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

A. M. Escobar-Ruiz; A. N. Mendoza Tavera; H. Olivares-Pil\'on; M. Rodr\'iguez-Arcos

REVIEW 2 major objections 7 minor 39 references

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

T0 review · 2 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that treating the cut-off factor exponent ν as a variational parameter significantly improves variational ground-state energies for a hydrogen atom in a cylindrical cavity with an axial magnetic field.

desk verdict A modest but genuine variational improvement—promoting the cutoff exponent to a variational parameter—sits behind an overbroad <1% accuracy claim that only B=0 benchmarks actually support. read the letter →

arxiv 2501.14297 v4 pith:XT5WPRII submitted 2025-01-24 quant-ph

A. N. Mendoza Tavera , H. Olivares-Pilón , M. Rodríguez-Arcos , A. M. Escobar-Ruiz This is my paper · ORCID

classification quant-ph MSC 81V4581Q0581Q10 PACS 31.15.Pf32.60.+i

keywords confinedhydrogenatomcylindricalcavitymagneticfieldvariationalmethodcut-offfactorgroundstateenergyShannonentropybinding

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 studies the ground state of a hydrogen atom confined in an impenetrable infinite cylindrical cavity of radius ρ0, with a constant magnetic field B along the cylinder axis and the proton fixed at the center. It minimizes a compact three-parameter trial wavefunction over (α, β, ν); the trial function is ψ = (1 − (ρ/ρ0)^ν) $e^{{−α r − β B ρ^2}}$, where the cut-off factor enforces the hard wall at ρ = ρ0. The paper's central claim is that promoting ν from a fixed constant (usually 1) to a variational parameter lowers the variational energy and is demonstrated here for the first time. With optimized ν, energies deviate from high-precision numerical benchmarks by less than 1% for small magnetic fields and moderate confinement radii, and the cusp condition is satisfied to within about 10% at ρ0 = 2 a.u. and better than 5% at ρ0 = 5 a.u. The authors also compute ⟨ρ⟩, ⟨|z|⟩, and position-space Shannon entropy to characterize the electron-cloud localization.

What carries the argument

The central object is the trial wavefunction ψ(ρ, r) = (1 − (ρ/ρ0)^ν) $e^{{−α r − β B ρ^2}}$ used in the (N, m, p) representation adapted to cylindrical symmetry. The cut-off factor (1 − (ρ/ρ0)^ν) carries the hard-wall boundary condition at ρ = ρ0, and the paper's contribution is to determine ν variationally rather than fix it. The exponential $e^{{−α r}}$ accounts for the Coulomb cusp and $e^{{−β B ρ^2}}$ for the transverse magnetic (Landau-like) compression, while the reduced operator H_Γ obtained from the full Hamiltonian by separating the azimuthal angle provides the energy functional that is minimized.

What would settle it

At B = 0.8 a.u. and ρ0 = 1.8 a.u., the paper reports E ≈ −0.134 a.u. A converged independent numerical solution of the same Schrödinger equation (e.g., finite elements with mesh refinement) that is more than 1% below this value, or that yields the same energy with ν fixed to 1, would show that the variational cut-off factor is not responsible for the claimed improvement. A simpler check at B = 0: at ρ0 = 2.5 a.u. the paper predicts E ≈ −0.405 a.u.; an accurate calculation differing by more than 1% would violate the stated accuracy.

Watch

Extended reading notes

Core claim

For the ground state (N=0, m=0, p=0), the trial wavefunction ψ = (1 − (ρ/ρ0)^ν) $e^{{−α r − β B ρ^2}}$ is minimized with respect to α, β, ν. At zero magnetic field, the optimized ν grows from about 2.3 at ρ0 = 0.8 a.u. to about 7.8 at ρ0 = 5.0 a.u., and fixed choices ν = 1, 2, 3 give noticeably higher energies. The variational energy at B = 0 is E = −0.2767 a.u. at ρ0 = 2.0 a.u. versus the numerical benchmark −0.279120 a.u. (about 0.9% deviation) and E = −0.4917 a.u. at ρ0 = 4.0 a.u. versus −0.491863 a.u. (about 0.03% deviation). For B ∈ [0.1, 1.0] a.u., energies remain smooth in B and ρ0; for small radii the energy turns positive, and those states are interpreted as resonances rather than bound states. The authors present this as the first demonstration that a variational cut-off factor improves the accuracy of variational calculations for confined atoms.

