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arxiv: 2605.16518 · v1 · pith:AB2H4MACnew · submitted 2026-05-15 · 🪐 quant-ph · math-ph· math.MP

Exact classical emergence from high-energy quantum superpositions

Juan A. Ca\~nas , Daniel A. Bonilla , J. Bernal , A. Mart\'in-Ruiz This is my paper

Pith reviewed 2026-05-20 18:47 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords infinite square wellclassical limitquantum superpositioncorrespondence principleprobability densityhigh-energy eigenstatesinterference terms
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The pith

In the infinite square well, the probability density of an equiprobable superposition of 2Δ+1 high-energy eigenstates converges exactly to the uniform classical distribution as Δ approaches infinity.

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

The paper establishes through exact analytic calculation that an equiprobable superposition of 2Δ+1 high-energy states in the infinite square well produces a probability density that becomes identical to the uniform classical distribution in the limit Δ → ∞. The derivation expands the interference terms into a geometric series of quantum Fourier coefficients, demonstrating that these terms become asymptotically equivalent and function as non-vanishing envelopes whose net effect yields the classical result. The same limit is shown to hold dynamically: the expectation value of position reproduces the classical triangular trajectory while residual quantum deviations remain confined to boundary layers whose relative width vanishes under macroscopic resolution.

Core claim

We prove the total probability density for a superposition of 2Δ+1 states converges exactly to the uniform classical distribution as Δ → ∞. Dynamically, the expectation value of position reproduces the classical triangular trajectory asymptotically. Residual quantum deviations remain confined to boundary layers whose relative width vanishes under macroscopic resolution.

What carries the argument

The interference terms ρ_α^a(x) expanded into a geometric series of quantum Fourier coefficients that become asymptotically equivalent in the large-n limit and serve as functional envelopes.

Load-bearing premise

The superposition must be exactly equiprobable among the chosen high-energy eigenstates so that the interference terms admit a geometric-series expansion whose terms become equivalent at large n.

What would settle it

Numerical integration of the absolute difference between the exact quantum probability density and the uniform classical density for increasing values of Δ; the difference must approach zero outside shrinking boundary layers if the claim holds.

Figures

Figures reproduced from arXiv: 2605.16518 by A. Mart\'in-Ruiz, Daniel A. Bonilla, J. Bernal, Juan A. Ca\~nas.

Figure 1
Figure 1. Figure 1: Dimensionless plots of QPD 𝐿𝜌qm 𝑛 (𝑥) (continuous gray), asymptotic distribution 𝐿𝜌a 𝑛 (𝑥) (continous blue) and CPD 𝐿𝜌cl(𝑥) (dashed red) for the quantum numbers 𝑛 = 5 (top) and 𝑛 = 15 (bottom). As discussed in [11], both the small-scale quantum oscillations and the anomalous behavior near the box walls arise from the relatively low values of 𝑛 considered here. In a truly classical regime, these features wo… view at source ↗
Figure 2
Figure 2. Figure 2: Plots of the interference term 𝐿𝜌qm 𝛼 (𝑥) (gray), its large-𝑛 asymptotic behavior 𝐿𝜌a 𝛼 (𝑥) (blue), and the 𝑛 → ∞ limit from eq. (23) (red dashed), for 𝛼 = 1 (top) and 𝛼 = 4 (bottom), 𝑛 = 15 in both cases. Unlike the case of pure eigen￾states, the asymptotic behavior does not represent a local average of the quantum oscillations; instead, it functions as a functional envelope for the interference terms. co… view at source ↗
Figure 3
Figure 3. Figure 3: Plots of probability distributions for the superposi￾tion of states defined in Eq. (28) (blue) for Δ = 10 (top), Δ = 40 (middle) and Δ = 100 (bottom). The corresponding CPD is shown for comparison (red dashed). The deviation between the quantum and classical distributions, more pronounced at lower Δ values, vanishes in the formal limit Δ → ∞. where the time-dependent state is obtained by evolving the super… view at source ↗
Figure 4
Figure 4. Figure 4: shows that the high-energy expectation value ⟨𝑥⟩(𝑡) closely follows the classical trajectory 𝑥(𝑡) over the entire period of motion. The most noticeable deviations occur near the turning points at the boundaries, where ⟨𝑥⟩(𝑡) exhibits a slightly anticipatory behavior, as if the quantum wave packet begins to reverse its motion marginally be￾fore the classical particle reaches the wall. Such behavior 0 1 2 3 … view at source ↗
read the original abstract

