Asymptotic Expansion for the Wave Function in a one-dimensional Model of Inelastic Interaction

We consider a two-body quantum system in dimension one composed by a test particle interacting with an harmonic oscillator placed at the position $a>0$. At time zero the test particle is concentrated around the position $R_0$ with average velocity $\pm v_0$ while the oscillator is in its ground state. In a suitable scaling limit, corresponding for the test particle to a semi-classical regime with small energy exchange with the oscillator, we give a complete asymptotic expansion of the wave function of the system in both cases $R_0 <a$ and $R_0 >a$.Comment: 23 page

A time-dependent perturbative analysis for a quantum particle in a cloud chamber

We consider a simple model of a cloud chamber consisting of a test particle (the alpha-particle) interacting with two other particles (the atoms of the vapour) subject to attractive potentials centered in $a_1, a_2 \in \mathbb{R}^3$. At time zero the alpha-particle is described by an outgoing spherical wave centered in the origin and the atoms are in their ground state. We show that, under suitable assumptions on the physical parameters of the system and up to second order in perturbation theory, the probability that both atoms are ionized is negligible unless $a_2$ lies on the line joining the origin with $a_1$. The work is a fully time-dependent version of the original analysis proposed by Mott in 1929.Comment: 23 page

Semiclassical wave-packets emerging from interaction with an environment

We study the quantum evolution in dimension three of a system composed by a test particle interacting with an environment made of N harmonic oscillators. At time zero the test particle is described by a spherical wave, i.e., a highly correlated continuous superposition of states with well localized position and momentum, and the oscillators are in the ground state. Furthermore, we assume that the positions of the oscillators are not collinear with the center of the spherical wave. Under suitable assumptions on the physical parameters characterizing the model, we give an asymptotic expression of the solution of the Schrodinger equation of the system with an explicit control of the error. The result shows that the approximate expression of the wave function is the sum of two terms, orthogonal in L-2(R3(N+1)) and describing rather different situations. In the first one, all the oscillators remain in their ground state and the test particle is described by the free evolution of a slightly deformed spherical wave. The second one consists of a sum of N terms where in each term there is only one excited oscillator and the test particle is correspondingly described by the free evolution of a wave packet, well concentrated in position and momentum. Moreover, the wave packet emerges from the excited oscillator with an average momentum parallel to the line joining the oscillator with the center of the initial spherical wave. Such wave packet represents a semiclassical state for the test particle, propagating along the corresponding classical trajectory. The main result of our analysis is to show how such a semiclassical state can be produced, starting from the original spherical wave, as a result of the interaction with the environment. (C) 2014 AIP Publishing LLC.We study the quantum evolution in dimension three of a system composed by a test particle interacting with an environment made of N harmonic oscillators. At time zero the test particle is described by a spherical wave, i.e., a highly correlated continuous superposition of stateswith well localized position and momentum, and the oscillators are in the ground state. Furthermore, we assume that the positions of the oscillators are not collinear with the center of the spherical wave. Under suitable assumptions on the physical parameters characterizing the model, we give an asymptotic expression of the solution of the Schr¨odinger equation of the system with an explicit control of the error. The result shows that the approximate expression of the wave function is the sum of two terms, orthogonal in L2(R3(N+1)) and describing rather different situations. In the first one, all the oscillators remain in their ground state and the test particle is described by the free evolution of a slightly