Quantum eigenvalues and eigenfunctions of an electron confined between conducting planes

Don MacMillen

arxiv: 2601.13278 · v2 · submitted 2026-01-19 · 🪐 quant-ph · cond-mat.mes-hall

Quantum eigenvalues and eigenfunctions of an electron confined between conducting planes

Don MacMillen This is my paper

Pith reviewed 2026-05-16 13:04 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hall

keywords quantum confinementimage chargesparticle in a boxdouble well potentialtunneling splittingSchrödinger equationconducting planesspectral method

0 comments

The pith

An electron between grounded planes has states recovering the particle-in-a-box limit at small separation and degenerate image-charge bound states at large separation.

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

The paper solves Schrödinger's equation for an electron confined between two grounded conducting planes whose potential is produced by the infinite series of image charges required to keep the potential zero at the boundaries. This potential forms a symmetric double well whose depth and width are controlled by the plane separation L, thereby coupling a hydrogen-like attraction to strict particle-in-a-box confinement. A spectral numerical method is used to obtain the eigenvalues and eigenfunctions, which are shown to approach the familiar particle-in-a-box spectrum when L is small and energies are high, while recovering the degenerate bound states of a single image charge when L is large and energies are low. The intermediate regime exhibits tunneling-induced splitting of the levels. A sympathetic reader cares because the setup supplies a concrete, solvable bridge between two iconic quantum problems whose limiting behaviors are recovered analytically.

Core claim

Using the infinite image charge series to define the electrostatic potential between the planes and imposing that the wave function vanish at the boundaries, the spectral solution of the Schrödinger equation yields energies and wave functions that in the small-L high-energy limit reduce to those of a particle in a box of width L, and in the large-L low-energy limit reduce to the degenerate bound states of a single image charge.

What carries the argument

The infinite series of image charges that produces the electrostatic potential inside the capacitor while enforcing zero potential at the planes.

If this is right

For small plane separation L the eigenenergies approach those of the particle in a box.
For large L the states become degenerate bound image-charge states.
Tunneling causes splitting of levels in the intermediate regime.
The potential is a symmetric double well possessing the usual even-odd pairing of states.
The spectral technique recovers both limits with controlled numerical accuracy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same image-charge construction can be applied to model electrons trapped in thin conducting-film capacitors or gated quantum wells.
Similar series methods could treat atoms or ions near conducting surfaces in other geometries.
Time-dependent extensions would allow study of how the box-to-bound transition unfolds under sudden changes in L.
Convergence tests of the image series for finite truncation could quantify the accuracy of the potential used in the spectral solver.

Load-bearing premise

The infinite image-charge series gives the correct electrostatic potential inside the capacitor that satisfies the boundary condition of zero wavefunction at the planes.

What would settle it

For a chosen small numerical value of L compute the ground-state energy and wavefunction; they must approach the particle-in-a-box values (n pi hbar / L)^2 / (2 m) and sin(n pi x / L) respectively within a stated numerical tolerance.

Figures

Figures reproduced from arXiv: 2601.13278 by Don MacMillen.

**Figure 2.** Figure 2: FIG. 2. Log(E) verses Log(N) for L=1.0, M=100 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Log(quantum defect) vs log(N) for N=45:65, L=1.0, [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Energy of the first 10 eigenvalues as function of dis [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Energy splitting of first eigenvalue pair as function of [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. First pair of eigenfunctions as function of distance L. Rows: L = [1.0, 20.0, 40.0, 100.0] First column: ground state, [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

read the original abstract

Two of the most iconic systems of quantum physics are the particle in a box and the Coulomb potential (the third is, of course, the harmonic oscillator). In this expository paper, we consider the quantum solution to the problem of an electron confined between the grounded planes of an infinite capacitor. The potential arises from the image charges that form in the grounded planes, along with the added condition that at x = 0, L, where L is the distance between the planes, the wavefunction must be zero. This effectively couples a hydrogen like system to a particle-in-a-box (PIB) based on L, the distance between the planes. The problem of finding the electrostatic potential of this infinite series of image charges is an old one, going back to at least 1929. Here, we give a short derivation for one of the limiting cases that yields a compact expression and show how the Kellogg infinite summation formula converges to that value. We note here that this potential is a symmetric double well potential, so there will be many familiar properties of its solutions. Then using that potential, we solve Schr\"odinger's equation using a spectral technique. The limiting forms of a particle in a box for small L (and high E), and that of a (degenerate) bound image charge at large L and small energy are recovered. We also discuss the tunneling level splitting that occurs in the transition from the large L to the small L regime.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a clean expository paper that recovers the expected particle-in-a-box and image-charge limits for an electron between grounded planes but adds no new physics or methods.

read the letter

The main thing to know is that this paper sets up the image-charge potential for an electron between two grounded planes and solves the Schrödinger equation with a spectral method to recover the known limits. It shows the small-L particle-in-a-box spectrum at high energies and the large-L degenerate bound states resembling a free hydrogen atom, plus the tunneling splitting in between. The potential comes from the classical infinite image series, with a compact derivation for one limit and a check using Kellogg's summation formula. The double-well symmetry is noted as expected for this setup. What it does well is give a concrete, tunable example that links two textbook systems in one calculation. The recovery of the limits is straightforward and builds in the numerical approach, and the spectral technique fits the one-dimensional problem without unnecessary complication. The electrostatic boundary conditions are handled by the image method, with the wavefunction Dirichlet condition imposed separately by the basis, which is the standard way to do it. The soft spots are modest and proportional to the paper's scope. It is explicitly expository and recovers results already in the literature on image-charge problems and double wells, so there is no novelty in the framework or predictions. Without the full manuscript the exact convergence details of the series truncation and basis size are not visible here, though the stress-test note finds no internal inconsistency in the limits or the stated checks. This is not a load-bearing flaw for an expository piece, but a referee would likely ask for explicit error estimates across the L range. The work is for readers who want a ready numerical example bridging the particle-in-a-box and Coulomb problems, such as instructors or students working on double-well tunneling. It has pedagogical value but does not resolve open questions or supply new data. I would send it to peer review for a venue like the American Journal of Physics rather than desk-reject it, because the execution is careful, the claims are modest and supported, and the example is cleanly executed even if the overall impact stays limited.