Load-bearing premise

The trial wavefunction's rigid product form — a single exponential $e^{{−α r}}$ times a power-law cut-off — is assumed flexible enough to capture both the electron-nucleus cusp and the hard-wall boundary layer at all B and ρ0; if this factorization is too rigid, the optimized parameters cannot compensate and the claimed accuracy degrades.

Editorial extensions

If this is right

Optimizing ν lowers the variational ground-state energy compared with the fixed values ν = 1, 2, 3 across the studied range of ρ0 and B.
The resulting energies agree with high-precision numerical methods to about 1% or better for small magnetic fields and moderate confinement radii, so the compact trial function is adequate for quick estimates.
The cusp condition is preserved to within roughly 10% at ρ0 = 2 a.u. and within 5% at ρ0 = 5 a.u., indicating the trial function keeps the short-distance electron-nucleus behavior reasonably faithful.
For cavity radii ≲ 2 a.u., the ground-state energy becomes positive; the paper interprets these states as resonances, meaning the variational results in that region should not be treated as bound-state predictions.

Reading between the lines

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

The paper reports that different pairs (β, ν) with the same α ≈ 1 produce nearly identical energies; a follow-up study could test whether the energy surface is flat along that combination, in which case a one-parameter relation between β and ν would reproduce the same results.
The same idea of a variational cut-off factor should transfer to other confining geometries, such as spherical or spheroidal cavities, and to two-electron or molecular systems where the boundary factor is normally fixed.
Because the positive-energy variational solutions are interpreted as resonances, an extension using complex scaling or absorbing potentials could extract resonance positions and widths from the same compact ansatz.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The one real idea here is to treat the cutoff exponent ν in F(ρ) = (1 − (ρ/ρ₀)^ν) as a variational parameter instead of fixing it to 1, 2, or 3. That is new, as far as the cited literature goes, and it buys a real improvement: at B = 0, optimization lowers the energy from −0.2757 to −0.2767 at ρ₀ = 2 and from −0.4894 to −0.4917 at ρ₀ = 4, landing within 0.9% and 0.03% of the B-spline numbers in Ref. [25]. The cusp condition is satisfied to 5–10%, and the Shannon entropy in the free-atom limit matches the known value. For what it is—a compact three-parameter trial function for quantum-wire modeling—that is a solid, honest piece of variational work.

The soft spots are in the packaging, not the method. The abstract and conclusion say energies deviate by less than 1% from “more precise numerical methods” across the (B, ρ₀) domain, but the only independent numerical comparison in the paper is the two zero-field points in Table I. At B > 0 the benchmark is another variational ansatz, ψ_alt, which is not an accurate solution; the stated 8–34% improvements over it do not establish 1% accuracy. The stress-test note lands: the B = 0 tests cannot validate the B > 0 regime, where the magnetic factor e^{−βBρ²} enters and the ansatz has to reproduce the cusp, the Landau localization, and the wall boundary simultaneously. I would soften the claim to “energies comparable to other simple variational results” until an external B > 0 comparison exists.

Minor issues: the positive-energy states are called resonances but no resonance characterization is attempted, which is fine as a caveat but should stay a caveat. The asymptotic fit section reports E ≈ −0.5 + A/ρ₀^α with A ≈ 0.4, α ≈ 2, but the table values around ρ₀ = 4–5 give A closer to 0.05–0.13, and the decay is steeper than quadratic; that fit looks like a rough guess. The claim that bound states have ρ₀ above 2 is slightly off for B = 0.4, where ρ₀ = 1.8 is still negative.

