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The Radial Momentum-position Uncertainty Relation in Equality Form of Hydrogen Atom and Hydrogen-like Ion

Luo Ling-xiao

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

The radial momentum-position uncertainty relation in equality form of hydrogen atom and hydrogen-like ion is deduced by using the operator theory.It is pointed out that the relation has something to do with the principal quantum number and the angular quantum number,but nothing to do with the nuclear electronic number and reduced mass.

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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
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 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.
The momentum-space wave functions have been obtained for the one-dimensional hydrogen atom and the symmetric and antisymmetric states of the one-dimensional hydrogen molecule ion with δ-function interactions between the particles. The uncertainty relation has been evaluated as a function of the distance between the nuclei in the molecule ion. For the symmetric state this is finite for the collapsed molecule. For the antisymmetric state the energy is positive for a nuclear separation less than a0=ℏ2/me2 and the uncertainty in the position of the electron is infinite.