Formalismo do operador translação dependente da posição: problemas unidimensionais

Abstract

In the present work, we study the formalism of the position-dependent translation operator (PDTO), a formalism obtained by trying to understand the consequences of space metrics in quantum mechanics (MQ). In view of this, two types of metrics were used, one quadratic and the other linear, through which two translation operators of length dx were defined, called quadratic infinitesimal translation operators Tγβ (dx), and linear Tγ (dx), which acting on state |x> takes it to another state |x+dx+γxdx+β2x2dx> and |x+dx+γxdx>, respectively. It was possible to observe that the results of these translations depend on the metrics, as they lead us to obtain two new moment operators, Pγβ and Pγ, which consequently give us new switching relations between these moment and position operators, and two modified Schrödinger equations. Finally, the two modified Schrödinger equations were applied to several one-dimensional MQ problems. When analyzing the results obtained through the quadratic metric, we observe that when they are independent of the β parameter, they are the same obtained by the linear metric, and also when the γ parameter tends to zero in the linear metric, we retrieve the usual MQ results.