Operador de translação dependente da posição nos formalismos de Schrödinger e Heisenberg da mecânica quântica

Abstract

This work discusses the position-dependent translation operator in the Schrödinger and Heisenberg formulation of quantum mechanics. In both representations, we discuss the importance of sufficient conditions for the moment operator to be hermitian in the metric space to which it is defined. After that, we see that in the Schrödinger's wave formulation we find a Schrödinger equation that governs the temporal evolution of wave functions. Likewise, in Heisenberg's matrix formulation we find a Heisenberg equation that governs the temporal evolution of the operators. Finally, as an application of the Heisenberg formalism we calculate the temporal evolution of the uncertainty in time and the broadening of wave packets of a particle in a one-dimensional space under null potential for a Euclidean metric and also for a metric with a first order term in its power series . On the other hand, we solve the one-dimensional space free particle for metrics with first and second term of its expansion in series of power in Schrödinger's formulation.