Airy eigenstates and their relation to coordinate eigenstates

TL;DR

This paper explores Airy eigenstates in quantum mechanics, deriving their properties, relations to coordinate eigenstates, and expressing the unit operator in terms of these states for a linear potential.

Contribution

It introduces Airy states in the context of linear potentials and establishes their connection to position and momentum eigenstates, providing new analytical tools.

Findings

Derived Airy states from the Schrödinger equation for linear potential

Expressed the unit operator using Airy states

Established relations between Airy states and eigenstates of position and momentum

Abstract

We study the eigenvalue problem for a linear potential Hamiltonian and, by writing Airy equation in terms of momentum and position operators define Airy states. We give a solution of the Schr\"odinger equation for the symmetrical linear potential in terms of the squeeze and displacement operators. Finally, we write the unit operator in terms of Airy states and find a relation between them and position and momentum eigenstates.