Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

European Journal of Physics

Comment on ‘Little-known non-orthogonality of hydrogenic radial wavefunctions: an ℓ -dependent operator perspective’

Gregory L. Bason, Bruce Cameron Reed

Published Jul 20, 2026• 0 citations
Publisher

Abstract

Abstract Following the recent discussion of Mankame (2026 Eur. J. Phys. 47 025405), we consider the orthogonality of the radial parts of the hydrogen atom wavefunctions, R n , ℓ ( r ) . As Mankame showed, these functions are not in general orthogonal. By this we mean that the overlap integral of the product of two of them weighted by the square of the radial coordinate may be non-zero if the two angular momentum quantum numbers differ. Orthogonality does hold in the case where the two angular momentum quantum numbers are the same, providing that the principal quantum numbers differ. Here we offer an alternate proof of this result, describe some analytic examples to complement Mankame’s numerical ones, and point out some further references on this issue.

DOI
10.1088/1361-6404/ae848c

Related papers

The little-known non-orthogonality of hydrogenic radial wavefunctions: an ℓ -dependent operator perspective

Devdatta Meghasham Mankame

Abstract It is often assumed that hydrogenic radial wavefunctions R nℓ ( r ) inherit universal orthogonality from the total wavefunctions ψ nℓm . This leads to the inference that all R nℓ are eigenfunctions of a single universal radial operator and thus form an orthonormal basis for L 2 ([0, ∞), r 2 d r ). We demonstrate this is false: orthogonality holds only within fixed- ℓ subspaces, where all R nℓ share the same self-adjoint operator h ˆ ℓ . For ℓ ≠ ℓ ′ , radial functions belong to distinct Sturm–Liouville families with operators differing in their centrifugal terms ∝ ℓ ( ℓ + 1)/ r 2 . This structural difference—not numerical artifact—explains the non-zero radial overlap (e.g. the 2 s –2 p overlap equals − 3 / 2 ). We classify all radial overlap integrals into four cases based on ( n , n ′ , ℓ , ℓ ′ ) , demonstrate non-orthogonality with explicit counterexamples, and explain the operator-family origin of this behavior. This perspective prevents subtle errors in basis expansions and perturbation theory for hydrogenic systems.

European Journal of Physics•Mar 12, 2026•1 citation

Reality of linear and angular momentum expectation values in bound states

Utpal Roy, Suranjana Ghosh, T. Shreecharan +1 more

In quantum mechanics textbooks the momentum operator is defined in the Cartesian coordinates and rarely the form of the momentum operator in spherical polar coordinates is discussed. Consequently one always generalizes the Cartesian prescription to other coordinates and falls in a trap. In this work we introduce the difficulties one faces when the question of the momentum operator in spherical polar coordinate comes. We have tried to point out most of the elementary quantum mechanical results, related to the momentum operator, which has coordinate dependence. We explicitly calculate the momentum expectation values in various bound states and show that the expectation value really turns out to be zero, a consequence of the fact that the momentum expectation value is real. We comment briefly on the status of the angular variables in quantum mechanics and the problems related in interpreting them as dynamical variables. At the end, we calculate the Heisenberg's equation of motion for the radial component of the momentum for the Hydrogen atom.

arXiv:0704.0373•Apr 3, 2007•3 citations

Generalized Radial Uncertainty Product for d-Dimensional Hydrogen Atom

Avoy Jana

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

arXiv:2502.03565•Feb 5, 2025•0 citations

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

Wen-Chao Qiang, Shi‐Hai Dong

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.

Journal of Physics A Mathematical and Theoretical•Jan 23, 2008•10 citations

The electron motion in the practically coulomb field

D. V. Kalinovsky, E. S. Savin

will be added. One can explain the nature of the term by examining the relativistic case. Below the formulas for the radial and angular parts of the wave function were evaluated. The term for the energetic levels of the hydrogen atom for specific quantum numbers was given. We calculated the values of several specific radial functions for different quantum numbers, as well as the values of the radial function at the origin of coordinates and wide apart. A number of mean values for different powers of r was calculated. For the continuous spectrum the radial eigenfunctions were calculated.

•Apr 1, 2013•0 citations

Radial position–momentum uncertainties for the Dirac hydrogen-like atoms

Wen-Chao Qiang, Shi‐Hai Dong

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 .

Journal of Physics A Mathematical and General•Jun 21, 2006•25 citations
Abstract Following the recent discussion of Mankame (2026 Eur. J. Phys. 47 025405), we consider the orthogonality of the radial parts of the hydrogen atom wavefunctions, R n , ℓ ( r ) . As Mankame showed, these functions are not in general orthogonal. By this we mean that the overlap integral of the product of two of them weighted by the square of the radial coordinate may be non-zero if the two angular momentum quantum numbers differ. Orthogonality does hold in the case where the two angular momentum quantum numbers are the same, providing that the principal quantum numbers differ. Here we offer an alternate proof of this result, describe some analytic examples to complement Mankame’s numerical ones, and point out some further references on this issue.
Abstract It is often assumed that hydrogenic radial wavefunctions R nℓ ( r ) inherit universal orthogonality from the total wavefunctions ψ nℓm . This leads to the inference that all R nℓ are eigenfunctions of a single universal radial operator and thus form an orthonormal basis for L 2 ([0, ∞), r 2 d r ). We demonstrate this is false: orthogonality holds only within fixed- ℓ subspaces, where all R nℓ share the same self-adjoint operator h ˆ ℓ . For ℓ ≠ ℓ ′ , radial functions belong to distinct Sturm–Liouville families with operators differing in their centrifugal terms ∝ ℓ ( ℓ + 1)/ r 2 . This structural difference—not numerical artifact—explains the non-zero radial overlap (e.g. the 2 s –2 p overlap equals − 3 / 2 ). We classify all radial overlap integrals into four cases based on ( n , n ′ , ℓ , ℓ ′ ) , demonstrate non-orthogonality with explicit counterexamples, and explain the operator-family origin of this behavior. This perspective prevents subtle errors in basis expansions and perturbation theory for hydrogenic systems.
In quantum mechanics textbooks the momentum operator is defined in the Cartesian coordinates and rarely the form of the momentum operator in spherical polar coordinates is discussed. Consequently one always generalizes the Cartesian prescription to other coordinates and falls in a trap. In this work we introduce the difficulties one faces when the question of the momentum operator in spherical polar coordinate comes. We have tried to point out most of the elementary quantum mechanical results, related to the momentum operator, which has coordinate dependence. We explicitly calculate the momentum expectation values in various bound states and show that the expectation value really turns out to be zero, a consequence of the fact that the momentum expectation value is real. We comment briefly on the status of the angular variables in quantum mechanics and the problems related in interpreting them as dynamical variables. At the end, we calculate the Heisenberg's equation of motion for the radial component of the momentum for the Hydrogen atom.
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.
will be added. One can explain the nature of the term by examining the relativistic case. Below the formulas for the radial and angular parts of the wave function were evaluated. The term for the energetic levels of the hydrogen atom for specific quantum numbers was given. We calculated the values of several specific radial functions for different quantum numbers, as well as the values of the radial function at the origin of coordinates and wide apart. A number of mean values for different powers of r was calculated. For the continuous spectrum the radial eigenfunctions were calculated.
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 .