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

Abstract

We analyze a one dimensional quantum system consisting of a test particle interacting with two harmonic oscillators placed at the positions a1 and a2, with a1>0 and ∣a2∣>a1, in the two possible situations: a2>0 and a2<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 Pn1n2−(t) [Pn1n2+(t)] that both oscillators are in the excited states labeled by n1 and n2>0 at time t>∣a2∣v0−1 when a2<0 (a2>0). We prove that Pn1n2−(t) is negligible with respect to Pn1n2+(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 [“The wave mechanics of α-ray tracks,” Proc. R. Soc. London, Ser. A 126, 79 (1929)], where the result was argued using euristic arguments in the framework of the time independent perturbation theory for the stationary Schrödinger 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 Schrödinger equation for the three-particle system, with no reference to the wave packet collapse postulate.