Nonspreading wave packets

TLDR

The Airy packet is classically associated with a rigidly translating parabolic family of orbits, making it uniquely capable of propagating without change of form. The authors solve Schrödinger’s equation for arbitrary time‑dependent forces F(t) and analyze the resulting motion for specific force profiles. We show that an Airy wave packet propagates in free space without distortion and with constant acceleration, its acceleration linked to the curvature of its caustic, and that it preserves its integrity even under a spatially uniform force F(t).

Abstract

We show that for a wave ψ in the form of an Airy function the probability density ‖ψ‖2 propagates in free space without distortion and with constant acceleration. This ’’Airy packet’’ corresponds classically to a family of orbits represented by a parabola in phase space; under the classical motion this parabola translates rigidly, and the fact that no other curve has this property shows that the Airy packet is unique in propagating without change of form. The acceleration of the packet (which does not violate Ehrenfest’s theorem) is related to the curvature of the caustic (envelope) of the family of world lines in spacetime. When a spatially uniform force F (t) acts the Airy packet continues to preserve its integrity. We exhibit the solution of Schrödinger’s equation for general F (t) and discuss the motion for some special forms of F (t).