A equação de Schroedinger em um cenário de comprimento mínimo.

Abstract

In order to quantize gravity several approaches have been proposed. It is interesting that all of them predict the existence of a minimal length in the nature. In this work, we carry out the quantization of the Schroedinger’s equation, that is, the transformation of the wave function into a field operator (second quantization), in a minimal-length scenario. In order to obtain the Schroedinger’s equation in minimal-length scenario we modify the de Broglie’s postulate, that is, the relation between the linear momentum and the wave vector is no longer linear. The Schroedinger’s equation obtained in this way is a differential equation of fourth order. For this reason, we study the classical field theory with derivatives of high-order, in particular the Noether’s Theorem and Ostrogradsky’s Method with aim of obtaining the conserved quantities and the Hamiltonian of the system. Although the Schroedinger’s equation permits the quantization using commutation or anti-commutation relations, we only employ the commutation relation between create and annihilation operators