The effects of the covariant generalized uncertainty principle on quantum mechanics

One can use the generalized uncertainty principle (GUP) to incorporate the minimum measurable length in quantum gravity. It may be interesting to have a minimal time interval as well as the minimal length in the relativistic version of quantum mechanics in the presence of the gravitational eﬀects. In this paper, we consider a covariant version of the generalized uncertainty principle to investigate the eﬀects of both the minimal time and the minimal length. Using the covariant GUP, the energy-momentum dispersion relation is modiﬁed. Starting with the modiﬁed dispersion relation, the corrections to the wave function are obtained and the problems of the particle in a box and the Hydrogen atom are revisited. Considering the minimal time may be an interesting new topic which is not widely studied before. It may lead to new results which can open a new window into quantum gravity.