REVIEW 6 minor 12 references
Comment on: "Some quantum aspects of a particle with electric quadrupole moment interacting with an electric field subject to confining potentials". Int. J. Mod. Phys. A 29 (2014) 1450117
T0 review · 0 major / 6 minor · reviewed 2026-08-27 · deepseek-v4-flash
Pith's one-line read A re-analysis of a quadrupole-particle model finds no quantization of the oscillator frequency: the reported allowed cyclotron frequencies come from a truncation artifact.
desk verdict This is a correct and useful comment: Bakke's allowed cyclotron frequencies are an artifact of polynomial truncation, and the comment shows it with real numerical evidence. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is a one-dimensional radial Schrödinger equation with potential $-\alpha/\xi+\xi^2$, obtained from the three-dimensional problem by separation in cylindrical coordinates. The original argument applies the Frobenius method, whose three-term recurrence can be truncated by imposing $g=2n$ and $a_{n+1}=0$; those two conditions make the power series a polynomial but are not equivalent to square integrability. The comment's argument instead treats the eigenvalue problem as a quasi-solvable system, meaning a model in which only exceptional eigenfunctions are polynomial while the rest of the spectrum is non-polynomial. To exhibit that full spectrum, the comment uses a variational calculation in the Gaussian basis $\{\xi^{|l|+j}e^{-\xi^2/2}\}$ with up to fifteen basis functions, independently checked by a continued-fraction summation method; the convergence tables show eigenvalues approaching their limits from above. The key conceptual move is to read the polynomial solutions as exceptional cases rather than as the definition of bound states.
What would settle it
Solve the radial equation (5) for a value not satisfying the truncation condition, say $\alpha=1$, $l=0$, with a high-accuracy numerical method; if no square-integrable eigenfunction exists, or if the three lowest eigenvalues do not match $W_{0,0}=-0.2085695649$, $W_{1,0}=4.601041510$, $W_{2,0}=8.834509671$, then the claim that bound states exist for every $\alpha$ would be refuted.
Extended reading notes
Core claim
The central claim is that the radial equation (5), written in terms of $\xi=\sqrt{m\omega}\,\rho$ and the coupling $\alpha=\tilde{Q}E_0/\sqrt{m\omega}$, has square-integrable solutions for every real $\alpha$ and every angular quantum number $l$. The eigenvalues $W_{\nu,l}(\alpha)$ are continuous curves in $\alpha$; only where $\alpha$ coincides with one of the roots $\alpha^{(i)}_{n,l}$ of $a_{n+1}=0$ with $g=2n$ does one eigenvalue sit exactly at the truncated value $W(n,l)=2n+2|l|+2$. Consequently, requiring $g=2n$ and $a_{n+1}=0$ does not determine the spectrum; it selects one exceptional state of one particular operator, and the same $\alpha$ still leaves infinitely many other bound states. The paper further argues that the original analysis overlooks that the three-dimensional motion is unbounded along $z$, so strictly speaking no bound states exist unless the motion is restricted to the $x$-$y$ plane. The claimed quantization of $\alpha$, and hence of the oscillator frequency $\omega$, is therefore presented as an artifact.
Load-bearing premise
A load-bearing premise is that the variational and continued-fraction calculations converge to the true eigenvalues for every real $\alpha$; the comment demonstrates this convergence empirically in tables, not by an a priori proof, so a failure of those digits would break the central claim.
Editorial extensions
If this is right
- The original paper's quantized cyclotron frequencies $\omega_{n,l}$ should be replaced by a continuous set: for fixed physical parameters, all oscillator frequencies are allowed.
- For a fixed coupling $\alpha$, the planar spectrum is an infinite ladder $W_{\nu,l}(\alpha)$ that varies continuously, so a measured transition cannot by itself reveal a discrete allowed-frequency spectrum.
- In the full three-dimensional problem, no normalizable bound states exist because of the free motion along $z$; any bound-state discussion must treat the planar model as an approximation or add a confining potential along $z$.
- The truncation method underestimates the spectrum: even where a polynomial solution exists, it produces only one of infinitely many eigenvalues for that same parameter value.
Reading between the lines
- This criticism likely transfers to other quasi-solvable models in which quantization of a field parameter is inferred from polynomial solutions; checking whether the full spectrum is continuous in the parameter would be a cheap diagnostic.
- A testable extension is to fix $l$ and scan $\alpha$ through one of the roots $\alpha^{(i)}_{n,l}$, computing the nearest-neighbor spacing $W_{\nu+1,l}-W_{\nu,l}$; the polynomial state should appear as one point on a smooth curve, not as an isolated resonance.
- If the planar model is intended to describe a physical trap, the lack of confinement along $z$ limits its validity to two dimensions; adding weak harmonic confinement in $z$ and taking the zero-frequency limit could reveal whether the three-dimensional spectrum remains continuous.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This Comment examines a published model of a particle with an electric quadrupole moment interacting with an electric field, in which the original author used polynomial truncation of a power series to derive bound-state energies and a set of 'allowed' cyclotron frequencies. The Comment makes two main claims. First, the full three-dimensional model does not support normalizable bound states at all, because the motion along the z axis is free and hence the wave function is not square-integrable over z. Second, even when attention is restricted to the x-y plane, the truncation conditions used in the original paper, g=2n and a_{n+1}=0, do not yield the full bound-state spectrum of the radial equation; they produce isolated polynomial eigenfunctions only for special values of the parameter alpha. Using Rayleigh-Ritz variational calculations with Gaussian basis functions and a Riccati-Padé cross-check, the Comment shows that the radial equation supports a complete discrete spectrum for every real alpha, with eigenvalues varying continuously with alpha, so alpha and hence the oscillator frequency are not quantized. The Comment therefore concludes that the original paper's allowed cyclotron frequencies are an artifact of the truncation method.
