Sobre sistemas de 2 níveis em mecânica quântica e suas conexões com o espaço físico

Abstract

Numerous experiments can be described through quantum concepts and interpretations, that is, expected outcomes no longer have a deterministic nature, and theoretical predictions are restricted to an irreducible probabilistic nature. The simplest possible quantum systems are called two-level systems, which will be thoroughly explored in this paper. Any and all experiment is conducted in a physical space, characterized by real parameters typically associated with laboratory equipment, therefore, is not a far stretch to say that, in some way or another, it would be possible to investigate quantum experiments through these parameters, and that there would be a connection between them and the mathematical/quantum interpretation of such experiments. This paper aims to provide an operational interpretation for the relationship between the dimension of the state space, where experiments are represented, and the physical dimension of the laboratory space itself. We will use popular examples of two-level systems for illustrative purposes, such as the Stern-Gerlach experiment and experiments involving polarized light. If on one hand, we can connect geometrically a four-dimensional sphere to a three-dimensional base space through Hopf fibration, on the other hand, we can provide a physical interpretation for this connection, which, at first, was purely mathematical.