Non-Dispersive Properties of the Airy Wave Function in a Free Potential*

The Airy function, which is the solution to the Stokes equation, has been found to have a myriad of physical and mathematical applications, ranging from optics and rainbows to fluid dynamics and probability. It is relevant when describing a number of quantum systems under a constant force, which correspond to linear potentials. The Airy function eigenstates of linear potentials have the intriguing property that they are non-dispersive when left to propagate in free space. A method by which one can prove this is the Feynman path integral, which propagates wave functions through time by integrating over all possible paths a particle can take when moving between points. We present a proof of this phenomenon utilizing this Feynman path integral formulation of the Airy wave function eigenstates of a linear potential to demonstrate their non-spreading behavior when let free, as discussed by M. V. Berry and N. L. Balazs.