Radial position–momentum uncertainties for the Dirac hydrogen-like atoms

We show how the radial position–momentum uncertainty product can be obtained analytically for the Dirac hydrogen-like atoms. Some interesting features for this system are found. First, for the same principal quantum number n , as the azimuthal quantum number l increases, the uncertainties Δ p r and Δ r Δ p r decrease. However, the uncertainty Δ r does not always decrease, which is different from the non-relativistic hydrogen-like atoms case, where it always decreases. Second, for the same l both Δ r and Δ r Δ p r increase as n increases. Third, all uncertainties for the same n and l = n − 1 are smallest in comparison with those for the same n but . Fourth, the uncertainty Δ r Δ p r is independent of the value of charge Z in the non-relativistic case, while it is related with the value of charge Z in the relativistic case. Fifth, the relativistic corrections to the non-relativistic values of uncertainties are very small when the values of charge Z are not too big, while the relativistic corrections to them will appear explicitly for a large value of charge Z .