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

TL;DR

This paper presents a time-dependent perturbative analysis of a quantum particle in a cloud chamber, showing ionization probabilities are significant only along specific atomic alignments, extending Mott's 1929 analysis.

Contribution

It introduces a fully time-dependent perturbative approach to analyze ionization in a cloud chamber, revealing geometric conditions for atom ionization.

Findings

Ionization probability is negligible unless atoms are aligned.

Second-order perturbation theory suffices for analysis.

The model extends Mott's original analysis to a time-dependent framework.

Abstract

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.