Significance. If correct, this Comment is a decisive correction of a published result in a mainstream journal. It identifies a common pitfall in quasi-exactly-solvable models: polynomial solutions are only exceptional eigenfunctions for exceptional parameter values, not the entire spectrum. The argument is supported by two independent numerical methods whose results converge monotonically to ten digits, and the analytic claim that the radial potential xi^2 - alpha/xi confines for every real alpha is consistent with standard Sturm-Liouville theory. The Comment also explains the earlier Vercin/Myrheim example correctly, strengthening its diagnosis. The paper is an appropriate and valuable contribution as a Comment, though its scope is limited to correcting the original work.
minor comments (6)
- [Abstract] The abstract contains a typo: 'interaction etween' should read 'interaction between'.
- [Body, after Eq. (8)] The phrase 'there cannot be any quantiz ation of alpha' contains a spacing typo; it should read 'quantization of alpha'.
- [Addendum] In the Addendum, 'Bake' appears instead of 'Bakke' in the sentence beginning 'It seems that Bake does not understand the problem'; this should be corrected.
- [Body, paragraph before Eq. (5)] The sentence 'In order to facilitate de following discussion' contains a typo: 'de' should be 'the'.
- [Notation, around Eq. (5) and Figure 2] The notation W(n,l) for the eigenvalues obtained by truncation and W_{nu,l}(alpha) for the actual eigenvalues is somewhat overloaded; renaming the truncation eigenvalues, for example E^{tr}_{n,l}, would make the distinction clearer.
- [Figure 2 and surrounding text] The claim that 'Any such vertical line will not meet more than one red circle, except the one at alpha = 0' is stated without proof. If this assertion is based on numerical observation, it should be phrased as such or supported by an argument, since it is not essential to the main conclusion.
Circularity Check
No significant circularity: the comment's critique of truncation-based frequency quantization is self-contained and checked by two independent numerical methods.
full rationale
The load-bearing claim is that Eq. (5) has bound states for all real alpha and that the truncation conditions (8) yield only isolated polynomial solutions. This is argued from the confining form of the potential, from textbook square-integrability criteria, from the Hellmann-Feynman theorem, and from two numerical computations, Rayleigh-Ritz with a complete Gaussian basis and Riccati-Padé. The Riccati-Padé check cites the author's own prior method [9], but the conclusion also rests on non-self-cited Rayleigh-Ritz tables and on analytic arguments, so that self-citation is not load-bearing. The only other self-citation, ref. [2] on dimensionless units, is a side remark. No equation of the comment is defined in terms of the result it derives, no fitted parameter is relabeled as a prediction, and no uniqueness theorem from the author's own prior work is invoked. The numerical convergence is shown empirically rather than proved, but that is a correctness-risk caveat, not circularity.
Assumptions & free parameters
assumptions (4)
- standard math A bound state requires a square-integrable wavefunction (Eq. 4); a plane-wave factor e^{ikz} from free motion along z makes the full 3D state non-normalizable, so the model has no 3D bound states.
- domain assumption The planar radial potential -alpha/xi + xi^2 confines the particle for every real alpha, so Equation (5) has a discrete spectrum bounded below for all alpha.
- domain assumption The Rayleigh-Ritz basis {xi^{|l|+j} exp(-xi^2/2)} is complete and converges to the exact eigenvalues from above; the Riccati-Pade method gives independent cross-checks.
- standard math In quasi-exactly solvable models, polynomial solutions of the three-term recurrence are isolated exact solutions, and all roots of a_{n+1}(g=2n, alpha)=0 are real (Child, Dong, Wang theorem).
Cite this review
Pith. "Pith review of Comment on: "Some quantum aspects of a particle with electric quadrupole moment interacting with an electric field subject to confining potentials". Int. J. Mod. Phys. A 29 (2014) 1450117." pith.science (2026). https://pith.science/paper/BGCT62GT
@misc{pith2026200910561,
author = {Pith},
title = {Pith review of: Comment on: "Some quantum aspects of a particle with electric quadrupole moment interacting with an electric field subject to confining potentials". Int. J. Mod. Phys. A 29 (2014) 1450117},
year = {2026},
howpublished = {\url{https://pith.science/paper/BGCT62GT}},
note = {Machine review of arXiv:2009.10561}
}
abstract
We analyze the results obtained from a model consisting of the interaction etween the electric quadrupole moment of a moving particle and an electric field. We argue that the system does not support bound states because the motion along the $z$ axis is unbounded. It is shown that the author obtains a wrong bound-state spectrum for the motion in the $x-y$ plane and that the existence of allowed cyclotron frequencies is an artifact of the approach.
Figures
Reference graph
Works this paper leans on
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F. M. Fern´ andez, Dimensionless equations in non-relativistic qua ntum me- chanics, arXiv:2005.05377 [quant-ph]
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Reviewed August 27, 2026 · model on record in the stance chip above.
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