Radial position–momentum uncertainties for Klein–Gordon hydrogen-like atoms

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