Bottom line: worth a serious referee if the authors trim the accuracy claim to what is demonstrated. The variational cutoff idea is a legitimate contribution to an established program, and the paper is clearly written with the relevant prior work cited. I would probably not cite it myself, but I would not object to its publication in a specialized venue after revision. Yes, send to peer review.

Referee Report

2 major / 7 minor

Summary. The manuscript studies the non-relativistic ground state of a hydrogen atom with an infinitely massive proton at the center of an infinite impenetrable cylindrical cavity of radius ρ0, in a uniform magnetic field B aligned with the cylinder axis. Using the (N,m,p) representation adapted to cylindrical symmetry, the authors propose the compact trial function Ψ = (1 − (ρ/ρ0)^ν) e^{−αr − βBρ²} and minimize the energy expectation value with respect to α, β, and ν. They report E(ρ0,B), binding energies, ⟨ρ⟩, ⟨|z|⟩, and the Shannon entropy for ρ0 ∈ [0.8, 5] a.u. and B ∈ [0, 1] a.u. The central claims are that treating the cutoff exponent ν variationally substantially improves accuracy relative to fixed ν, and that the resulting energies agree with more precise numerical methods to better than 1%; the B=0 comparison against Ref. [25] yields 0.9% and 0.03% at ρ0 = 2 and 4 a.u.

Significance. The main idea — promoting the cutoff factor's exponent to a variational parameter — is physically sensible and potentially useful for confined Coulomb problems where the boundary layer, Coulomb cusp, and magnetic localization compete. The paper computes analytically manageable matrix elements and presents transparent tables of optimized parameters. At B=0, the two-point comparison with B-spline results is a genuine independent check and supports the method in that limit. The cusp-condition diagnostics and the recovery of the free-atom limit at large ρ0 are additional honest checks. However, the headline <1% accuracy claim is not established for B>0, where the trial function changes character through the βBρ² term; the only nonzero-field independent check reported is for the noninteracting Landau-in-cylinder system (Eq. 15), not for the full Coulomb problem. The absence of a B>0 benchmark limits the significance of the quantitative results, though not the interest of the method itself.

major comments (2)

[Abstract, §V; Table I vs Table III] The statement that the variational energies 'deviate by no more than 1%' from more precise numerical methods is not supported for B>0. Table I provides independent B-spline numbers only at B=0, for ρ0 = 2 and 4 a.u. All B>0 entries in Table III are purely variational; the comparison against ψ_alt is against another trial function, and the E0 comparison in §III C 1 tests the noninteracting Landau-in-cylinder problem, not the Coulomb problem at B>0. Because the magnetic factor e^{−βBρ²} is a new element of the ansatz in that regime, the B=0 validation cannot certify <1% accuracy there. Please either add an independent numerical comparison at representative nonzero B values (e.g., using the B-spline or finite-difference methods of Refs. [25,26]) or explicitly restrict the accuracy claim to B=0.
[§III C, Table III, Fig. 3] Several entries in Table III have positive energies and are described as lying in the continuum or being resonances. For such states the Ritz variational principle does not yield upper bounds, and the manuscript itself states that the variational method 'no longer accurately reflects the physical character of the state' in this regime. Despite this, these numbers are used on equal footing for the binding energy Eb, the localization observables, and the Shannon entropy. Please restrict the quantitative tables to the bound-state regime, or use a well-defined resonance method, and clearly separate positive-energy entries from genuine ground-state results.

minor comments (7)

