Relativistic effects on information measures for hydrogen-like atoms

• Published in: Journal of computational and applied mathematics Vol. 233; no. 6; pp. 1399 - 1415
• Main Authors: Katriel, Jacob, Sen, K.D.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Kidlington Elsevier B.V 15.01.2010; Elsevier
• Subjects: Exact sciences and technology; Fisher information measure; Heisenberg uncertainty relation; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Operator theory; Relativistic H atom; Sciences and techniques of general use; Shannon and Rényi entropies; Statistical complexity; Shannon and Rényi entropies; Relativistic H atom; Statistical complexity; Heisenberg uncertainty relation; Fisher information measure; Numerical analysis; Applied mathematics; Entropy; Complexity measure; Self adjoint operator; Variance
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2008.04.039

Position and momentum information measures are evaluated for the ground state of the relativistic hydrogen-like atoms. Consequences of the fact that the radial momentum operator is not self-adjoint are explicitly studied, exhibiting fundamental shortcomings of the conventional uncertainty measures in terms of the radial position and momentum variances. The Shannon and Rényi entropies, the Fisher information measure, as well as several related information measures, are considered as viable alternatives. Detailed results on the onset of relativistic effects for low nuclear charges, and on the extreme relativistic limit, are presented. The relativistic position density decays exponentially at large r , but is singular at the origin. Correspondingly, the momentum density decays as an inverse power of p . Both features yield divergent Rényi entropies away from a finite vicinity of the Shannon entropy. While the position space information measures can be evaluated analytically for both the nonrelativistic and the relativistic hydrogen atom, this is not the case for the relativistic momentum space. Some of the results allow interesting insight into the significance of recently evaluated Dirac–Fock vs. Hartree–Fock complexity measures for many-electron neutral atoms.