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Journal of Physics A Mathematical and Theoretical

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

Wen-Chao Qiang, Shi‐Hai Dong

Published Jan 23, 2008• 10 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8113/41/5/055306

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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.
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 .
This paper presents a comprehensive analysis of the generalized radial uncertainty product for the d-dimensional non-relativistic Hydrogen atom in position space. Utilizing the framework of quantum mechanics in d-dimensional spherical coordinates, the study extends the standard radial uncertainty relation to higher dimensions. Taking the solution of the radial Schrodinger equation, the normalized radial wave functions, expectation values, and uncertainties in both position and momentum space are rigorously evaluated. The analytical derivations reveal the dependence of the uncertainty product on the principal and angular quantum numbers, as well as the dimensional parameter d. The results provide deeper insight into the role of dimensionality in quantum uncertainty relations and their implications for higher-dimensional quantum systems
Abstract We present the analytical expression of the product of the radial position and momentum uncertainties and for the infinite spherical well. We find a few interesting features. First, the uncertainty increases with the radius R and the quantum number n , the n th root of the spherical Bessel function, but finally arrives at a constant for a large n and decreases with the angular momentum number l . It is seen that the becomes imaginary, which arises from the fact that the moving particle is shifted to axis y suddenly from the original axis x as the quantum number n increases. Furthermore, the becomes zero when the n increases for a given l . This means that the particle is spiraling around a circle whose radius r < R changes between a varying radius and a constant but with an increasing radius r as the n increases. Finally, the particle moves around a circle with a maximum radius R . Second, the relative dispersion is independent of the radius R , and it increases with the quantum number n but decreases with the quantum number l . Third, the radial momentum uncertainty decreases with the radius R and increases with the quantum numbers l > 0 and n . We notice that there exists a turning point for the uncertainty when l = 1 and n > 1. This also leads to the product . Fourth, the product is independent of the radius R and increases with the quantum numbers l > 0 and n . Finally, we obtain the analytical expression of the Fisher entropy and notice that it decreases with the radius R but increases with the quantum numbers l and n . We also find that the Cramer–Rao uncertainty relation is satisfied, and it increases with the quantum numbers l and n .
This paper presents a detailed analysis of the radial uncertainty product for quantum systems with spherically symmetric potentials. Using the principles of quantum mechanics, the study derives the radial uncertainty relation analogous to the Cartesian form and investigates its implications for three key spherically symmetric potentials: the Hydrogen atom, the infinite spherical potential well, and the spherical harmonic oscillator, all within the non-relativistic regime. Employing the Schrodinger equation in spherical coordinates, the paper rigorously evaluates the normalized radial wave functions, expectation values, and uncertainties associated with both position and momentum. Analytical derivations and numerical computations highlight the dependence of the uncertainty product on quantum numbers and system-specific parameters.
Position and momentum information measures are evaluated for the ground state of the \emph{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 rrr, but is singular at the origin. Correspondingly, the momentum density decays as an inverse power of ppp. 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.