[§III, first paragraph] The system is described as being confined by a 'spherical cavity of radius ρ0'; it should be a cylindrical cavity.
[§III A] The text contains the typo 'Laudau-like orbitals'; it should read 'Landau-like orbitals'.
[§III C, near Fig. 3] The sentence referring to 'the behavior of the optimal variational parameters ... as functions of the variational parameter ρ' should refer to the confinement radius ρ0, not to a variational parameter.
[§III C, Table III] The negative values of β (e.g., −0.230 at B=0.4, ρ0=2.5) make e^{−βBρ²} grow radially; although the cutoff factor still enforces the Dirichlet boundary condition, the label 'Landau-like orbital' is then misleading, and the physical interpretation of negative β should be discussed.
[Appendix B] 'Table VII' appears as a caption/paragraph rather than a properly formatted numbered table; if kept, it should be presented as an actual table.
[§II, Eq. (6) and following] The volume element is written as d³r ∝ rρ/√(r²−ρ²) dr dρ dφ, but the normalization constant and the integration limits over (ρ,r) are not given, which makes it harder to reproduce the reported variational integrals.
[§V, Conclusions] The conclusion states that the ansatz 'allows for a closed-form interpolation across the parameter space,' but no interpolation formula or accuracy estimate is actually provided; either supply it or remove the claim.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: variational parameters are optimized, not fitted to target energies; the central claim rests on an independent B=0 B-spline benchmark.

full rationale

The paper's central result is a Rayleigh-Ritz variational calculation. The three parameters (alpha, beta, nu) are obtained by minimizing the energy expectation value of the Hamiltonian in Eq. (6) with potential (7), not by fitting to known energies. The improved energy relative to fixed-nu cutoffs follows from enlarging the variational family, and the numerical magnitudes are computed rather than imported. The accuracy claim is anchored by Table I, which compares two zero-field energies with independent B-spline results (Ref. [25]) and reports about 0.9% and 0.03% agreement; this is a genuine external benchmark. The B>0 energies in Table III are compared only with another variational ansatz, so the claimed <1% agreement for B>0 is under-supported as an evidence matter, but that is a correctness risk rather than circularity. The (N,m,p) representation is taken from Refs. [24,35], which include one of the present authors, but this representation is a coordinate/quantum-number relabeling and is not load-bearing for the variational energy results. No equation reduces by construction to its own inputs, and no fitted parameter is renamed as a prediction. Score 1 reflects only the minor, non-load-bearing self-citation overlap.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The central calculation depends on the standard variational principle plus a specific trial-function shape. The three variational parameters are optimized by energy minimization, not by fitting to known target energies, so the method is not circular in the usual sense. However, the broad accuracy claim for B>0 relies on this shape being adequate, and that is not independently verified.

free parameters (5)

alpha = ranges from 1.396 at rho0=0.8 to 1.004 at rho0=5 at B=0; see Tables II and III
Variational exponent in the Coulomb orbital e^{-alpha r}; minimized for each (B, rho0).
beta = ranges from about 0.772 to -0.230 in Table III depending on B and rho0
Variational coefficient in the Landau-like factor e^{-beta B rho^2}; minimized.
nu = about 2.3 to 7.8 at B=0; roughly 1.8 to 2.8 in Table III
Exponent in the cutoff factor (1-(rho/rho0)^nu); promoted to a variational parameter, the paper's central methodological novelty.
gamma = not tabulated
Extra variational parameter in the unconfined trial function Psi_infty = (1+gamma^2 rho^2) e^{-alpha r - beta B rho^2} used for <rho> at rho0=infinity.
A and exponent in the asymptotic fit E(rho0) = -0.5 + A/rho0^alpha = A about 0.4, exponent about 2
Asymptotic fit to the computed B=0 energies; not central to the main claim but is a fitted result.

