Perturbation hydrogen-atom spectrum in deformed space with minimal length

Abstract

We study energy spectrum for hydrogen atom with deformed Heisenberg algebra leading to minimal length. We develop correct perturbation theory free of divergences. It gives a possibility to calculate analytically in the 3D case the corrections to s-levels of hydrogen atom caused by the minimal length. Comparing our result with experimental data from precision hydrogen spectroscopy an upper bound for the minimal length is obtained.

Hydrogen atom in momentum space with a minimal length

“…[16][17][18][19][20], where an upper bound for the minimal length was found to be about 0.01 − 0.1 fm. The estimation of this bound was mainly obtained by two methods.…”

“…(6) Finally, Stetsko and Tkachuk [19] and [20] proposed another perturbative method by using the following position representation of the operators X i and P i :…”

“…(15), the condition η = n, which is not sufficient to assure that the wave function be single valued. In the perturbative treatment of the hydrogen atom [17][18][19][20], β is the perturbation parameter, and, naturally the first-order correction is proportional to this deformation parameter.…”

“…I, several papers have been devoted to the study of the hydrogen atom problem in quantum mechanics with a generalized uncertainty relation, based on the following deformed Heisenberg algebra [16][17][18][19][20]:…”

“…However, this is not often possible, especially when the potential depends, in a not too straightforward manner, on the position operators as in the case of the hydrogen atom potential. In the literature, this problem is the most studied in this modified version of quantum mechanics [16][17][18][19][20]. This is natural because this system has a particular interest.…”