Hydrogen atom on curved noncommutative space

Abstract

We have calculated the hydrogen atom spectrum on curved noncommutative space defined by the commutation relations $\left[ \hat {x}^{i},\hat{x}^{j}\right] =i\theta\hat{\omega}^{ij}\left( \hat {x}\right) $, where $\theta$ is the parameter of noncommutativity. The external antisymmetric field which determines the noncommutativity is chosen as $\omega^{ij}(x) =\varepsilon^{ijk}{x}_{k}f\left( {x_i}x^{i}\right) $. In this case the rotational symmetry of the system is conserved, preserving the degeneracy of the energy spectrum. The contribution of the noncommutativity appears as a correction to the fine structure. The corresponding nonlocality is calculated: $\Delta x\Delta y \geq \frac{\theta^2}{4} |m\langle f^2\rangle| $, where $m$ is a magnetic quantum number.