The uncertainty in the position operator of the Dirac electron is closely investigated based upon the method of Huang's expectation value. It is found that the obtained results are consistent with that of the previous paper.
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 .
The positive helium ion consists of two protons and one orbital electron to be viewed as a hydrogenic atom. The use of ions has very potential, especially in the development of spintronic devices and plasma-based sterilization. This study aims to determine the expected value and position uncertainty of He^+ ion electrons. Determination of the expected value and position uncertainty is done by analytical calculation through the Schrodinger method approach, especially the radial part at quantum number n≤4, and numerical calculation with the help of a simulation program. The results showed that the main quantum number and orbital n, l influence the expectation value and position uncertainty. The greater the main quantum number n, the expectation value of electron position decreases and the uncertainty of electron position increases. At the same time, the increase in the orbital quantum number l has an impact on the value, which tends to go down. This research is in accordance with de Broglie's hypothesis about the nature of dualism and the principle of uncertainty in the review of position space.