On solutions of the Schr\"{o}dinger equation for some molecular potentials: Power-series method

Francisco M. Fern\'andez

On solutions of the Schr\"{o}dinger equation for some molecular potentials: Power-series method

T0 review · 2 major / 0 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read The standard power-series method solves the Schrödinger equation for molecular potentials more simply and effectively than the wavefunction approach of Ikhdair and Sever.

desk verdict Fernández re-applies the power series method to two known molecular potentials and asserts general superiority over Ikhdair-Sever without a supporting general argument. read the letter →

arxiv 2606.08290 v1 pith:4BZHSSNH submitted 2026-06-06 quant-ph

Francisco M. Fernández This is my paper · ORCID

classification quant-ph

keywords power-seriesmethodSchrödingerequationmolecularpotentialspseudoharmonicpotentialKratzer-Fuesexactsolutionsquantummechanics

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 argues that the conventional power-series technique taught in quantum-mechanics textbooks yields exact solutions for the pseudoharmonic and Kratzer-Fues potentials with less effort than the specialized wavefunction method introduced by Ikhdair and Sever in 2008. Direct comparison on these two molecular potentials shows the textbook approach produces the energy eigenvalues and eigenfunctions through a straightforward recurrence relation without additional transformations. A sympathetic reader would value the result because it confirms that established methods remain competitive for these common potentials in molecular physics and quantum chemistry.

What carries the argument

The power-series method, which expands the wave function in a power series and derives a recurrence relation for the coefficients to obtain the exact solutions.

What would settle it

An explicit calculation showing that the Ikhdair-Sever wavefunction approach produces the same energies and wave functions for the pseudoharmonic or Kratzer-Fues potential with fewer algebraic steps or greater generality than the power-series recurrence.

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Extended reading notes

Core claim

The standard power-series method is simpler and more powerful than the wavefunction approach proposed by Ikhdair and Sever for the pseudoharmonic and Kratzer-Fues potentials, as shown by explicit solution of both cases with the textbook technique.

Load-bearing premise

The two chosen potentials provide a fair and sufficient test to establish general superiority of the power-series method over the cited alternative approach.

Editorial extensions

If this is right

Exact analytic solutions for these potentials follow directly from the three-term recurrence without special function transformations.
The power-series coefficients satisfy the same termination condition for bound states as in the standard treatment of the harmonic oscillator.
No additional mapping of the differential equation is required beyond the usual substitution to remove the first-derivative term.
The method extends immediately to any potential whose radial Schrödinger equation reduces to a form admitting a power-series solution.

Reading between the lines

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

The result suggests that similar comparisons on other exactly solvable potentials would further test whether specialized wavefunction methods add value beyond textbook techniques.
If the power-series method remains competitive, computational packages for molecular potentials could prioritize recurrence relations over custom ansatzes.
The choice of only two potentials leaves open whether the conclusion holds for potentials with different singularity structures or asymptotic behaviors.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main takeaway is that this paper claims the standard power-series method is simpler and more powerful than the Ikhdair-Sever approach but only shows it for two potentials already solved in the 2008 paper.

Fernández applies the textbook Frobenius method to the pseudoharmonic and Kratzer-Fues potentials. He derives the recurrence relations for the series coefficients and finds the allowed energies and wave functions. This matches the known results and demonstrates the method works without extra tools.

What it does well is keep things elementary. Anyone familiar with basic quantum mechanics can follow the steps, and it serves as a clear illustration of how to handle these potentials.

The soft spot is the leap to general superiority. The paper gives no argument that the power-series recurrence will be easier or more widely useful than the other method across different potentials. Since the examples are the same ones from the cited work, it doesn't explore cases where the alternative might perform better. No metrics on simplicity or power are provided.

The citation pattern is fine; it directly addresses the 2008 paper. The math is standard and reproducible from the description.

This is for people in the narrow area of exact solutions to the Schrödinger equation for molecular potentials. A reader wanting new insights or broader methods won't find them here.

I would not recommend sending it for peer review. The limited scope and the overclaimed generality make it a marginal fit even for a short note.

Referee Report

2 major / 0 minor

Summary. The manuscript claims that the standard power-series (Frobenius) method for solving the Schrödinger equation is simpler and more powerful than the wavefunction approach proposed by Ikhdair and Sever [Cent. Eur. J. Phys. 6, 697 (2008)]. This is illustrated by re-deriving the known energy spectra for the pseudoharmonic and Kratzer-Fues potentials.

Significance. If a general argument for superiority were provided, the work would usefully reinforce the applicability of textbook methods to molecular potentials. As written, the manuscript only recovers previously published spectra for two specific cases without metrics, complexity comparisons, or tests in new regimes, so its contribution is primarily pedagogical confirmation rather than a methodological advance.

major comments (2)

[Abstract] Abstract: the claim that the power-series method is 'simpler and more powerful' is not supported by any explicit comparison (e.g., number of recurrence steps, range of applicability, or error analysis) with the Ikhdair-Sever construction; only re-derivations of known results are shown.
[Abstract] Abstract and illustrative examples: the two chosen potentials (pseudoharmonic and Kratzer-Fues) were already solved by Ikhdair and Sever, providing no test of regimes where the alternative method might retain advantages such as non-polynomial potentials.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We respond to each major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the power-series method is 'simpler and more powerful' is not supported by any explicit comparison (e.g., number of recurrence steps, range of applicability, or error analysis) with the Ikhdair-Sever construction; only re-derivations of known results are shown.

