Perturbation hydrogen-atom spectrum in deformed space with minimal length

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

Figure 1

Comparison of different results for the energy of $1s$ states with $\eta =1∕3$ (upper graph) and $\eta =1$ (lower graph). Solid lines represent our results and dotted lines correspond to the results of Benczik (see Ref. 12).

Figure 2

Energy of the $2s$ level. The solid line shows the result when $\eta =1$ and the dash-dotted line corresponds to the parameter $\eta =1∕3$.

Figure 3

Constraints on the minimal length as a function of dimensionless parameter $\eta$. The solid line represents the results obtained here and the dotted line shows the results of Ref. 12, where the experimental and theoretical data for Lamb shift from Refs. 16, 17 were used. The dashed line represents the results of the present work obtained using experimental and theoretical data for the Lamb shift from Ref. 18.