Lets assume the classic quantum harmonic oscillator (HO) with a Hamiltonian $$H(t) = - \frac{\hbar^2}{2m} \frac{d^2}{d x^2} + \frac{m \omega^2 x^2}{2} + F(t)$$ where $$F(t)$$ is a time-dependent force defined via $$F(t)=\frac{F_0 \tau / \omega}{\tau^2 + t^2}$$ At time $$t \to -\infty$$ the particle with mass $$m$$ is in the ground state $$| 0 \rangle$$ of the HO potential. Since this is a time-dependent problem, I try to use time-dependent perturbation theory to first order to obtain the probability that at $$t \to \infty$$ the particle is in the first excited state $$|1\rangle$$. From the lecture we have obtained the (slightly more general) formula $$|c_f (t)|^2 = \frac{1}{\hbar^2} \left| \int_{t_0}^t V_{fi} (t') e^{i \omega_{fi} t'} dt' \right|^2$$ which we already have applied to the kicked oscillator as shown here (I used the same notation to make life easier for the readers here). But, how can I (can I?) apply this approach to this problem? As I see it, I need to somehow use $$F(t)$$ to get an expression for $$V_{fi}(t')$$ that gives me a calculable expression.