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On the two-dimensional hydrogen atom in a circular box in the presence of an electric field

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A two-dimensional hydrogen atom in a circular box has an exact degenerate point that an electric field splits.

desk verdict A verifiable exact ground state and clean symmetry arguments sit inside a numerical RRM calculation whose convergence is never certified; the degeneracy claim needs support before it carries weight. read the letter →

arxiv 2504.18928 v1 pith:HUF5GYOK submitted 2025-04-26 quant-ph

Paolo Amore , Francisco M. Fernández This is my paper · ORCID
classification quant-ph
keywords two-dimensionalhydrogenatomcircularboxconfinementStarkeffectRayleigh-Ritzmethodaccidentaldegeneracyexactgroundstateavoidedcrossings
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

This paper studies the quantum spectrum of a two-dimensional hydrogen atom in an impenetrable circular box, with the nucleus fixed at the centre and a uniform electric field applied. The main analytical result is that when the dimensionless box-strength parameter is $\beta=3/4$, the zero-field ground state is exactly $R_{00}(r)=(1-r)e^{(1-r)/2}/\sqrt{3e-8}$ with energy $E_{00}=-1/8$, and at the same parameter the levels $E_{02}$ and $E_{10}$ cross, producing a three-fold degeneracy that the reflection symmetry of the problem does not require. The paper presents Rayleigh-Ritz eigenvalues for nonzero field and shows that this accidental degeneracy splits when the field is turned on, with avoided crossings appearing only between states of the same parity. If the exact solution and the degeneracy are genuine, the model offers a concrete confined-Stark system where a closed-form state and a degeneracy-splitting mechanism can be studied quantitatively.

What carries the argument

The load-bearing objects are the accidental degeneracy at $\beta=3/4$ and the exact radial function $R_{00}(r)$. The degeneracy is produced by the crossing of the energy curves $E_{02}(\beta)$ and $E_{10}(\beta)$, and the exact ground state is a closed-form eigenfunction at that same point. The numerical engine is the Rayleigh-Ritz method with the polynomial basis sets $r^i(r_0-r)\cos(j\varphi)$ and $r^i(r_0-r)\sin(j\varphi)$; because the method yields eigenvalues that approach the exact ones from above, the apparent crossings and splittings have a variational guarantee of accuracy within the chosen truncation.

What would settle it

Substitute $R_{00}(r)=(1-r)e^{(1-r)/2}/\sqrt{3e-8}$ into the radial Schrödinger equation at $\beta=3/4$ and confirm $E=-1/8$; then compute $E_{10}-E_{02}$ at $\beta=3/4$ with an independent high-accuracy method, such as finite differences or a basis with exponential asymptotics. If the gap does not vanish to numerical precision, the three-fold accidental degeneracy is an artifact of the Rayleigh-Ritz truncation.

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Extended reading notes

Core claim

For the dimensionless Hamiltonian $H=-\tfrac12\nabla^2-\beta/r$ on the unit disk, the authors exhibit, at $\beta=3/4$, the closed-form radial ground state $R_{00}(r)=(1-r)e^{(1-r)/2}/\sqrt{3e-8}$ with eigenvalue $E_{00}=-1/8$; this function satisfies the wall condition $R_{00}(1)=0$ and also decays at infinity. Their Rayleigh-Ritz spectrum shows that this same coupling is where the states $E_{02}$ and $E_{10}$ cross, giving a three-fold accidental degeneracy: $E_{02}$ carries $|m|=2$ and is therefore two-fold, while $E_{10}$ carries $m=0$. The numerical results further suggest the stronger pattern $E_{n0}=E_{n-1,2}$ for all $n\ge1$ at this value of $\beta$. When the electric field with dimensionless strength $\lambda$ is turned on, the degeneracy splits; because the eigenvalues are even functions of $\lambda$, the spectrum is symmetric under reversal of the field, and avoided crossings occur only between states with the same parity under $\phi\to-\phi$.

Load-bearing premise

The numerical eigenvalues are treated as converged for every box radius and field strength shown, even though no truncation-error estimate or independent check is given for nonzero field and the polynomial basis is acknowledged to be unreliable for large boxes.

Editorial extensions

If this is right

  • At $\beta=3/4$ the confined atom has at least one exactly known eigenstate, so the model provides a closed-form benchmark for testing numerical methods on confined Stark systems.
  • The three-fold accidental degeneracy means a uniaxial electric field splits the level into distinct components, giving a concrete prediction for the Stark pattern of a confined two-dimensional hydrogen atom.
  • If the conjectured pattern $E_{n0}=E_{n-1,2}$ at $\beta=3/4$ holds, the spectrum at that box size has a hidden degeneracy structure absent at other sizes.
  • The symmetry $E(-\lambda)=E(\lambda)$ implies Stark shifts are invariant under reversing the field direction, a testable property of the confined model.
  • In the small-box limit $r_0^2 E_{n\nu}$ approaches the eigenvalues of a particle in a unit circular box, so the box radius acts as a tunable parameter interpolating between hard-wall and free-atom behaviour.

