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. 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.