We examine the correspondence principle for an equiprobable superposition of high-energy eigenstates of the infinite square well using a fully analytical Fourier-based approach. We derive a closed-form asymptotic expression for the interference terms $\rho_{\alpha}^{\text{a}}(x)$ by expanding them into a geometric series of quantum Fourier coefficients. We show these terms act as functional envelopes that do not vanish individually but become asymptotically equivalent in the large-$n$ limit. Furthermore, we prove the total probability density for a superposition of $2\Delta+1$ states converges exactly to the uniform classical distribution as $\Delta \to \infty$. Dynamically, the expectation value of position reproduces the classical triangular trajectory asymptotically. Residual quantum deviations remain confined to boundary layers whose relative width vanishes under macroscopic resolution. These results establish a rigorous asymptotic realization of the classical limit for isolated bound systems in both static and dynamical contexts.

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.

Referee Report

1 major / 2 minor

Summary. The paper examines the correspondence principle for an equiprobable superposition of 2Δ+1 high-energy eigenstates of the infinite square well. Using a Fourier-based approach, it derives a closed-form asymptotic expression for the interference terms ρ_α^a(x) via geometric series expansion of the quantum Fourier coefficients of the sine eigenfunctions. The central claims are that these terms become asymptotically equivalent in the large-n limit, that the total probability density converges exactly to the uniform classical distribution as Δ → ∞, that the position expectation value reproduces the classical triangular trajectory, and that residual quantum deviations are confined to boundary layers whose relative width vanishes under macroscopic resolution.

Significance. If the convergence proof holds with controlled remainders, the work supplies a fully analytical, parameter-free example of exact classical emergence from a finite superposition of bound eigenstates. This strengthens the correspondence principle for isolated systems in both static and dynamical settings and provides a concrete benchmark for discussions of the quantum-to-classical transition.

major comments (1)
  1. [Geometric series expansion of ρ_α^a(x) and subsequent summation over 2Δ+1 states] The expansion of each interference term ρ_α^a(x) into a geometric series of Fourier coefficients (detailed in the derivation following Eq. (12) or equivalent) asserts asymptotic equivalence to the classical envelope in the large-n limit. However, the manuscript supplies no explicit remainder bound, uniform convergence estimate, or dominated-convergence argument that controls the tail uniformly in x. Without such control, it is not shown that the summed oscillations vanish faster than 1/Δ inside the boundary layers, which is required for the exact convergence claim to the uniform density.
minor comments (2)
  1. Notation for the indices α and a in the interference terms ρ_α^a(x) is introduced without an explicit definition table or equation reference at first use.
  2. The boundary-layer width scaling is stated qualitatively; an explicit expression for the layer thickness as a function of Δ would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comment on the convergence analysis. We address the major concern point by point below and outline the revisions we will make to strengthen the rigor of the proof.

read point-by-point responses

  1. Referee: The expansion of each interference term ρ_α^a(x) into a geometric series of Fourier coefficients (detailed in the derivation following Eq. (12) or equivalent) asserts asymptotic equivalence to the classical envelope in the large-n limit. However, the manuscript supplies no explicit remainder bound, uniform convergence estimate, or dominated-convergence argument that controls the tail uniformly in x. Without such control, it is not shown that the summed oscillations vanish faster than 1/Δ inside the boundary layers, which is required for the exact convergence claim to the uniform density.

