Harmonic states for the free particle

TL;DR

This paper introduces new families of quantum states for the free particle, derived from harmonic oscillator solutions via the Quantum Arnold Transformation, including coherent and squeezed states with potential experimental applications.

Contribution

It presents novel normalizable, multi-localized solutions of the free Schrödinger equation, expanding the set of available quantum states and their applications.

Findings

New multi-localized free particle states constructed

Definition of free particle coherent and squeezed states

Extension to higher-dimensional states like Hermite-Gauss

Abstract

Different families of states, which are solutions of the time-dependent free Schr\"odinger equation, are imported from the harmonic oscillator using the Quantum Arnold Transformation introduced in a previous paper. Among them, infinite series of states are given that are normalizable, expand the whole space of solutions, are spatially multi-localized and are eigenstates of a suitably defined number operator. Associated with these states new sets of coherent and squeezed states for the free particle are defined representing traveling, squeezed, multi-localized wave packets. These states are also constructed in higher dimensions, leading to the quantum mechanical version of the Hermite-Gauss and Laguerre-Gauss states of paraxial wave optics. Some applications of these new families of states and procedures to experimentally realize and manipulate them are outlined.