400 research outputs found
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 , ,
with , , in the two possible situations: and . 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
(resp. ) that both oscillators are in the
excited states labelled by , at time when
). We prove that is
negligible with respect to , 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.Comment: 26 page
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