Introdução à mecânica quântica relativística: a equação de Klein-Gordon

Abstract

The end of the 19th century and the beginning of the 20th century was marked by the emergence of two theories that revolutionized Physics, the theory of special relativity and quantum mechanics. The theory of special relativity provides a generalization of Newton's ideas about space and time, imposing a speed limit for interactions in nature given by the speed of light in free space, making Newton's laws of motion compatible with the laws of James Clerk Maxwell for electromagnetism. Quantum mechanics, on the other hand, was formulated from the necessity to build a theory applicable to atomic phenomena, impacting the way of thinking about science in a very significant way, explaining the stability of atoms and stars and providing support for the description of phenomena and technological applications in modern electronics and telecommunications. Although there are some incompatibilities between these two theories, their unification provide a very consistent theory named as relativistic quantum mechanics to approach the behavior of elementary particles in high-energy physics. Since this field is quite broad, in this work we restrict ourselves to relativistic wave equations, considering their most introductory development, given by the Klein-Gordon equation. For this purpose, we introduce the fundamental concepts of special relativity, such as its postulates, the implications related to the structure of space-time for high-speed regime, the four-vectors notation, and we also discuss the concepts of covariance, contravariance and invariance. Regarding quantum mechanics, we restrict ourselves to the description of the free-particle Schrödinger wave equation. The relativistic Klein-Gordon wave equation was introduced from the necessity of modification of the Schrödinger equation to make it covariant under Lorentz transformations. Considering the free-particle Klein-Gordon equation we discuss what is meant by a relativistic wavenfuncion, an equivalent continuity equation, the postulate of existence of antiparticles and how the Schrödinger wave equation can be recovered considering the non-relativistic limit, showing that the Klein-Gordon equation can be considered as a direct relativistic generalization of the quantum mechanics Schrödinger equation for spin-zero particles.