Pith. sign in

REVIEW 2 major objections 5 minor 23 references

Generalized Radial Uncertainty Product for d-Dimensional Hydrogen Atom

T0 review · 2 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper derives a single exact closed form, Eq. (82), for the radial uncertainty product ΔrΔp_r of a d-dimensional hydrogen-like atom, valid for d ≥ 2 (except d = 2, ℓ = 0).

desk verdict A clean, mostly correct derivation of the d-dimensional radial uncertainty product; the main formula checks out, but the paper's own 2D reduction contains a wrong displayed factor that needs fixing. read the letter →

arxiv 2502.03565 v1 pith:NSFPKKKD submitted 2025-02-05 quant-ph

Avoy Jana This is my paper · ORCID
classification quant-ph
keywords radialuncertaintyproductd-dimensionalhydrogenatommomentumoperatorhyper-sphericalcoordinatesassociatedLaguerrepolynomialsHellmann-Feynmantheoremvirialrelation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper derives a single closed-form expression, Eq. (82), for the radial uncertainty product ΔrΔp_r of a non-relativistic hydrogen-like atom in d-dimensional space, as a function of the principal quantum number n, angular quantum number ℓ, and dimension d. The derivation is carried out in position space using the d-dimensional radial Schrödinger equation, the Hermitian radial momentum operator p_r = -iℏ(∂/∂r + (d-1)/(2r)), and the virial and Hellmann-Feynman theorems to obtain the required expectation values. If correct, the formula gives an exact benchmark for how quantum uncertainty in the radial direction grows with dimensionality, and it reduces to known three-dimensional results as a special case. The expression holds for all d ≥ 2 except the d = 2, ℓ = 0 state, where the squared radial momentum expectation value is not defined within this operator choice.

What carries the argument

The load-bearing object is the Hermitian radial momentum operator p_r = -iℏ(∂/∂r + (d-1)/(2r)) (Eq. 10), chosen so that [r, p_r] = iℏ in any dimension; squaring it gives p_r² = -ℏ²(∂²/∂r² + (d-1)/r ∂/∂r + (d-1)(d-3)/(4r²)), which connects to the radial Laplacian ∇_r² through a dimension-dependent 1/r² term. That connection fixes the effective potential Veff(r) = V(r) + [ℓ(ℓ+d-2) + (d-1)(d-3)/4]ℏ²/(2µr²), whose expectation value is evaluated using the Hellmann-Feynman theorem to get ⟨1/r²⟩. The uncertainty product is then assembled from ⟨r⟩ and ⟨r²⟩, computed by recursion relations for associated Laguerre polynomials, and from ⟨p_r²⟩, computed by combining the virial theorem with the energy spectrum En = -µZ²ℏ²/(2ν²a₀²).

What would settle it

Compute Δp_r independently by Fourier-transforming the d-dimensional radial wavefunction to momentum space and evaluating the variance of the radial momentum component; if the result differs from Eq. (81) for any d ≥ 3 state, the closed form is wrong.

Watch

Extended reading notes

Core claim

The central claim is that the generalized radial uncertainty product is exactly given by ΔrΔp_r = (ℏ/(4ν)) √S √(1 - ((d-1)(d-3)+4ℓ(ℓ+d-2))/((2n+d-3)(2ℓ+d-2))), with ν = n + (d-3)/2 and S the degree-four polynomial in d, n, ℓ displayed in Eq. (50). The formula is obtained by computing the normalized d-dimensional radial wavefunction, evaluating ⟨r⟩ and ⟨r²⟩ through Laguerre-polynomial recursion and orthogonality, and computing ⟨p_r²⟩ from the energy and the effective-potential expectation value rather than by direct integration. A notable structural point is that the radial momentum operator in d dimensions contains the term (d-1)/(2r), which contributes an extra (d-1)(d-3)/(4r²) term in p_r² and therefore enters the uncertainty product through the 1/r² expectation value.

