[QUE/QM-24007]

A particle of charge $e$ is confined to a two dimensional square box of sides $L$. An electric field $\vec{E}$ given below is applied to the system. $$  \vec{E} = \left\{ \begin{array}{cl} 0, & t< 0 \\ \vec{E_0} \exp(-\alpha t), & t > 0 \end{array} \right. $$ where $\alpha>0$. The vector $E_0$ is perpendicular to one of the sides of the box. To the lowest order in $E_0$ calculate the probability that the charged particle, in the ground state at time $t=0$, is excited to the first state at time $t=\infty$.

Source{Bransden and Jochain 9.1*}