[QUE/QM-24005]

Consider a particle of charge \(q\) and mass \(m\), which is in simple harmonic motion along the \(x-\)axis with 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 \(\mathcal{E}(t)\) is applied along the \(x\)-axis \[\Eca(t)= \mathcal{E}_0 e^{-(t/\tau)^2}\] where \(\mathcal{E}_0\) and \(\tau\) are constants. 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-9.2}