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Revista Mexicana de Física

Relating the free particle with the harmonic oscillator

G. F. Torres del Castillo

Published May 1, 2025• 1 citation
Publisher

Abstract

We show that by means of a coordinate transformation in the extended configuration space the problem of a free particle can be related to that of a harmonic oscillator in classical mechanics and in quantum mechanics.
DOI
10.31349/revmexfis.71.030701

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We construct the coherent states of a free particle by implementing a coordinate transformation in the extended configuration space, which establishes a correspondence between the solutions of the Schrödinger equation for a harmonic oscillator and those for a free particle. Using this framework, we derive analytical expressions for the coherent states of a free particle, avoiding the complexities of non-normalizable fiducial states, integrals of motion, or group-theoretic approaches. Our method provides a systematic way to characterize these states at any instant while ensuring that they satisfy the Robertson-Schrödinger uncertainty relation.