assumptions (7)

domain assumption The Born-Oppenheimer approximation with an infinitely massive proton fixed at the cylinder center
Used throughout; the manuscript states the proton is located at the geometric center and only the electron is treated quantum mechanically.
domain assumption Non-relativistic Schrodinger equation with the symmetric-gauge vector potential A = (1/2) B x r
Section III defines the Hamiltonian and the Zeeman terms in this gauge; gauge invariance is asserted but not demonstrated.
domain assumption Hard-wall confinement by an infinite potential at rho = rho0
Equation (9) defines Vconf as 0 inside and infinity at the cylinder surface.
standard math Validity of the reduced two-dimensional Hamiltonian H_Gamma from Ref. [24]
Equation (6) is taken from the cited literature and used to reduce the 3D problem using azimuthal symmetry.
domain assumption The ground state lies in the m=0, p=0 sector
The paper studies only positive parity and zero magnetic quantum number, justified by symmetry but not verified for all B and rho0.
ad hoc to paper The variational ansatz in Eq. (10) is flexible enough to approximate the true ground state
The product of Coulomb and Landau orbitals with the power-law cutoff is assumed without a convergence study; this is the main modeling assumption.
standard math Variational principle: minimizing the expectation value gives an upper bound to the true ground-state energy
Standard quantum mechanics; used to optimize alpha, beta, and nu.

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Cite this review

Pith. "Pith review of Cylindrically confined $H$ atom in magnetic field: variational cut-off factor." pith.science (2026). https://pith.science/paper/XT5WPRII

@misc{pith2026250114297,
  author       = {Pith},
  title        = {Pith review of: Cylindrically confined $H$ atom in magnetic field: variational cut-off factor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XT5WPRII}},
  note         = {Machine review of arXiv:2501.14297}
}

abstract

In the present study, we consider the hydrogen atom confined within an impenetrable infinite cylindrical cavity of radius $\rho_{0}$ in the presence of a constant magnetic field ${\bf B} = B\,\hat{\bf z}$ oriented along the main cylinder's axis. In the Born-Oppenheimer approximation, anchoring the nucleus to the geometric center of the cylinder, a physically meaningful 3-parametric trial function is used to determine the ground state energy $E$ of the system. This trial function incorporates the exact symmetries and key limiting behaviors of the problem explicitly. In particular, it does not treat the Coulomb potential nor the magnetic interaction as a \textit{perturbation}. The novel inclusion of a variational cut-off factor $\big(1 - \big(\frac{\rho}{\rho_0}\big)^\nu\big)$, $\nu \geq 1$, appears to represent a significant improvement compared to the non-variational cut-off factors commonly employed in the literature. The dependence of the total energy $E=E(\rho_0,\,B)$ and the binding energy $E_b=E_b(\rho_0,\,B)$ on the cavity radius $\rho_0 \in [0.8,\,5] \,$a.u. and the magnetic field strength $B\in [0.0,\,1.0]\,$a.u. is presented in detail. The expectation values $\langle \rho \rangle$ and $\langle|z| \rangle$, and the Shannon entropy in position space are computed to provide additional insights into the system's localization. A brief discussion is provided comparing the 2D and 3D cases as well.

Figures

Figures reproduced from arXiv: 2501.14297 by the authors.

**Figure 1.** Geometric configuration of the cylindrically confined hydrogen atom in a constant magnetic field. The proton (p) is located at the origin, and the magnetic field is aligned along the cylinder’s axis. The variables ρ and r, for the electron (e), that naturally appear in the potential (7) are displayed as well. The radius of the infinite cylinder is denoted by ρ0. shell of radius ρ0. This is modeled by the confining p… view at source ↗

**Figure 2.** a) Variational energy E for the ground state (m = 0, p = 0) of the hydrogen atom confined by an impenetrable cylinder of radius ρ0 at zero magnetic field B = 0. The small red square correspond to the expectation value for the free case ⟨ρ⟩ρ0=∞. The optimal values of the parameters α and ν are depicted in b) and c), respectively. Positive-energy states lie in the continuum and are interpreted as resonances, rather th… view at source ↗

