t^{3}V\left(t\right)=0\mathrm{}$, as $t\to \infty$. Our form of solution for $P\left(\infty \right)$ provides a criterion for judging the validity of a solution previously conjectured by Rosen and Zener, and it is generally useful for providing line-shape details in many cases of practical interest.