    Authors: We agree that an explicit uniform remainder bound would make the argument more complete. The geometric series expansion yields an exact closed-form expression for the partial sum over the 2Δ+1 states for any finite Δ. In the revised manuscript we will insert a new subsection that bounds the tail of the geometric series uniformly for x in any compact subinterval of (0,1) away from the boundaries, showing that the remainder is O(1/n) with a constant independent of x. We will further demonstrate that the measure of the boundary layers where this bound fails scales as O(1/√n), so that their contribution to the L1 norm vanishes as Δ → ∞. This supplies the missing control and confirms that the oscillations are suppressed faster than 1/Δ in the interior under the macroscopic resolution stated in the paper. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from standard Fourier analysis of infinite-well eigenstates

full rationale

The paper's central derivation begins with the known sine eigenfunctions of the infinite square well and expands the interference terms ρ_α^a(x) into geometric series of their standard quantum Fourier coefficients. The claimed exact convergence of the total probability density to the uniform classical distribution as Δ → ∞ follows from asymptotic analysis of these series sums in the large-n limit. No step reduces the target result to a fitted parameter, a self-citation that bears the load, or a definitional equivalence; the argument is self-contained within ordinary quantum mechanics and limit-taking procedures that do not presuppose the classical limit being derived.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard mathematical tools of Fourier analysis for the infinite square well and the domain assumption of equiprobable high-energy superpositions; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The superposition of high-energy eigenstates is equiprobable.
    Explicitly stated in the abstract as the starting point for the analysis.
  • standard math Interference terms can be expanded into a geometric series of quantum Fourier coefficients that become asymptotically equivalent for large n.
    Invoked in the derivation of the closed-form asymptotic expression.

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    We derive a closed-form asymptotic expression for the interference terms ρ_α^a(x) by expanding them into a geometric series of quantum Fourier coefficients... prove the total probability density... converges exactly to the uniform classical distribution as Δ→∞

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Works this paper leans on

37 extracted references · 37 canonical work pages

  1. [1]

    A. J. Makowski, A brief survey of various formulations of the correspondence principle, European Journal of Physics 27 (5) (2006) 1133–1139.doi:10.1088/0143-0807/27/5/012. URLhttp://dx.doi.org/10.1088/0143-0807/27/5/ 012

    work page doi:10.1088/0143-0807/27/5/012 2006

  2. [2]

    R. L. Liboff, The correspondence principle revisited, Physics Today 37 (2) (1984) 50–55.doi:10.1063/1.2916084. URLhttp://dx.doi.org/10.1063/1.2916084

    work page doi:10.1063/1.2916084 1984

  3. [3]

    R. W. Robinett, Quantum and classical probability distributions for position and momentum, American Journal of Physics 63 (9) (1995) :Preprint submitted to Elsevier Page 10 of 12 823–832.doi:10.1119/1.17807. URLhttp://dx.doi.org/10.1119/1.17807

    work page doi:10.1119/1.17807 1995

  4. [4]

    R. W. Robinett, Visualizing the solutions for the circular infinite well in quantum and classical mechanics, American Journal of Physics 64 (4) (1996) 440–446.doi:10.1119/1.18188. URLhttp://dx.doi.org/10.1119/1.18188

    work page doi:10.1119/1.18188 1996

  5. [5]

    M. A. Doncheski, R. W. Robinett, Comparing classical and quantum probability distributions for an asymmetric infinite well, European Journal of Physics 21 (3) (2000) 217–228.doi:10.1088/ 0143-0807/21/3/303. URLhttp://dx.doi.org/10.1088/0143-0807/21/3/ 303

    work page doi:10.1088/0143-0807/21/3/ 2000

  6. [6]

    R. W. Robinett, Visualizing classical and quantum probability den- sities for momentum using variations on familiar one-dimensional potentials, European Journal of Physics 23 (2) (2002) 165–174. doi:10.1088/0143-0807/23/2/310. URLhttp://dx.doi.org/10.1088/0143-0807/23/2/ 310

    work page doi:10.1088/0143-0807/23/2/310 2002

  7. [7]

    G. W. Yoder, Using classical probability functions to illuminate the relation between classical and quantum physics, American Journal of Physics 74 (2006) 404–411. URLhttps://api.semanticscholar.org/CorpusID: 122279845

    work page 2006

  8. [8]

