Sobre a validade da solução de Penrose para o problema da iluminação dentro do contexto da mecânica quântica

Abstract

In 1958, Sir Roger Penrose proposed a solution to a mathematical problem of the time known as “the illumination problem”. The problem, initially related to a two-dimensional geometric optics system, can be summarized by the following question: “given a point light source, placed in a room with an arbitrary geometry and walls completely mirrored, is it possible that there are still dark regions within the same?”. Penrose solved this problem by proposing a room with some elliptical walls and demonstrated that regardless of where the light source was placed, there would always be at least one dark region inside the room. Later, research groups tested the Penrose solution in another limit, by taking it into the scope of physical optics, where the wave nature of light is already taken into account. Their results have shown that, due to diffraction, a small light wave intensity is detected in places where light rays would never reach, showing that Penrose’s solution cannot be fully extended to that limit. In this work, we consider the geometric Penrose solution in the quantum limit and verify its validity within it. In order to do so, we apply computational techniques to obtain numerical solutions of the time-dependent Schrödinger equation, using the split-operator method. This is done by propagating an electron, simulated by a gaussian wave packet, through a quantum potential with a geometry that reproduces the shape of Penrose’s unilluminable room. Our results suggests that the solution proposed by Penrose should also not be able to be hold in this limit, at least for the specific parameters used in our simulations.