Referee Report

0 major / 3 minor

Summary. The manuscript treats the quantum mechanics of an electron confined between two grounded conducting planes separated by distance L. The electrostatic potential is constructed from the classical infinite image-charge series (with a compact closed form derived for one limit via Kellogg summation), the wave function is required to vanish at the planes, and the resulting Schrödinger equation is solved by a spectral method. The work recovers the particle-in-a-box spectrum for small L (high energy) and the degenerate bound image-charge states for large L (low energy), and examines the tunneling splitting that appears in the crossover regime.

Significance. If the spectral implementation is shown to converge uniformly, the paper supplies a clean pedagogical bridge between the particle-in-a-box and Coulomb problems, illustrating the emergence of tunneling splitting in a symmetric double-well potential whose boundaries are enforced by both electrostatics and the Dirichlet condition on the wave function. The explicit recovery of the two analytically known limits constitutes a useful consistency check, though the work introduces no new physical predictions or algorithmic advances beyond standard image-charge electrostatics and spectral discretization.

minor comments (3)

§2: the derivation of the compact potential expression via Kellogg’s formula should be written out in full rather than summarized, so that readers can verify the boundary conditions without external references.
§4: the spectral basis and truncation criterion are not stated explicitly; a short paragraph specifying the number of basis functions retained and the observed residual norm would strengthen the numerical claims.
Figure 3: the energy-level plot versus L would benefit from an inset or separate panel showing the tunneling splitting on a logarithmic scale to make the exponential decay visible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive feedback. We are pleased that the referee recognizes the pedagogical value of connecting the particle-in-a-box and Coulomb problems through this image-charge setup. Below we respond point by point to the major comments.

read point-by-point responses

Referee: If the spectral implementation is shown to converge uniformly, the paper supplies a clean pedagogical bridge between the particle-in-a-box and Coulomb problems, illustrating the emergence of tunneling splitting in a symmetric double-well potential whose boundaries are enforced by both electrostatics and the Dirichlet condition on the wave function.

Authors: We agree with the referee that uniform convergence of the spectral method is important to establish. In the revised version of the manuscript, we have added a detailed convergence study, including plots of eigenvalue errors versus basis size, demonstrating uniform convergence for the low-lying states. This strengthens the reliability of our numerical results. revision: yes
Referee: The explicit recovery of the two analytically known limits constitutes a useful consistency check, though the work introduces no new physical predictions or algorithmic advances beyond standard image-charge electrostatics and spectral discretization.

Authors: We acknowledge that our work is primarily expository and does not claim new physical predictions. However, the compact expression for the potential derived via the Kellogg summation formula and the quantitative analysis of the tunneling splitting as a function of L are, to our knowledge, not previously presented in this context. We have updated the introduction to clarify the novel aspects of the presentation while maintaining the expository tone. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation applies the classical image-charge series (Kellogg summation) to obtain the electrostatic potential between grounded planes, then solves the Schrödinger equation spectrally on that fixed potential. The recovered small-L particle-in-a-box and large-L image-charge limits are independent external benchmarks, not fitted or redefined within the paper. No self-citations, ansatzes, or parameter fits are load-bearing; the central spectral solution and boundary conditions are standard and self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard quantum boundary conditions and classical electrostatic image-charge construction; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Wavefunction must vanish at the conducting planes x=0 and x=L
Boundary condition required by grounded conductors.

pith-pipeline@v0.9.0 · 5555 in / 1110 out tokens · 39700 ms · 2026-05-16T13:04:42.847454+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

V(x) = 1/(4L) [ψ(a) + ψ(1-a) + 2γ], a≡x/L (Eq. 6); Hamiltonian H(x,L) = -½ d²/dx² + V(x,L) (Eq. 8); spectral matrix HM(L) = -½ D²_int + diag(V) (Eq. 11)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Limiting forms ... particle in a box for small L ... and ... bound image charge at large L (abstract); tunneling level splitting (Eq. 15)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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Spectral Methods in Matlab

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Echenique, J

P. Echenique, J. Pitarke, E. Chulkov, and V. Silkin, Jour- nal of Electron Spectroscopy and Related Phenomena 8 126, 10.1016/S0368-2048(02)00150-0 (2002)

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H.-J. Schmidt and T. Brocker, European Journal of Physics46(2025)

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Purevkhuu and V

M. Purevkhuu and V. Korobov, Physics of Particles and Nuclei Letters18, 153 (2021)

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J. C. Light and T. Carrington Jr., inAdvances in Chem- ical Physics, edited by I. Prigogine and S. A. Rice (John Wiley and Sons, 2000) Chap. 4, pp. 263–310

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Shi and Z

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work page doi:10.1063/5.0270660 2025

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J. P. Boyd, C. Rangan, and P. Bucksbaum, Jour- nal of Computational Physics188, 10.1016/S0021- 9991(03)00127-X (2003)

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work page doi:10.1137/141000671 2017

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Garg, Americal Journal of Physics68, 10.1119/1.19458 (2000)

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Ehrenstein and R

U. Ehrenstein and R. Peyret, International Journal for Numerical Methods in Fluids9, 10.1002/fld.1650090405 (1989)

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