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 $a_1 >0$, $|a_2|>a_1$, 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 $t > |a_2| v_0^{-1}$ 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.