Single particle nonlocality, geometric phases and time-dependent boundary conditions

Abstract

We investigate the issue of single particle nonlocality in a quantum system subjected to timedependent boundary conditions. We discuss earlier claims according to which the quantum state of a particle remaining localized at the center of an infinite well with moving walls would be specifically modified by the change in boundary conditions due to the wall's motion. We first prove that the evolution of an initially localized Gaussian state is not affected nonlocally by a linearly moving wall: as long as the quantum state has negligible amplitude near the wall, the boundary motion has no effect. This result is further extended to related confined time-dependent oscillators in which the boundary's motion is known to give rise to geometric phases: for a Gaussian state remaining localized far from the boundaries, the effect of the geometric phases is washed out and the particle dynamics shows no traces of a nonlocal influence that would be induced by the moving boundaries.

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

“…Indeed, near the wall, the initial wavefunction is nonzero, since φ n (x ≈ L 0 , t = 0) n (L 0 − x), and is substantially modified when the wall moves. By the arguments given in [16] (or simply by the continuity of the logarithmic derivative noting that the potential remains unchanged except at x = L(t)) we then know that at an infinitesimal time t = ε we will have ψ n (x, ε) − ψ n (x, 0) = 0 at any x, although we expect this quantity to be large near x = L(ε) and smaller in the regions away from the moving wall.…”

“…Nevertheless it should be mentioned that formally, the proper way of obtaining the basis solutions (see [16] and Refs. therein) given by Eq.…”

“…Such systems are delicate to handle because from a formal point of view a different Hilbert space needs to be defined for each time t, so that a simple operation like taking the time derivative ∂ t ψ is not straightforward. We will introduce the system we will deal with -a particle in an expanding infinite well -in the context of recent works [14][15][16] involving time dependent boundary conditions in Sec. 2.…”

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

“…It should be noted however that, in the static cases (the standard particle in a box problem or a harmonic oscillator confined by static walls [32]), these peculiarities are not known to lead to any unphysical results. Second, dealing with time-dependent boundary conditions involves formally [14,23] a different Hilbert space at each time t. The time-dependent unitary mapping to a standard problem-a problem with fixed boundary conditions defined in a single Hilbert space-yields a Hamiltonian with a time-dependent mass, so that the (local) boundary conditions are mapped to a (delocalized) time-dependent parameter. We remark that mapping the time-dependence of parameters in the original system to their parameters in the transformed system is a generic mathematical property of time-dependent unitary transformations, and this mapping has in general no bearing on the physical properties of the original system.…”

“…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].…”