Nonlocality and local causality in the Schrödinger equation with time-dependent boundary conditions

Abstract

We investigate the nonlocal dynamics of a single particle placed in an infinite well with moving walls. It is shown that in this situation, the Schrödinger equation (SE) violates local causality by causing instantaneous changes in the probability current everywhere inside the well. This violation is formalized by designing a gedanken faster-than-light communication device which uses an ensemble of long narrow cavities and weak measurements to resolve the weak value of the momentum far away from the movable wall. Our system is free from the usual features causing nonphysical violations of local causality when using the (nonrelativistic) SE, such as instantaneous changes in potentials or states involving arbitraily high energies or velocities. We explore in detail several possible artifacts that could account for the failure of the SE to respect local causality for systems involving time-dependent boundary conditions.

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

“…The phase µ n [Equation (12)] is a property of the entire wavefunction, although part of the phase increment is due to the walls' motion. To see this, we compare two oscillators that have exactly the same potential everywhere except in the vicinity of the outer wall.…”

“…The total phase increment after one period µ I n and µ II n is given by Equation (12). Case III, however, does not respect Equation ( 10) and therefore does not fit in the framework developed in Section 2.2.…”

“…Then, to discriminate the phase of the different cases mentioned above, it is crucial that µ I n and µ II n differ significantly. Indeed, µ ad n appears as a correction to µ II n [Equation ( 27)], so that ensuring that µ ad n ≈ µ II n = µ I n while still having L 1 (t) ≈ L 2 (t) typically implies high values of n and/or small values of m [see Equation (12)].…”

“…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].…”

“…It was indeed conjectured [9,10] that the moving wall could nonlocally change the phase of a wavepacket at the center of the box that remained localized far from the wall. While this conjecture proved to be incorrect [14], it was recently noted [12] that, when a quantum state had a non-zero probability amplitude near the wall, a linearly expanding wall induced instantaneously a current density at any point of the box. We note for completeness that systems with moving boundaries are of current interest in practical schemes in the field of quantum engines or in atomic spectroscopy [15][16][17].…”