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European Journal of Physics

The general wavefunction for a particle under uniform force

F. Gori, Massimo Santarsiero, Riccardo Borghi, Giorgio Guattari

Published Nov 1, 1999• 13 citations
Publisher

Abstract

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.

DOI
10.1088/0143-0807/20/6/313

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American Journal of Physics•Feb 1, 1994•1 citation
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.
The classical solution is used to provide appropriate variables for the quantum-mechanical free particle time-dependent Schrödinger wave equation, which then becomes separable giving nonenergy eigenfunctions of closed form. Solutions derived in this fashion may fail to satisfy the homogeneous Schrödinger wave equation at one point in space/time; if this should occur, an inhomogeneous term is produced and a corresponding restricted solution reduces to the free particle propagator. In general, it is shown that the nonenergy eigenfunctions may also be interpreted as eigenfunctions of a time-independent transformed momentum operator. The similarity transformation, by means of which is constructed the transformed momentum operator, the transformed position operator, and the transformed Hamiltonian, are explicitly determined. The relationship of the nonenergy eigenfunctions to the energy eigenfunctions is also exhibited.