Uma proposta de operador de translação dependente da posição para a Mecânica Quântica Relativística

Abstract

In the context of non-relativistic Quantum Mechanics, there are spatial translation and time evolution operators that indicate the final state of the system through the application of these operators in an initial state. The present work aims to define the properties of a translation operator to be used in Relativistic Quantum Mechanics in Minkowski's spacetime. Thus, it was possible to obtain the 4-moment operator's form in the position representation and propose a solution for the translation operator through the differential equation that has been found. By applying the 4-moment operator contracted in itself, we obtained a Klein-Gordon equation for the translation operator. Inserting, thus, the previously proposed solution into the aforementioned equation, we calculated the energy-momentum relation of the particle under study. Afterwards, using Pauli's gamma matrices, we found a Dirac-type equation for the translation operator and, through minimal coupling, we were able to find the energy of an electron in the presence of an external electromagnetic field, in which we verified the spin-orbit coupling term and the correct gyromagnetic factor of the aforementioned particle. Subsequently, we constructed a position-dependent translation operator for Minkowski's spacetime. The new terms of the metric and the modified 4-moment operator were calculated. By proposing a variable change, it was possible to analyze the solution of the modified Klein-Gordon equation found in order to obtain the energy-momentum relation of the particle. This particle is now valid only in a domain defined from the characteristics imposed on the position-dependent metric elements. Through the non-relativistic limit, and returning to the concept of time as a parameter and no longer as a coordinate, we prove that the modified Klein-Gordon equation reduces itself to the modified Schrodinger equation obtained earlier by Costa Filho et al. Thereafter, we obtained a modified Dirac equation and its free solutions. In order to understand the interpretation of the modified wave function obtained, we calculated the continuity equation and analyzed the quantities related to the possible density and probability current. Finally, the infinity potential well problem was resolved by means of a modified Dirac equation. By imposing that the flux of probability in the well’s walls was null, we obtained a transcendental equation whose solution is related to the Compton wavelength and that supplies the energy-momentum relation. Taking this same equation in the non-relativistic limit, we obtain the same discrete energy levels calculated by Costa Filho et al using the modified Schrodinger equation.