[QUE/QM-24006]

Consider a particle of charge \(q\) and mass \(m\), which is in simple harmonic motion along the \(x\)-axis so that Hamiltonian is given by \[ H_0 = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + \frac{1}{2}m\omega^2 x^2\] A homogeneous electric field \(\mathcal{E}(t)\) along the \(x\)-axis is switched on at time \(t=0\), so that the  system is perturbed by the interaction \[H^\prime(t)= -q x \mathcal{E}(t).\] If \(\mathcal{E}(t)\) has the form \[\mathcal{E}(t)= \mathcal{E}_0 e^{-t/\tau}\] where \(\mathcal{E}_0\) and  \(\tau\) are constants and if the oscillator is in the ground state at \(t=0\), find the probability it will be in the \(n^\text{th}\) excited state as \(t\to\infty\).

Source{Bransden and Jochain 9.1*}