Formulação da mecânica quântica por integrais de caminho de Feynman

Abstract

The formulation of quantum mechanics by Feynman path integrals arises as a third representation of the quantum theory, similarly the other two representations, Schrödinger and Heisenberg, so that they are mathematically equivalent. However, the representation of the Feynman Hamiltonian formalism is replaced by the Lagrangian formalism as a mathematical tool used in the construction of quantum theory. In functional integration of quantum mechanics, operators no longer the main objects of study, such as they are in the other two representations, giving place to the propagators, which has an intrinsic relationship with time evolution operator, being directly responsible for the evolution of the quantum state. This study begins with a mathematical analysis of the Lagrangian formalism, fundamental to the development of quantum theory of Feynman path integrals. Following the introduction of the quantum propagator from the de nition of the time evolution operator for a time independent Hamiltonian (isolated system). From these concepts , it introduces the idea of path integral, where now has that all paths are possible, unlike the classical action in which the only possible way would be one in which the action is a minimum. The study continues with the path integral and its analogy with the Riemann integral. Then, it investigates the rules for two successive events, and also, the rules for several successive events. Finally, it uses the knowledge obtained from the Feynman path integral to solve the Simple Harmonic Oscillator and Forced Harmonic Oscillator by calculating its eigenvalues ??and eigenfunctions through quantum propagator, obtaining similar results to those found by Schrödinger and Heisenberg, proving the equivalence of this representation with the other two representations of quantum mechanics.