Non-locality and time-dependent boundary conditions: A Klein-Gordon perspective

Abstract

The dynamics of a particle in an expanding cavity is investigated in the Klein-Gordon framework in a regime in which the single-particle picture remains valid. The cavity expansion represents a time-dependent boundary condition for the relativistic wave function. We show that this expansion induces a non-local effect on the current density throughout the cavity. Our results indicate that a relativistic treatment still contains apparently spurious effects traditionally associated with the unbounded velocities inherent in non-relativistic solutions obtained from the Schrödinger equation. Possible reasons for this behaviour are discussed.

Relativistic spin-0 particle in a box: bound states, wavepackets, and the disappearance of the Klein paradox

“…The disappearance of Klein tunneling in the infinite well limit should also be of interest to recent works that have studied the Klein-Gordon equation in a box with moving walls [14][15][16] (the special boundary conditions chosen in these works were indeed not justified).…”

Klein paradox for bosons, wave packets and negative tunnelling times

“…As is well known, spin-zero particles are described in relativistic quantum theory by the Klein-Gordon equation (KGE) (see the textbooks 8,12 and Refs. 13,14 for very recent work). Constructing a broad wave packet from the scattering solution of the Klein-Gordon equation (KGE), and choosing the mean energy to be a half of the barrier's height, E = V /2 , we find the particle not only transmitted without reflection, but also advanced relative to free propagation by twice the barrier's width d (see Fig.…”

“…Similarly, it can be shown that the last WP moving to the left emerged at x = 0 at about the same time the incident WP arrived there. Now we can reconstruct the scenario described by the divergent MRE (13). Initially, there is not only a WP approaching the barrier from the left, but also another wave packet, already trapped in the barrier region.…”

“…Now the potential varies over a region δx ∼ 1/b, and the Klein tunnelling persists for as long as δx is small compared to the particle's Compton wavelength, δx < /mc. Using the results of [20] it can be shown (see Methods) that for a smooth potential (19), with δx << d, one can continue using the MRE (13), with the pole in T n (p, −q) moved into the complex p-plane,…”

Nonlocal Interferences Induced by the Phase of the Wavefunction for a Particle in a Cavity with Moving Boundaries

“…The first reason for choosing this system is that analytic solutions of the time-dependent Schrödinger equation are known [8]. The second, less mundane, reason is that such systems, and in particular their simplest variant (a box with a linearly moving wall), have long been suspected of manifesting some form of nonlocality [9][10][11][12][13].…”