Dynamics of a particle confined in a two-dimensional dilating and deforming domain

Abstract

Some recent results concerning a particle confined in a one-dimensional box with moving walls are briefly reviewed. By exploiting the same techniques used for the 1D problem, we investigate the behavior of a quantum particle confined in a two-dimensional box (a 2D billiard) whose walls are moving, by recasting the relevant mathematical problem with moving boundaries in the form of a problem with fixed boundaries and time-dependent Hamiltonian. Changes of the shape of the box are shown to be important, as it clearly emerges from the comparison between the "pantographic" case (same shape of the box through all the process) and the case with deformation.

Resonant Transitions Due to Changing Boundaries

“…[50] to treat changes of the domain with deformation. It is appropriate to mention here that in [50] we assume knowledge of the dynamics in the pantographic case -the relevant Hamiltonian describing pantographic changes is considered as the unperturbed one -and then treat the terms coming from deformation as a perturbation. Nevertheless, knowledge of the dynamics in the pantographic case is quite limited, and exact dynamics is known only for the case of uniformly moving domain.…”

“…[46] and Ref. [50], shows that when the problem of a system with moving boundaries is mapped into a problem with static boundaries the Hamiltonian inducing the evolution in the new picture becomes time-dependent. Even the very kinetic energy contains a time-dependent factor, so that the particle appears as a particle with changing mass.…”

“…In Ref. [50] we have shown that the problem of a particle confined in a two-dimensional (or three-dimensional) box with moving boundaries can be traced back to the problem of a particle with a varying mass, confined within a static box and subjected to a time-dependent potential. Following such an approach, we recast the original problem into the problem of a particle in a static box governed by a time-dependent effective Hamiltonian.…”

Breakdown of separability due to confinement

“…Over the decade, many papers appeared dedicated to the problem of quantum systems with boundaries. Several works are devoted to the problem of particles confined in a box, sometimes with moving walls [1][2][3][4][5][6][7] or specific shapes [8,9]. The confinement means the restriction on the motion of randomly moving particles, e.g.…”

Fermionic edge states and new physics

“…Replacing the boundary with δ and δ potential enables physical meaning of these considerations. The effect of moving boundaries has been investigated briefly in the literature, see [23,24,25,26,27], and these are of relevance in diverse physical situations like an evolving black hole horizon or the time dependent cosmological horizon in an expanding universe, the Casimir effect in the presence of moving boundaries to name a few.…”