Mecânica quântica não relativística na fita de Möbius

Abstract

The description of quantum dynamics of restricted particles on surfaces has attracted considerable attention from condensed matter physicists in recent years. We owe this mainly to the fact that research on two-dimensional nanostructures, such as graphene, fullerene, and carbon nanotubes, has taken great prominence in recent decades. However, this subject is not new. There has been an attempt to understand how curvature affects quantum mechanics in two-dimensional surfaces immersed in Euclidean spaces. The research that contemplated this problem followed two distinct perspectives. In one of them, quantum dynamics of one particle is intrinsically approached, considering only the surface geometry. In the other one, the behavior of the particle is examined extrinsically, considering both the method by which the particle is restricted on the surface and the geometry of the immersion space. Considering these two approaches, we will investigate the quantum mechanical properties of a Möbius tape-like surface. With this aim, we obtain the Schrödinger equation for a spin-restricted particle in the Möbius strip in the absence of external fields. We do this intrinsically, by modifying the Laplacian operator to a two-dimensional curvilinear system, defined on the Möbius strip. Working extrinsically, we use the confining potential formalism, where, in addition to modifying the aplacian of the Schrödinger equation, we add the action of a curvature-dependent potential called da Costa potential. Due to the geometric properties of the Möbius strip, we cannot perform a separation of variables in the wave function, so we fix one of the coordinates of the curvilinear system, and consequently restrict the particle movement in two possible directions. Either the particle will move in a ring around the Möbius strip, or in a curve across the width of the strip. We obtain effective hermitian Hamiltonians for each direction considered. We intrinsically analyze the movement of particles around the Möbius strip, and extrinsically, for particles that only move in the center of the strip. We obtain the normalized wave function and the corresponding energy spectrum for each case. Finally, we summarize our conclusions and point out our future perspectives for this work.