Reading between the lines

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

  • If the pattern $E_{n0}=E_{n-1,2}$ at $\beta=3/4$ is exact, the model likely possesses a hidden dynamical symmetry beyond its geometric symmetries; the closed form of $R_{00}$ resembles the ground state of a shape-invariant potential and may be the first rung of an exact ladder of solutions.
  • A direct extension is to compute the small-$\lambda$ Stark splitting of the degenerate manifold: a linear splitting would measure an effective dipole moment of the doublet, while a quadratic splitting would measure a polarizability, both quantities being checkable with the same Rayleigh-Ritz code at finer truncations.
  • Because the polynomial basis is designed for moderate box sizes, the large-$r_0$ portions of the spectrum would be a natural place to repeat with a basis carrying the correct exponential decay; the avoided crossings at large $r_0$ are the most likely features to be affected by truncation error.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper revisits the two-dimensional hydrogen atom confined to a circular impenetrable box, with the nucleus clamped at the origin, in the presence of a uniform electric field. Working with dimensionless Hamiltonians obtained by different length scalings, the authors derive scaling limits (r0→0 reduces to the particle in a circular box; r0→∞ recovers the free 2D hydrogen energies), prove that the eigenvalues are even functions of the field strength λ under a unitary transformation, and separate the even/odd parity sectors. The main numerical tool is a Rayleigh-Ritz method (RRM) with a truncated polynomial basis. At zero field, for the dimensionless box parameter β=3/4, the authors find a three-fold accidental degeneracy among the states (0,0), (0,2), and (1,0), and give an exact ground-state wavefunction R00(r) with energy -1/8 (Eq. 13). They conjecture the wider ladder E_{n0}=E_{n-1,2} at this β. For nonzero field, they present spectra showing the splitting of the degenerate levels and avoided crossings.

Significance. The paper contains several verifiable analytical results: the exact ground state in Eq. (13) checks by direct substitution; the small-box limit in Table 1 agrees with independent particle-in-a-box eigenvalues; and the symmetry E(-λ)=E(λ) is proven. If the accidental degeneracy is confirmed, the model is a useful benchmark for confined Stark systems. However, the central degeneracy claim currently rests on a single truncated RRM calculation with no convergence certification, which limits the paper's significance.

major comments (3)
  1. [Section 4, Figure 1] The three-fold degeneracy E00=E02=E10 at β=3/4 is the paper's central novel feature, but it is established only by the truncated N=12 RRM calculation. The exact state (13) proves only E00=-1/8; it does not constrain E02 or E10. Since RRM eigenvalues are upper bounds and the convergence rates differ between the ν=0 and ν=2 sectors, the apparent crossing in Fig. 1 could shift or disappear with increasing N. Please provide a convergence study (e.g., a table of E02(3/4) and E10(3/4) for increasing N, or an independent shooting-method calculation) before presenting the degeneracy as a fact in Section 5.
  2. [Section 3, Figures 2-4] No truncation-error estimate or convergence test is provided for any λ≠0 eigenvalue. The authors state that the polynomial basis (9) is 'not expected to be suitable for too large values of r0', yet Figs. 3 and 4 extend to r0=10. The assertion that the N=12 basis provides 'sufficiently accurate eigenvalues' is unsupported. Please provide convergence data for representative (r0,λ) points and either truncate the plotted range to the converged regime or quantify the error bars.
  3. [Section 2/5] The claimed discussion of the large-box-radius limit is incomplete for λ≠0. For λ>0, the potential -λ r cosφ is unbounded below as r→∞, so no bound-state limit r0→∞ exists for fixed λ; the large-r0 parts of Figs. 3-4 therefore require interpretation or removal. The paper discusses the r0→∞ limit only for f=0 (Eq. 8), which should be stated explicitly.
minor comments (4)
  1. [Section 2, Eq. (5)] The paper should state explicitly that after scaling L=ℏ²/(m_e K) the box radius becomes r0 (in units of L) and that this quantity coincides with the parameter β introduced in Section 4.
  2. [Section 4, Eq. (13)] The normalization constant is printed as '√3e − 8', which is ambiguous; it should read √(3e−8). Also, the remark 'R(r→∞)=0' is confusing for a finite-box problem and should be rephrased as 'the function (13) decays as r→∞' or removed.
  3. [Section 4 and Table 1] Typos: 'stisfies' should be 'satisfies' (Section 4); 'higly' should be 'highly' (Introduction); 'r2_0' should be 'r_0^2' in the Table 1 caption and in the text.
  4. [Section 5] For reproducibility, please include a small table of the lowest eigenvalues for representative parameters (e.g., r0=3/4 with λ=0 and λ=1) instead of relying entirely on figures without numerical data.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the exact-state check, scaling limits, and symmetry arguments are independent of the numerical fitting.