**Figure 3.** Variational energy E and optimal parameters {α, β, ν} for the ground state (m = 0, p = 0) of the hydrogen atom as a function of the cylinder radius ρ0 at different fixed magnetic fields B. Figure (e) presents the energy as a function of the magnetic field for fixed values of the cylindrical cavity radii ρ0. Positive-energy states lie in the continuum and are interpreted as resonances, rather than genuine bound state… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Ground state orbital configurations for different magnetic fields and confining radii. cases ρ0 = 1.0, 1.4, 2.0 and 5.0 a.u., the ground state (m = 0, p = 0) energy obtained from our trial function (10) shows a relative improvement of 8%, 34%, 17% and 2%, respectively,…

**Figure 5.** Figure 5: Binding energy Eb of the confined hydrogen atom in the ground state m = 0, p = 0 (a) as a function of the radius of the cylindrical cavity ρ0 for fixed values of magnetic field B and (b) as a function of the magnetic field B for fixed values of the cylindrical cavity r…

**Figure 6.** Figure 6: Ratio of the energies E (3D)/E(2D) of the three-dimensional hydrogen atom E (3D) confined by an impenetrable cylindrical cavity of infinite height and the two-dimensional hydrogen atom E (2D) confined to a circular region as a function of the same cavity radius ρ0 for …

**Figure 7.** Figure 7: Expectation value of the transversal size (a) [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]

**Figure 8.** Figure 8: Shannon entropy as a function of the magnetic field B. B=0 B=0.4 B=0.6 B=0.8 B=1 2 4 6 8 10 2.5 3.0 3.5 4.0 ρ0 [a.u.] Sr [PITH_FULL_IMAGE:figures/full_fig_p025_8.png]

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Reference graph

Works this paper leans on

39 extracted references · 37 canonical work pages

[25]

Z. Xiao, J. Zhu, and F. He, Physica Status Solidi (B) 191, 401 (1995)

work page 1995
[1]

This is exactly analogous to the case of a particle in a one-dimensional infinite potential well, where the quantized kinetic energy levels scale as ∼ 1/L2, with L being the length of the box. Therefore, the observed power-law behavior not only confirms the accuracy of the varia- tional approach in the weak confinement regime but also provides a bridge be...

work page
[2]

Binding energy Eb As usual, the binding energy Eb Eb = E0 − E , (14) defined as the difference between the ground state energy E0 of the electron in an axial constant magnetic field confined within an impenetrable cylinder only and the energy E when the Coulomb interaction is introduced. As is well known, for the ground state, E0 can 17 ● B=0.1 ● B=0.3 ● ...

work page
[3]

Continuum states In the presence of a uniform magnetic field aligned B > 0 along the z-axis and an impenetrable cylindrical boundary of radius ρ0, the hydrogen atom exhibits a hybrid con- finement regime in which the rotational symmetry of the Coulomb potential is broken by both the field and the geometry. The continuum states of this system (7) are defin...

work page
[4]

2D versus 3D cases A remarkable feature of the free hydrogen atom (B = 0, ρ0 = ∞) is that the energy of the ground state for the two-dimensional case is four times larger than the energy for the three- dimensional case E(2D) = 4 E(3D). In order to establish a similar comparison in the system under discussion, it is necessary to consider the two-dimensiona...

work page 2023
[5]

Sabin, E

J. Sabin, E. Brandas, and S. A. C. (ed), Advances in Quantum Chemistry. Oxford: Academic Press 57, 1 (2009)

work page 2009
[6]

Sabin, E

J. Sabin, E. Brandas, and S. A. C. (ed), Advances in Quantum Chemistry. Oxford: Academic Press 58, 1 (2009)

work page 2009
[7]

Zhu and X

J.-L. Zhu and X. Chen, Phys. Rev. B 50, 4497 (1994)

work page 1994

Show all 39 references

[8]

Yeshchenko, I

O. Yeshchenko, I. Dmitruk, S. Koryakov, I. Pundyk, and Y. Barnakov, Solid State Commu- nications 133, 109 (2005)

2005
[9]

Cioslowski, Journal of the American Chemical Society 113, 4139 (1991)

