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European Journal of Physics

Converting states of a particle under uniform or elastic forces into free particle states

C. Bernardini, F. Gori, Massimo Santarsiero

Published Mar 1, 1995• 23 citations
Publisher

Abstract

We show that there is an explicit one-to-one correspondence between the wavefunctions of a free particle and those of a particle subjected to a uniform force field. A similar correspondence can be established between free particle and harmonic oscillator or, more generally, displaced harmonic oscillator. Simple applications to the evaluation of propagators are presented and elementary examples are discussed.

DOI
10.1088/0143-0807/16/2/002

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RECEN - Revista Ciências Exatas e Naturais•Dec 14, 2009•1 citation
We show that there is an explicit one-to-one correspondence between the wavefunctions of a free particle and those of a particle subjected to a uniform force field. A similar correspondence can be established between free particle and harmonic oscillator or, more generally, displaced harmonic oscillator. Simple applications to the evaluation of propagators are presented and elementary examples are discussed.
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.
We present a simple approach for finding the general solution of the time-dependent Schrödinger equation for a particle moving under the action of a uniform force. From such a general solution we derive the eigenfunctions of the energy as well as those of the momentum. Next we obtain the propagator and show that the time evolution of an arbitrarily chosen initial wavefunction is described by a Fresnel transformation. This establishes a useful link with optical phenomena and allows us to link the case of a particle moving under a uniform force to that of the free particle.