Load-bearing premise

The whole formula rests on accepting that the radial momentum operator in d dimensions is p_r = -iℏ(∂/∂r + (d-1)/(2r)); a different Hermitian quantization of radial momentum would change the (d-1)(d-3)/(4r²) term and hence the uncertainty product.

Editorial extensions

If this is right

  • For d = 3, Eq. (82) reduces to the standard radial uncertainty product of the three-dimensional hydrogen atom, so the formula contains the known result as a special case.
  • For fixed n and ℓ, the uncertainty product increases with d for large d, showing that radial confinement sharpens with dimensionality.
  • The formula provides closed-form expectation values ⟨r⟩, ⟨r²⟩, ⟨1/r⟩, and ⟨1/r²⟩ for hydrogenic states in any dimension, which can serve as inputs for other d-dimensional atomic calculations.
  • The exceptional case d = 2, ℓ = 0 is excluded because ⟨p_r²⟩ diverges there, meaning the radial momentum operator as defined cannot support a finite uncertainty product for that state.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same operator identity and Hellmann-Feynman route could in principle be applied to other spherically symmetric potentials with known d-dimensional solutions, such as the isotropic harmonic oscillator, to produce dimension-dependent uncertainty products.
  • The d = 2, ℓ = 0 failure suggests that a different self-adjoint extension of the radial momentum operator, or a different radial quantization, would be needed to define a sensible uncertainty product in that sector; testing this is a natural next step.
  • If the formula is used as a benchmark, numerical diagonalization in d = 4 or d = 5 on a radial grid would provide a direct check of each individual expectation value, not just the product.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The manuscript derives a closed-form expression for the radial uncertainty product Δr Δp_r of the d-dimensional non-relativistic hydrogen atom in position space. The derivation introduces the radial momentum operator p_r = -iℏ(∂/∂r + (d-1)/(2r)), obtains the normalized d-dimensional radial wavefunctions, evaluates ⟨r⟩, ⟨r²⟩, ⟨p_r⟩, and ⟨p_r²⟩ using virial and Hellmann–Feynman theorems, and combines them into the general result Eq. (82). The paper also lists reductions to two and three dimensions and provides graphical captions showing the dependence of the uncertainty on the dimension.

Significance. If correct, the central formula Eq. (82) is an exact, parameter-free generalization of the radial uncertainty relation to arbitrary dimension d ≥ 2, valid for all allowed n, ℓ except d=2, ℓ=0. The derivation is explicit and checkable: it reproduces the standard 3D expectation values, reduces correctly to 2D non-s states when Eq. (81) is used, and uses standard theorems without adjustable parameters. The paper's value is primarily as a reference formula for higher-dimensional hydrogenic systems; it is not a conceptual advance. The presence of two displayed errors—the integration measure in Eqs. (18)–(19) and the 2D Δp_r radicand in §IV.B—means the manuscript currently contains internally inconsistent formulas, even though the central result Eq. (82) appears to be correct.

