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

TL;DR

This paper derives a detailed asymptotic expansion of the wave function for a one-dimensional quantum system involving a test particle and an oscillator, focusing on semi-classical regimes and energy exchange.

Contribution

It provides a novel asymptotic expansion of the wave function in a semi-classical limit for a two-body quantum system with inelastic interactions.

Findings

Asymptotic expansion valid for initial positions R_0 < a and R_0 > a

Explicit formulas for wave function behavior in different regimes

Insights into energy exchange in semi-classical limit

Abstract

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 $R_0 >a$.