full rationale

The paper's argument chain is not circular. The dimensionless Hamiltonian follows from a scale transformation, and the symmetry E(λ)=E(−λ) is derived from the unitary transformation U2, not assumed. The r0→0 limit in Eq. (10) is checked against the independent particle-in-a-circular-box spectrum in Table 1. The exact state in Eq. (13), R00(r)=(1−r)e^{(1−r)/2}/√(3e−8) with E00=−1/8 at β=3/4, is a direct solution of the radial Schrödinger equation and can be verified by substitution; it is not fitted to the degeneracy E02=E10. The accidental-degeneracy observation is presented as a numerical finding from Rayleigh-Ritz eigenvalues, and the wider ladder E_{n0}=E_{n−1,2} is explicitly labeled a conjecture that the authors do not attempt to prove. Thus the central degeneracy claim is not imported as a premise. The self-citations in the paper concern dimensionless-equation scaling, variational upper-bound properties, and a comparison harmonic-oscillator model; none of them carries the load of the accidental-degeneracy claim. The lack of a convergence certificate for the N=12 truncation is a numerical rigor or correctness risk, not a circularity, because the Rayleigh-Ritz upper-bound property does not make the apparent crossing an input. Therefore no circular step is exhibited, and the score is 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities and fits no physical constants. It depends on the standard model assumptions (2D Coulomb potential, clamped nucleus, hard wall, uniform field) plus the unverified completeness of the polynomial basis. The only hand-chosen numerical parameter is the truncation N=12.

free parameters (1)
  • RRM basis truncation N = N=12 (i,j up to 12)
    Chosen by hand; all reported RRM eigenvalues depend on this truncation, and no systematic convergence test is provided.
assumptions (5)
  • domain assumption The Hamiltonian with 2D Coulomb potential -1/r and Stark term -lambda r cosphi is the model.
    Equation (6) sets the physical model; no derivation of the potential or field is given.
  • domain assumption The nucleus is clamped at the origin.
    Section 2 states this explicitly; it avoids nuclear motion and is an idealization relative to the treatment in Longo et al.
  • domain assumption The circular box has an impenetrable wall, so psi(r0,phi)=0.
    Section 2, boundary condition; basis functions are built to satisfy it exactly, but the infinite wall is an idealization.
  • standard math Rayleigh-Ritz upper-bound theorem: approximate eigenvalues converge from above.
    Invoked in Section 1 with references [12,13]; standard variational theorem.
  • ad hoc to paper The polynomial basis (9) truncated at N=12 is complete enough for the reported low-lying eigenvalues.
    No convergence proof or error estimate is supplied; accuracy is asserted and only the r0-to-0 limit is checked, so this is the load-bearing numerical assumption.

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Cite this review

Pith. "Pith review of On the two-dimensional hydrogen atom in a circular box in the presence of an electric field." pith.science (2026). https://pith.science/paper/HUF5GYOK

@misc{pith2026250418928,
  author       = {Pith},
  title        = {Pith review of: On the two-dimensional hydrogen atom in a circular box in the presence of an electric field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HUF5GYOK}},
  note         = {Machine review of arXiv:2504.18928}
}
read the original abstract

We revisit the quantum-mechanical two-dimensional hydrogen atom with an electric field confined to a circular box of impenetrable wall. In order to obtain the energy spectrum we resort to the Rayleigh-Ritz method with a polynomial basis sets. We discuss the limits of large and small box radius and the symmetry of the solutions of the Schr\"{o}dinger equation. An interesting feature of the model is the appearance of accidental degeneracy and the splitting of degenerate energy levels due to the presence of the electric field.

Figures

Figures reproduced from arXiv: 2504.18928 by the authors.

Figure 1
Figure 1. Dimensionless eigenvalues Enν(β) for some values of β. Each line is labelled by the numbers (n, ν) [11] Szabo A and Ostlund N S 1996 Modern Quantum Chemistry (Dover Pub￾lications, Inc., Mineola, New York). [12] MacDonald J K L 1933 Phys Rev. 43 830. [13] Fern´andez F M 2025 J. Math. Chem. 63 911. arXiv:2206.05122 [quant-ph] [14] Fern´andez F M 2024 J. Math. Chem. 62 2083. arXiv:2405.10340 [quant-ph] [15] Fern´andez … view at source ↗
Figure 2
Figure 2. Lowest eigenvalues for r0 = 3/4 and some values of λ 0 2 4 6 8 10 -4 -3 -2 -1 0 1 2 3 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Lowest eigenvalues for even states with λ = 1 and some values of r0 10 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Lowest eigenvalues for odd states with λ = 1 and some values of r0 11 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Cylindrically confined $H$ atom in magnetic field: variational cut-off factor

    quant-ph 2025-01 conditional novelty 6.0 of 10

    A variational calculation for a hydrogen atom in a cylindrical cavity with a magnetic field shows that making the cutoff exponent an adjustable parameter improves ground-state energies.

Reference graph

Works this paper leans on

16 extracted references · 13 canonical work pages · cited by 1 Pith paper

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