Classical-quantum Interface of a Particle in a Time-dependent Linear Potential

Abstract

We use a wave packets approach to analyze the non-trivial time-dependence solution of quantum mechanical systems of a particle in a linear potential. We relate the system of a free particle with that of a particle in a time-dependent linear potential by the use of the de Broglie hypothesis which is one important aspect of the study of the classical-quantum interface. Closed-form analytic results have been obtained as: (i) nonspreading Airy wave packets, (ii) Gaussian wave packets. Keywords Time-dependent Schrödinger equation · de Broglie hypothesis · Wave packetsThere has been considerable interest in the system of a particle in a linear potential (with time -dependent parameters) where the exact propagator has long been known [1]. This model has eigenfunctions (wave packets) described by the Airy function [2]. This system and the Airy wave functions on a half-line have been used to model the production of high harmonic generation in the laser irradiation of rare gases [3,4], and the edge electron gas [5][6][7]. The model on piecewise domains and the wave functions have been frequently used to model various physical systems [8,9]. Recent work [10][11][12][13][14][15][16][17][18][19][20][21][22] focus further on exact solutions and their properties and several method have been used to solve this system.Using a particular unitary transformation operator, which resembles the one responsible for the existence of coherent states in harmonic oscillators, Song [22] relates the system of a particle in a linear potential to that of a free particle and found a general wave packet described by an Airy function. The Schrödinger equation for a free particle has also long been interesting in that the equation is formally identical to the wave equation of a beam of light in the paraxial approximation [23][24][25][26]. In the present work, the relation between the two systems naturally arises. The unitary operator is not assumed beforehand, it will naturally appear in the derivation.