    Bernal, A

    J. Bernal, A. Martín-Ruiz, J. C. García-Melgarejo, A simple math- ematical formulation of the correspondence principle, Journal of Modern Physics 04 (01) (2013) 108–112.doi:10.4236/jmp. 2013.41017. URLhttp://dx.doi.org/10.4236/jmp.2013.41017

    work page doi:10.4236/jmp 2013

  9. [9]

    Martín-Ruiz, J

    A. Martín-Ruiz, J. Bernal, A. Frank, A. Carbajal-Dominguez, The classical limit of the quantum kepler problem, Journal of Modern Physics 04 (06) (2013) 818–822.doi:10.4236/jmp.2013. 46112. URLhttp://dx.doi.org/10.4236/jmp.2013.46112

    work page doi:10.4236/jmp.2013 2013

  10. [10]

    J. A. Cañas, J. Bernal, A. Martín-Ruiz, Exact classical limit of the quantum bouncer, The European Physical Journal Plus 137 (12) (dec 2022).doi:10.1140/epjp/s13360-022-03529-2. URLhttp://dx.doi.org/10.1140/epjp/ s13360-022-03529-2

    work page doi:10.1140/epjp/s13360-022-03529-2 2022

  11. [11]

    J. A. Cañas, J. Bernal, A. Martín-Ruiz, On the classical limit of freely falling quantum particles, quantum corrections and the emergence of the equivalence principle, Universe 10 (9) (2024).doi:10.3390/ universe10090351. URLhttps://www.mdpi.com/2218-1997/10/9/351

    work page 2024

  12. [12]

    J. A. Cañas, A. Martín-Ruiz, J. Bernal, Emergent universality of free fall from quantum mechanics, International Journal of Mod- ern Physics D 33 (15) (2024) 2441004.arXiv:https:// doi.org/10.1142/S0218271824410049,doi:10.1142/ S0218271824410049. URLhttps://doi.org/10.1142/S0218271824410049

    work page doi:10.1142/s0218271824410049 2024

  13. [13]

    K. G. Hernandez, S. E. Aguilar-Gutierrez, J. Bernalc, On the correspondence principle for the klein-gordon and dirac equations, Journal of Theoretical and Applied Physics 16 (4) (Nov. 2023). doi:10.30495/JTAP.162244. URLhttps://oiccpress.com/jtap/article/view/ 1948

    work page doi:10.30495/jtap.162244 2023

  14. [14]

    Y . Y . Fein, P. Geyer, P. Zwick, F. Kiałka, S. Pedalino, M. Mayor, S. Gerlich, M. Arndt, Quantum superposition of molecules beyond 25 kda, Nature Physics 15 (2019) 1242–1245.doi:10.1038/ s41567-019-0663-9

    work page 2019

  15. [15]

    M. Bild, M. Fadel, Y . Yang, U. von Lüpke, P. Martin, A. Bruno, Y . Chu, Schrödinger cat states of a 16-microgram mechanical oscillator, Science 380 (6642) (2023) 274–278.arXiv:https: //www.science.org/doi/pdf/10.1126/science. adf7553,doi:10.1126/science.adf7553. URLhttps://www.science.org/doi/abs/10.1126/ science.adf7553

    work page doi:10.1126/science 2023

  16. [16]

    J. von. Neumann, N. A. Wheeler, Mathematical foundations of quan- tum mechanics, Princeton University Press : Princeton University Press, 2018

    work page 2018

  17. [17]

    W. H. Zurek, Decoherence and the transition from quantum to classi- cal, Physics Today 44 (1991) 36–44.doi:10.1063/1.881293

    work page doi:10.1063/1.881293 1991

  18. [18]

    E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, I.-O. Stamatescu, Decoherence and the Appearance of a Classical World in Quantum Theory, 2nd Edition, Springer Berlin Heidelberg, Berlin, Heidelberg, 2003, softcover reprint of the hardcover second edition. doi:10.1007/978-3-662-05328-7. URLhttps://link.springer.com/book/10.1007/ 978-3-662-05328-7

    work page doi:10.1007/978-3-662-05328-7 2003

  19. [19]

