Linear Potentials, Airy Wave Packets and Airy Transform

TL;DR

This paper explores solutions to the Schrödinger equation with linear potentials, focusing on Airy wave packets, their properties, and applications, highlighting algebraic methods and the Airy transform analogy.

Contribution

It introduces algebraic techniques to explicitly solve the Schrödinger equation with time-dependent linear potentials and discusses conditions for non-spreading Airy wave packets.

Findings

Explicit solutions in terms of Airy functions for linear potentials

Conditions for non-spreading Airy wave packets

Analogy between Airy and Gauss-Weierstrass transforms

Abstract

The solution of the Schrodinger equation with a linear potential is considered. We use algebraic methods to obtain the explicit form of the solution for the explicitly time dependent Hamiltonian and discuss the general conditions which allow us to get solutions in terms of the Airy functions, yelding non spreading wave packets. We analyze the relevant physical meaning of these solutions and give examples of their applications. We discuss the analogy between Airy and Gauss-Weirestrass transform.