Joint excitation probability for two harmonic oscillators in dimension one and the Mott problem

We analyze a one dimensional quantum system consisting of a test particle interacting with two harmonic oscillators placed at the positions $a_1$, $a_2$, with $a_1>0$, $\mathrm{\mid }{a}_2\mathrm{\mid }>a_1$, in the two possible situations: $a_2>0$ and $a_2 <0$. At time zero the harmonic oscillators are in their ground state and the test particle is in a superposition state of two wave packets centered in the origin with opposite mean momentum. %$\pm M{v}_0$. Under suitable assumptions on the physical parameters of the model, we consider the time evolution of the wave function and we compute the probability ${\mathcal{P}}_{n_1{n}_2}^{-}(t)$ (resp. ${\mathcal{P}}_{n_1{n}_2}^{+}(t)$) that both oscillators are in the excited states labelled by $n_1$, $n_2>0$ at time $t>\mathrm{\mid }{a}_2\mathrm{\mid }{v}_0^{-1}$ when $a_2<0$ (resp. $a_2>0$). We prove that ${\mathcal{P}}_{n_1{n}_2}^{-}(t)$ is negligible with respect to ${\mathcal{P}}_{n_1{n}_2}^{+}(t)$, up to second order in time dependent perturbation theory. The system we consider is a simplified, one dimensional version of the original model of a cloud chamber introduced by Mott in \cite{m}, where the result was argued using euristic arguments in the framework of the time independent perturbation theory for the stationary Schr"{o}dinger equation. The method of the proof is entirely elementary and it is essentially based on a stationary phase argument. We also remark that all the computations refer to the Schr"{o}dinger equation for the three-particle system, with no reference to the wave packet collapse postulate.