Unitary transformation for the system of a particle in a linear potential

– A unitary operator which relates the system of a particle in a linear potential with time-dependent parameters to that of a free particle, has been given. This operator, closely related to the one which is responsible for the existence of coherent states for a harmonic oscillator, is used to find a general wave packet described by an Airy function. The kernel (propagator) and a complete set of Hermite-Gaussian type wave functions are also given. Introduction. – The existence of coherent and squeezed states for a simple harmonic oscillator [1,2] can be understood from the fact that there exist unitary transformations which leave the time-dependent Schödinger equation (formally) invariant under the transformations. These transformations have been found as the relations between harmonic oscillators of time-dependent parameters [3, 4], while the transformations between the same simple harmonic oscillators can be applied to a stationary states to give the coherent and squeezed states [4]. On the other hand, Feynman and Hibbs show that the kernel of a general quadratic system is described by the classical solutions of the system [5]. Since the wave function might be derived from the kernel, this suggests that the unitary transformations are described by classical solutions, as explicitly shown in the quadratic systems [6]. There has been considerable interest for the system of a particle in a linear potential (with time-dependent parameters) [7, 8, 9]. This model has eigenfunctions (wave packets) described by the Airy function [10]. The model 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 [11], and the edge electron gas [12, 13]. The model on piecewise domains and the wave functions have been frequently used to model various physical systems [14]. The Schödinger 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 [15]. In this article, we will show that there exists a unitary relation between the system of a particle in a linear potential and that of a free particle. Indeed, time-dependent unitary relations have been known to be useful in analyzing quantum systems. Unitary transformations have been extensively used in showing that the Dirac theory goes to the Pauli theory in the non-relativistic limit [16], and a unitary relation between the system of …