major comments (2)
  1. [Section III.C, Eqs. (18)–(19)] The displayed expectation-value integrals for ⟨p_r⟩ and ⟨p_r²⟩ use the measure r² dr, but for a d-dimensional position space the correct measure is r^{d-1} dr, as used in Eqs. (15)–(17) and in §IV.A. The subsequent computation in Eq. (52) correctly uses r^{d-1} dr, so the final results are not affected; nevertheless, the formulas as displayed contradict the paper's own definition of expectation values and need to be corrected so that the derivation can be followed as written.
  2. [Section IV.B, 2D Δp_r] The bullet point for the 2D radial momentum uncertainty states Δp_r = (Zℏ/((n−1/2)a0)) sqrt(1 − (2|ℓ|−1)/(2|ℓ|(2n−1))). Substituting d=2 into the general formula Eq. (81) gives the radicand 1 − (4ℓ²−1)/(2ℓ(2n−1)) = 1 − ((2ℓ−1)(2ℓ+1))/(2ℓ(2n−1)); the displayed expression is missing the factor (2ℓ+1). Direct integration for n=2, ℓ=1 yields ⟨p_r²⟩ = 2Z²ℏ²/(9a0²), i.e. radicand 1/2, matching Eq. (81) and not the displayed 5/6. This false displayed result undermines the paper's own lower-dimensional cross-validation of Eq. (82) and must be fixed.
minor comments (5)
  1. [Section IV.B, 2D ⟨1/r²⟩] The 2D formula for ⟨1/r²⟩ is stated to hold for |ℓ| ≠ 0; a brief explanation that the defining integral diverges for ℓ = 0 would clarify the exclusion of d=2, ℓ=0 in Eqs. (81)–(82).
  2. [Section III.A, Eqs. (29)–(30)] The principal quantum number ν is introduced in Eq. (29) and then replaced by n through Eq. (30) without a clear statement that ν = n + (d−3)/2; this relation should be highlighted before Eq. (29) is used in the wavefunction.
  3. [Introduction] The phrase 'We have searched (Ref. [1])' is awkward; consider rewording to describe the prior work more naturally.
  4. [General presentation] The manuscript lists figure captions for Figs. 1–9 but does not include the actual plots in the visible text; the final version should ensure that all figures are present and legible.
  5. [Section III.A, Eq. (42)] The alternative normalization formula based on Griffiths' version of the orthogonality relation is mentioned but not used; removing it would reduce confusion about which convention is adopted.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Eq. (82) is derived from external wavefunction and standard theorems, not assumed or fitted.

full rationale

The central formula Eq. (82) is obtained by explicit integration of the known d-dimensional hydrogen radial wavefunction Eq. (43) (taken from Ref. [11]), using standard Laguerre identities (Eqs. (32)-(34)), the virial theorem, and the Hellmann-Feynman theorem. The radial-momentum operator Eq. (10) is introduced from Refs. [1,2] and independently rederived in the paper via the divergence theorem; it is a definitional input, not the target result. No parameter is fit to the uncertainty product, and no prediction is statistically forced. The only self-citation is Ref. [1], used for the Laguerre orthogonality convention in Eq. (32) ('while this will not affect in our calculation (Ref. [1])'); this is not load-bearing because the identities are standard and the integrals are recomputed in the paper. No uniqueness theorem or ansatz is imported from the author's prior work, and no known result is merely renamed. The apparent mismatch between the §IV B 2D radicand and Eq. (81) is an internal algebraic discrepancy, not a circular step; it bears on correctness rather than circularity. The derivation therefore does not reduce to its own inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted; all constants are physical. The derivation leans on known wavefunctions and standard theorems. The radial momentum operator form is the most consequential assumption, as it determines the d-dependent term in p_r².

assumptions (5)
  • domain assumption The d-dimensional hydrogen radial wavefunction is given by Eq. (43), including the associated Laguerre polynomials and the principal quantum number shift ν = n + (d-3)/2.
    The paper imports this solution from Ref [11] rather than deriving it from the Schrödinger equation.
  • domain assumption The radial momentum operator is p_r = -iℏ(∂/∂r + (d-1)/(2r)).
    This operator is Hermitian and satisfies [r,p_r]=iℏ, but its form fixes the d-dependent quantum correction in p_r² and hence the final uncertainty product.
  • standard math The virial theorem for Coulomb potentials: ⟨T⟩ = -1/2⟨V⟩, so E = 1/2⟨V⟩.
    Used in Eq. (61) and (78) to express ⟨p_r²⟩ without direct integration.
  • standard math Hellmann-Feynman theorem.
    Used in Appendix B to derive ⟨1/r⟩ and ⟨1/r²⟩; requires differentiability of eigenvalues with respect to parameters.
  • standard math The eigenvalue of the squared angular momentum operator on hyper-spherical harmonics is ℓ(ℓ+d-2)ℏ².
    Standard result for the Laplace-Beltrami operator on S^{d-1}, used throughout.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Generalized Radial Uncertainty Product for d-Dimensional Hydrogen Atom." pith.science (2026). https://pith.science/paper/NSFPKKKD

@misc{pith2026250203565,
  author       = {Pith},
  title        = {Pith review of: Generalized Radial Uncertainty Product for d-Dimensional Hydrogen Atom},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NSFPKKKD}},
  note         = {Machine review of arXiv:2502.03565}
}
read the original abstract

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

Figures

Figures reproduced from arXiv: 2502.03565 by the authors.

Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figures from the paper (4 more)
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 6
Figure 6. Figure 6: FIG. 6. ∆ [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. ∆ˆp [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. ∆ˆr [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 19 canonical work pages

  1. [1]

    [14]) uses the orthogo- nality property as Z ∞ 0 ρae−ρLa b (ρ)La c (ρ)dρ = Γ(a + b + 1) [Γ(b + 1)]3 δbc while this will not affect in our calculation (Ref

    Orthonormal property Z ∞ 0 ρae−ρLa b (ρ)La c (ρ)dρ = Γ(a + b + 1) Γ(b + 1) δbc (32) 5 Many authors like Griffith’s (Ref. [14]) uses the orthogo- nality property as Z ∞ 0 ρae−ρLa b (ρ)La c (ρ)dρ = Γ(a + b + 1) [Γ(b + 1)]3 δbc while this will not affect in our calculation (Ref. [1])

    work page

  2. [2]

    Recursive property ρLa b (ρ) = (a + 2b + 1)La b (ρ) − b + 1 a + b + 1La b+1(ρ) − (a + b)2La b−1(ρ) (33)

    work page

  3. [3]

    Derivative property d dρ La b (ρ) = La+1 b (ρ) = 1 ρ [bLa b (ρ) − (b + a)La b−1(ρ)] (34) Let’s introduce a dimensionless parameter ρ as ρ = 2Z n + d−3 2 a0 r (35) With this substitution, the radial wave function takes the form Rnℓ(ρ) = Nnℓe− ρ 2 ρℓL2ℓ+d−2 n−ℓ−1 (ρ) (36) To ensure proper normalization, we redefine the proba- bility density as P (r) = n + d...

    work page

  4. [4]

    Avoy Jana, Radial Uncertainty Product for Spherically Symmetric Potential in Position Space, arXiv:2501.14831 [quant-ph] (2025), https://arxiv.org/abs/2501.14831 16

    work page Pith review arXiv 2025

  5. [5]

    On the connection between the radial momentum operator and the Hamiltonian in n dimensions

    Gil Paz, On the connection between the radial mo- mentum operator and the Hamiltonian in n dimen- sions, European Journal of Physics, 22(4), 337 (2001), doi:10.1088/0143-0807/22/4/308. https://arxiv.org/ abs/quant-ph/0009046

    work page Pith review arXiv 2001

  6. [6]

    https://doi.org/10.1142/S0217979222500722

    Anzor Khelashvili and Teimuraz Nadareishvili, Gener- alized Heisenberg uncertainty relation in spherical co- ordinates, International Journal of Modern Physics B, 36(15), 2250072 (2022), doi:10.1142/S0217979222500722. https://doi.org/10.1142/S0217979222500722

    work page doi:10.1142/s0217979222500722 2022

  7. [7]

    https://doi.org/ 10.1119/1.3534840

    Christian Bracher, Uncertainty relations for angular momentum eigenstates in two and three spatial di- mensions, American Journal of Physics, 79(3), 313– 319 (2011), doi:10.1119/1.3534840. https://doi.org/ 10.1119/1.3534840

    work page doi:10.1119/1.3534840 2011

  8. [8]

    Al-Jaber, Uncertainty Relations for Some Cen- tral Potentials in N-Dimensional Space , Applied Math- ematics, 7(6), March 2016, doi:10.4236/am.2016.76047

    Sami M. Al-Jaber, Uncertainty Relations for Some Cen- tral Potentials in N-Dimensional Space , Applied Math- ematics, 7(6), March 2016, doi:10.4236/am.2016.76047. https://doi.org/10.4236/am.2016.76047

    work page arXiv 2016

Show all 23 references
  1. [9]

