A particle of charge $e$ moves freely within a cubical box of side $a$. Its initial wavefunction is

$$(2 / a)^{-\frac{3}{2}} \sin (\pi x / a) \sin (\pi y / a) \sin (\pi z / a) .$$

A uniform electric field $\mathcal{E}$ in the $x$ direction is switched on for a time $T$. Derive from first principles the probability, correct to order $\mathcal{E}^{2}$, that after the field has been switched off the wave function will be found to be

$$(2 / a)^{-\frac{3}{2}} \sin (2 \pi x / a) \sin (\pi y / a) \sin (\pi z / a) .$$