Estudo de sistemas quânticos unidimensionais usando o método da matriz de transferência: uma abordagem usando Python

Abstract

The scientific and technological development of a society is closely related to the under standing of fundamental concepts of essential areas of knowledge. The peculiar properties of electron transmission in electric potential barriers in the context of Quantum Mechanics, for example, served as the basis for developing several devices based on heterostructures composed of semiconductors. Furthermore, multiple quantum wells and superlattices have also become important and attractive structures for applications in optical and electronic devices. Different theoretical approaches have emerged to model such structures. The transfer matrix technique, for example, is a useful and affordable method for studying a variety of one-dimensional quantum systems that are based on multiple barriers and well systems. In addition to the scope of Condensed Matter Physics, physical systems in several other areas, such as Statistical Mechanics and Optics, also use this powerful and compact formalism. The methodology used in this work will be based on the analytical development of the transfer matrix of a rectangular multi-barrier system within the context of non-relativistic quantum mechanics, using Schrödinger’s equation. The solutions for an arbitrary number of symmetric barriers will be generalized and implemented in the Python programming language, with computational codes made available in a didactic and detailed way. The analyzed results start from the simple barrier structure, which is widely studied in Introduction to Modern Physics courses, to a system of infinite barriers (superlattices) where the transmission properties and energy spectrum are analyzed. We hope this material can be useful as a starting point for students to expand their knowledge about the properties of these quantum systems and open doors to new studies