The Feynman path integral formulation of non-dispersive Airy wave packets and their applications to the heavy meson mass spectra and ultra-cold neutrons

TL;DR

This work links the non-dispersive evolution of Airy wave packets in linear potentials to practical quantum systems by deriving the linear kernel via the Feynman path integral and solving Airy eigenstates for several linear-potentials. The authors map Airy function zeros to the confinement contribution in heavy-meson 1S-2S mass gaps within a 1+1D absolute linear potential, introducing a vacuum-energy offset $a\approx 141$ MeV that yields agreement with light-front calculations in the heavy-quark regime. They also apply the Airy solutions to ultra-cold neutrons bouncing in Earth's gravity, predicting quantized heights $h_n$ that align with experimental data and WKB results. Overall, the paper provides a unified 1+1D framework connecting Airy function mathematics to hadron spectroscopy and gravitational quantum states, with clear avenues for incorporating relativistic corrections and broader QCD-inspired models.

Abstract

We demonstrate the non-spreading behavior of Airy wave packets utilizing the Feynman path integral formulation of a linear potential, the Airy functions' zeros correspondence to heavy-meson mass spectroscopy, and their implications to the eigenstates of ultra-cold neutrons in Earth's gravitational field. We derive the linear kernel, and utilize the Feynman path integral time evolution to show that Airy function wave packets are non-dispersive in free space. We then model the confining contribution to 1S - 2S heavy meson mass gaps as a 1+1D absolute linear potential and look at the correspondence of the Airy function zeros. In doing so, we predicted the confining contribution to the mass gap of heavy mesons with a good accuracy when compared to calculations performed in the light front. Furthermore, we used these Airy function solutions to model the quantum states of a neutron under Earth's gravity. We show that the measured heights of a neutron can be modeled by the zeros of the Airy function, and compare to experimental data and predictions utilizing the WKB approximation.

Figures (6)

• Figure 1: Graph of Airy $\mathsf{Ai}(x)$ and $\mathsf{Bi}(x)$ functions.
• Figure 2: The probability density $\rho(q=x/x_0,t)$ of an Airy wave packet at 3 different times. The probability density shows no dispersion, but instead accelerates along the $q$ axis. Here, $t$ is in units of $2m x_0^2/\hbar$.
• Figure 3: Graph of $\mathsf{Ai}(x)$, with marked zeroes for $\mathsf{Ai}(x)$ and $\mathsf{Ai}'(x)$.
• Figure 4: The first four eigenstates of the absolute linear potential, scaled to prevent overlap.
• Figure 5: The first four eigenstates of the one sided linear potential, scaled to prevent overlap.