    J. S. Dehesa, Spherical-Symmetry and Spin Effects on the Uncertainty Measures of Multidimensional Quan- tum Systems with Central Potentials , Entropy, 23(5), 607 (2021), doi:10.3390/e23050607. https://doi.org/ 10.3390/e23050607

    2021 doi

  2. [10]

    J. S. Dehesa and D. Puertas-Centeno, Multidimensional hydrogenic states: position and momentum expectation values, Journal of Physics B: Atomic, Molecular and Optical Physics, 54(6), 065006 (2021), doi:10.1088/1361- 6455/abcdee. http://dx.doi.org/10.1088/1361-6455/ abcdee

    2021 doi

  3. [11]

    A. Smirnov, View on N-dimensional spherical harmon- ics from the quantum mechanical P¨ oschl-Teller poten- tial well , arXiv:1901.06711 [math-ph] (2019), https:// arxiv.org/abs/1901.06711

    2019 arXiv

  4. [12]

    Cohl and Ernie G

    Howard S. Cohl and Ernie G. Kalnins, Fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry, arXiv:1201.4406 [math-ph] (2012), https://arxiv.org/abs/1201.4406

    2012 arXiv

  5. [13]

    O. L. Trinhammer and G. Olafsson, The Full Laplace- Beltrami operator on U(N) and SU(N) , arXiv:math- ph/9901002 (2012), https://arxiv.org/abs/math-ph/ 9901002

    2012

  6. [14]

    S. M. Al-Jaber, Hydrogen Atom in N Dimensions , International Journal of Theoretical Physics, 37, 1289–1298 (1998), doi:10.1023/A:1026679921970. https: //doi.org/10.1023/A:1026679921970

    1998 doi

  7. [15]

    Bransden and C

    B. Bransden and C. Joachain, Introduction to Quantum Mechanics, Wiley, New York, 1989

    1989

  8. [16]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun, Handbook of Math- ematical Functions with Formulas, Graphs, and Math- ematical Tables , 9th printing, Dover, New York, 1972. Chapter 22: Orthogonal Polynomials, 771–802

    1972

  9. [17]

    Griffiths and Darrell F

    David J. Griffiths and Darrell F. Schroeter, Introduction to Quantum Mechanics , 3rd ed., Cambridge University Press, Cambridge, 2018

    2018

  10. [18]

    Nouredine Zettili, Quantum Mechanics: Concepts and Applications, 2nd ed., Wiley, Hoboken, NJ, 2013

    2013

  11. [19]

    A. B. Gupta, Fundamentals of Classical Mechanics , 3rd Edition, Paperback, New Central Book Agency, 2023

    2023

  12. [20]

    Supriadi, A

    B. Supriadi, A. Harijanto, M. Maulana, Z. R. Ridlo, W. D. Wisesa, and A. Nurdiniaya, The function of the radial wave of a hydrogen atom in the principal quan- tum numbers (n) 4 and 5 , Journal of Physics: Con- ference Series, 1211, 012052 (2019), doi:10.1088/1742- 6596/1211/1/012052

    2019 doi

  13. [21]

    L. M. B. C. Campos and M. J. S. Silva, On hyperspherical associated Legendre functions: the extension of spheri- cal harmonics to N dimensions, arXiv:2005.09603 (2020), https://arxiv.org/abs/2005.09603

    2020 arXiv

  14. [22]

    J. G. Esteve, F. Falceto, and C. Garc ´ ıa Canal, Generalization of the Hellmann–Feynman theo- rem, Physics Letters A, 374(6), 819–822 (2010), doi:10.1016/j.physleta.2009.12.005. http://dx.doi. org/10.1016/j.physleta.2009.12.005

    2010 doi

  15. [23]

    X. L. Yang, S. H. Guo, F. T. Chan, K. W. Wong, and W. Y. Ching, Analytic solution of a two-dimensional hydrogen atom. I. Nonrela- tivistic theory , Phys. Rev. A, 43(3), 1186–1196 (1991), doi:10.1103/PhysRevA.43.1186. https: //link.aps.org/doi/10.1103/PhysRevA.43.1186

    1991 doi

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.