Dynamic pilot wave bound states

Abstract

Motivated by recent parallels between classical bouncing droplet experiments and quantum bound states, we explore the lessons that droplet experiments might teach us about the dynamics of quantum solutions. Since the classical experiment is periodically driven, we examine periodic driving of the integer spin Klein-Gordon equation. We find that an exact solution can be obtained, and surprisingly this solution necessarily produces “half-integer” orbital angular momentum. We stress that these findings are strictly mathematical; nevertheless, this and other physical implications are intriguing and suggest further study.

Hydrodynamic quantum field theory: the free particle

“…Borghesi [33] proposed an elastic pilot-wave model wherein a point-particle is allowed to move within a nondissipative elastic substrate. Shinbrot [34] examined the influence of periodic driving of the Klein-Gordon equation, and found bound-state solutions with half-integer spin composed of spin-less particles. A number of investigators have taken inspiration from the walking droplets to inform and advance their theoretical modeling of quantum stochastic dynamics [35][36][37][38].…”

Hydrodynamic Quantum Field Theory: The Onset of Particle Motion and the Form of the Pilot Wave

“…As may be confirmed numerically by direct simulation of Equation (22), the static state is stable for 0 < κ < κ c and unstable for κ > κ c . Similar lateral oscillations were observed in the simulations of Dagan and Bush [19], who described free particle motion in terms of lateral oscillations at the Compton frequency, superimposed on a slowly varying net drift.…”

“…We note that the novelty of HQFT is the wave-particle coupling, as manifest in the forcing of the Klein-Gordon equation and the particle trajectory equation. It is this coupling that distinguishes our work from the numerous studies of the Klein-Gordon equation with a potential [21,22]. The strength of the wave-particle coupling is governed by the free parameter α.…”

“…where the reality condition forx p is X −n = X * n for all n, and * denotes complex conjugation. We substitute the Floquet ansatz (Equation 23) into (22) and evaluate the resultant integrals analytically, before grouping together powers of e it to form an infinite-dimensional system of equations for the unknown coefficients, X n . The critical threshold, κ c , is the value of κ at which the X n coefficients have a non-trivial solution, for which the determinant of the corresponding matrix vanishes.…”

“…After substituting Equation 23into (22), following the aforementioned procedure and evaluating integrals using the identity [28] ∞ 0 e ±iµs J 1 (s)…”