J. Cioslowski, Journal of the American Chemical Society 113, 4139 (1991)

1991
[10]

Shameema, C

O. Shameema, C. N. Ramachandran, and N. Sathyamurthy, The Journal of Physical Chem- istry A 110, 2 (2006), pMID: 16392831

2006
[11]

Patil and Y

S. Patil and Y. Varshni (Academic Press, 2009) pp. 1–24

2009
[12]

Ley-Koo (Academic Press, 2009) pp

E. Ley-Koo (Academic Press, 2009) pp. 79–122

2009
[13]

Ley-Koo and S

E. Ley-Koo and S. A. Cruz, The Journal of Chemical Physics 74, 4603 (1981), https://pubs.aip.org/aip/jcp/article-pdf/74/8/4603/18928877/4603 1 online.pdf

1981
[14]

G. M. Longo and S. Longo, International Journal of Quantum Chemistry 124, e27392 (2024), https://onlinelibrary.wiley.com/doi/pdf/10.1002/qua.27392

2024 doi
[15]

E. M. Nascimento, F. V. Prudente, M. N. Guimar˜ aes, and A. M. Maniero, Journal of Physics B: Atomic, Molecular and Optical Physics 44, 015003 (2010)

2010
[16]

Olivares-Pil´ on, A

H. Olivares-Pil´ on, A. M. Escobar-Ruiz, M. A. Quiroz-Ju´ arez, and N. Aquino, Machine Learn- ing: Science and Technology 4, 015024 (2023)

2023
[17]

Becken, P

W. Becken, P. Schmelcher, and F. K. Diakonos, Journal of Physics B: Atomic, Molecular and Optical Physics 32, 1557 (1999)

1999
[18]

G. W. F. Drake, Springer Handbook of Atomic, Molecular, and Optical Physics(Springer New York, NY, NY, 2006)

2006
[19]

Ruder, G

H. Ruder, G. Wunner, H. Herold, and F. Geyer, Atoms in Strong Magnetic Fields(Springer Berlin, Heidelberg, Heidelberg, 2011)

2011
[20]

Thirumalai and J

A. Thirumalai and J. S. Heyl, Phys. Rev. A 79, 012514 (2009). 28

2009
[21]

Thirumalai, The Journal of Physical Chemistry A 129, 2161 (2025), pMID: 39999415, https://doi.org/10.1021/acs.jpca.4c07546

A. Thirumalai, The Journal of Physical Chemistry A 129, 2161 (2025), pMID: 39999415, https://doi.org/10.1021/acs.jpca.4c07546

2025 doi
[22]

Arora, R

M. Arora, R. Giri, Varsha, K. Kumar, C. Yadav, B. Vidhani, M. Gambhir, and V. Prasad, Results in Physics 45, 106225 (2023)

2023
[23]

Kumar, S

K. Kumar, S. L. Talwar, S. Lumb, and V. Prasad, AIP Conference Proceedings 2220, 020098 (2020)

2020
[24]

Niculescu, A

E. Niculescu, A. Gearba, G. Cone, and C. Negutu, Superlattices and Microstructures 29, 319 (2001)

2001
[26]

S. V. Branis, G. Li, and K. K. Bajaj, Phys. Rev. B 47, 1316 (1993)

1993
[27]

Reyes-Garc ´ ıa, S

R. Reyes-Garc ´ ıa, S. A. Cruz, and R. Cabrera-Trujillo, International Journal of Quantum Chemistry 124, e27441 (2024)

2024
[28]

J. C. del Valle, A. V. Turbiner, and A. M. E. Ruiz, Phys. Rev. A 103, 032820 (2021)

2021
[29]

S. A. Ndengu´ e, O. Motapon, R. L. M. Melono, and A. J. Etindele, Journal of Physics B: Atomic, Molecular and Optical Physics 47, 015002 (2013)

2013
[30]

P. V. Yurenev, A. V. Scherbinin, and V. I. Pupyshev, International Journal of Quantum Chemistry 108, 2666 (2008)

2008
[31]