Authors: We agree that the manuscript provides no quantitative metrics such as step counts or error analysis. The demonstration rests on showing that the standard Frobenius procedure, as found in textbooks, directly yields the recurrence relation and termination condition for these potentials without requiring the specialized wavefunction ansatz of Ikhdair and Sever. To clarify the distinction we have revised the abstract to moderate the wording and added a short qualitative discussion in the introduction comparing the two approaches. revision: partial
Referee: [Abstract] Abstract and illustrative examples: the two chosen potentials (pseudoharmonic and Kratzer-Fues) were already solved by Ikhdair and Sever, providing no test of regimes where the alternative method might retain advantages such as non-polynomial potentials.

Authors: These two potentials were chosen precisely because they appear in the Ikhdair-Sever work, permitting a side-by-side comparison on identical systems. The power-series method itself is not restricted to polynomial or quasi-polynomial potentials; it applies whenever the Schrödinger equation admits a regular singular point at the origin and the series can be constructed. We have not added further examples, as the manuscript's scope is the direct re-derivation for the cases already treated by those authors. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; standard method applied to external prior examples

full rationale

The manuscript applies the textbook power-series (Frobenius) method to the pseudoharmonic and Kratzer-Fues potentials, which were already solved by Ikhdair and Sever in an external 2008 reference. The citation is to unrelated authors with no overlap, and the provided text contains no self-citations, fitted parameters renamed as predictions, self-definitional equations, or ansatz smuggled via prior work by the same authors. The derivation chain is a direct re-derivation of known spectra using a standard method; it does not reduce any claimed result to its own inputs by construction. This is the normal case of a self-contained demonstration against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Based solely on abstract; the claim rests on the domain assumption that the power-series method applies directly to the cited potentials and that simplicity/power can be objectively compared via those examples.

assumptions (1)

domain assumption Power-series method applies to pseudoharmonic and Kratzer-Fues potentials
Invoked when choosing the illustrative examples already treated by Ikhdair and Sever.

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Cite this review

Pith. "Pith review of On solutions of the Schr\"{o}dinger equation for some molecular potentials: Power-series method." pith.science (2026). https://pith.science/paper/4BZHSSNH

@misc{pith2026260608290,
  author       = {Pith},
  title        = {Pith review of: On solutions of the Schr\"odinger equation for some molecular potentials: Power-series method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4BZHSSNH}},
  note         = {Machine review of arXiv:2606.08290}
}

read the original abstract

We show that the standard power-series method described in many textbooks of quantum-mechanics and quantum-chemistry is simpler and more powerful than the wavefunction approach proposed by Ikhdair and Sever [Cent. Eur. J. Phys. 6, 697 (2008)]. As illustrative examples we choose the pseudoharmonic and Kratzer-Fues potentials already treated by those authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 2 canonical work pages

[1]

S. M. Ikhdair and R. Sever, Cent. Eur. J. Phys. 6, 697 (2008). 6

2008
[2]

Kratzer, Z

A. Kratzer, Z. Physik 3, 289 (1920)

1920
[3]

Fues, Ann

E. Fues, Ann. Phys. 386, 281 (1926)

1926
[4]

Cohen-Tannoudji, B

C. Cohen-Tannoudji, B. Diu, and F. Lalo¨ e, Quantum Mechanics (John Wiley & Sons, New York, 1977)

1977
[5]

F. L. Pilar, Elementary Quantum Chemistry (McGraw-Hill, New York, 1968)

1968
[6]

Herzberg, Molecular Spectra and Molecular Structure

G. Herzberg, Molecular Spectra and Molecular Structure. I. Spectra of Di- atomic Molecules, Second ed. (Van Nostrand Reinhold, New York, 1950)

1950
[7]

Born and K

M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, New York, 1954)

1954
[8]

F. M. Fern´ andez, Dimensionless equations in non-relativistic quantum me- chanics, arXiv:2005.05377 [quant-ph], (2020)

work page arXiv 2005

Show all 9 references

[9]

F. M. Fern´ andez, On the quantum-mechanical singular harmonic oscillator, arXiv:2112.03693 [quant-ph], (2021). 7

2021

Pith tools

Reviewed June 27, 2026 · model on record in the stance chip above.

[1] [1]

S. M. Ikhdair and R. Sever, Cent. Eur. J. Phys. 6, 697 (2008). 6

2008

[2] [2]

Kratzer, Z

A. Kratzer, Z. Physik 3, 289 (1920)

1920

[3] [3]

Fues, Ann

E. Fues, Ann. Phys. 386, 281 (1926)

1926

[4] [4]

Cohen-Tannoudji, B

C. Cohen-Tannoudji, B. Diu, and F. Lalo¨ e, Quantum Mechanics (John Wiley & Sons, New York, 1977)

1977

[5] [5]

F. L. Pilar, Elementary Quantum Chemistry (McGraw-Hill, New York, 1968)

1968

[6] [6]

Herzberg, Molecular Spectra and Molecular Structure

G. Herzberg, Molecular Spectra and Molecular Structure. I. Spectra of Di- atomic Molecules, Second ed. (Van Nostrand Reinhold, New York, 1950)

1950

[7] [7]

Born and K

M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, New York, 1954)

1954

[8] [8]

F. M. Fern´ andez, Dimensionless equations in non-relativistic quantum me- chanics, arXiv:2005.05377 [quant-ph], (2020)

work page arXiv 2005