Equivalence between free and harmonically trapped quantum particles

TL;DR

This paper demonstrates a mathematical mapping that relates solutions of the free-particle Schrödinger equation to those of the harmonic oscillator, enabling efficient analysis of quantum transitions between these states.

Contribution

It introduces a coordinate transformation method that simplifies the transition analysis between free particles and harmonic oscillators, improving computational efficiency.

Findings

Mapping allows transition analysis with simple coordinate change

Method is more efficient than traditional state-projection techniques

Can connect harmonic oscillators with different spring constants

Abstract

It is shown that general solutions of the free-particle Schroedinger equation can be mapped onto solutions of the Schroedinger equation for the harmonic oscillator. This is done in such a way that the time evolution of a free particle subjected to a sudden transition to a harmonic potential can be described by a simple coordinate transformation applied at the transition time. This procedure is computationally more efficient than either state-projection or propagator techniques. A concatenation of the map and its inverse allows us to map from one harmonic oscillator to another with a different spring constant.