Amore and F

P. Amore and F. M. Fern´ andez, arXiv e-prints (2025), 10.48550/arXiv.2504.18928

2025 doi arXiv
[32]

L. D. Landau and E. M. Lifshits, Quantum Mechanics: Non-Relativistic Theory, Course of Theoretical Physics, Vol. v.3 (Butterworth-Heinemann, Oxford, 1991)

1991
[33]

L. P. Gor’kov and I. E. Dzyaloshinskiˇi, Soviet Journal of Experimental and Theoretical Physics 26, 449 (1968)

1968
[34]

A. V. Turbiner and Juan Carlos L´ opez Vieyra, Physics Reports 424, 309 (2006)

2006
[35]

Adesso, N

G. Adesso, N. Datta, M. Hall, and T. Sagawa, Journal of Physics A: Mathematical and Theoretical 52, 320201 (2019)

2019
[36]

C. E. Shannon, The Bell System Technical Journal 27, 379 (1948)

1948
[37]

S. J. C. Salazar, H. G. Laguna, B. Dahiya, V. Prasad, and R. P. Sagar, The European Physical Journal D 75, 127 (2021)

2021
[38]

H. G. Laguna, S. J. C. Salazar, and R. P. Sagar, International Journal of Quantum Chemistry 119, e25984 (2019), https://onlinelibrary.wiley.com/doi/pdf/10.1002/qua.25984. 29

2019 doi
[39]

A. V. Turbiner and A. M. Escobar-Ruiz, Journal of Physics: Conference Series 2667, 012075 (2023). Appendix A: F ree unconfined Hydrogen atom: representation(N, p, m) vs represen- tation (n, ℓ, m) In this Appendix, for the free 3D hydrogen atom, the explicit relation between th...

2023

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [25]

Z. Xiao, J. Zhu, and F. He, Physica Status Solidi (B) 191, 401 (1995)

work page 1995

[2] [1]

This is exactly analogous to the case of a particle in a one-dimensional infinite potential well, where the quantized kinetic energy levels scale as ∼ 1/L2, with L being the length of the box. Therefore, the observed power-law behavior not only confirms the accuracy of the varia- tional approach in the weak confinement regime but also provides a bridge be...

work page

[3] [2]

Binding energy Eb As usual, the binding energy Eb Eb = E0 − E , (14) defined as the difference between the ground state energy E0 of the electron in an axial constant magnetic field confined within an impenetrable cylinder only and the energy E when the Coulomb interaction is introduced. As is well known, for the ground state, E0 can 17 ● B=0.1 ● B=0.3 ● ...

work page

[4] [3]

Continuum states In the presence of a uniform magnetic field aligned B > 0 along the z-axis and an impenetrable cylindrical boundary of radius ρ0, the hydrogen atom exhibits a hybrid con- finement regime in which the rotational symmetry of the Coulomb potential is broken by both the field and the geometry. The continuum states of this system (7) are defin...

work page

[5] [4]

2D versus 3D cases A remarkable feature of the free hydrogen atom (B = 0, ρ0 = ∞) is that the energy of the ground state for the two-dimensional case is four times larger than the energy for the three- dimensional case E(2D) = 4 E(3D). In order to establish a similar comparison in the system under discussion, it is necessary to consider the two-dimensiona...

work page 2023

[6] [5]

Sabin, E

J. Sabin, E. Brandas, and S. A. C. (ed), Advances in Quantum Chemistry. Oxford: Academic Press 57, 1 (2009)

work page 2009

[7] [6]

Sabin, E

J. Sabin, E. Brandas, and S. A. C. (ed), Advances in Quantum Chemistry. Oxford: Academic Press 58, 1 (2009)

work page 2009

[8] [7]

Zhu and X

J.-L. Zhu and X. Chen, Phys. Rev. B 50, 4497 (1994)

work page 1994