    M. Schlosshauer, Decoherence and the Quantum-to-Classical Transition, 1st Edition, The Frontiers Collection, Springer Berlin Heidelberg, Berlin, Heidelberg, 2007, corrected third printing, 2008. doi:10.1007/978-3-540-35775-9. URLhttps://link.springer.com/book/10.1007/ 978-3-540-35775-9

    work page doi:10.1007/978-3-540-35775-9 2007

  20. [20]

    Hornberger, Introduction to decoherence theory, in: A

    K. Hornberger, Introduction to decoherence theory, in: A. Buchleit- ner, C. Viviescas, M. Tiersch (Eds.), Entanglement and Decoherence: Foundations and Modern Trends, V ol. 768 of Lecture Notes in Physics, Springer Berlin Heidelberg, Berlin, Heidelberg, 2009, pp. 221–276.doi:10.1007/978-3-540-88169-8_5. URLhttps://link.springer.com/chapter/10.1007/ 978-3-...

    work page doi:10.1007/978-3-540-88169-8_5 2009

  21. [21]

    W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Rev. Mod. Phys. 75 (2003) 715–775. doi:10.1103/RevModPhys.75.715. URLhttps://link.aps.org/doi/10.1103/ RevModPhys.75.715

    work page doi:10.1103/revmodphys.75.715 2003

  22. [22]

    W. H. Zurek, Quantum theory of the classical: Einselection, envari- ance, quantum darwinism and extantons, Entropy 24 (11) (2022). doi:10.3390/e24111520. URLhttps://www.mdpi.com/1099-4300/24/11/1520

    work page doi:10.3390/e24111520 2022

  23. [23]

    Tirandaz, F

    A. Tirandaz, F. Taher Ghahramani, A. Asadian, M. Golshani, Classicalization of quantum state of detector by amplification process, Physics Letters A 383 (15) (2019) 1677–1682. doi:https://doi.org/10.1016/j.physleta.2019. 02.039. URLhttps://www.sciencedirect.com/science/ article/pii/S0375960119301860

    work page doi:10.1016/j.physleta.2019 2019

  24. [24]

    Rosen, The relation between classical and quantum mechanics, American Journal of Physics 32 (1964) 597–600.doi:10.1119/ 1.1970870

    N. Rosen, The relation between classical and quantum mechanics, American Journal of Physics 32 (1964) 597–600.doi:10.1119/ 1.1970870

    work page 1964

  25. [25]

    D. Home, S. Sengupta, Classical limit of quantum mechanics. a paradoxical example, Il Nuovo Cimento B Series 11 82 (1984) 214– 224.doi:10.1007/BF02732874

    work page doi:10.1007/bf02732874 1984

  26. [26]

    Kümmel, Zur quantentheoretischen begründung der klassischen physik, Il Nuovo Cimento 1 (1955) 1057–1078.doi:10.1007/ BF02731413

    H. Kümmel, Zur quantentheoretischen begründung der klassischen physik, Il Nuovo Cimento 1 (1955) 1057–1078.doi:10.1007/ BF02731413

    work page 1955

  27. [27]

    G. G. Cabrera, M. Kiwi, Large quantum-number states and the correspondence principle, Phys. Rev. A 36 (1987) 2995–2998.doi: 10.1103/PhysRevA.36.2995. URLhttps://link.aps.org/doi/10.1103/PhysRevA. 36.2995

    work page doi:10.1103/physreva.36.2995 1987

  28. [28]

    Korsch, R

    H. Korsch, R. Möhlenkamp, A note on multidimensional wkb wave- functions: Local and global semiclassical approximations, Physics Letters A 67 (1978) 110–112.doi:10.1016/0375-9601(78) 90035-X

    work page doi:10.1016/0375-9601(78 1978

  29. [29]

    Albrecht, J

    I. Albrecht, J. Herrmann, A. Mariani, U.-J. Wiese, V . Wyss, Bouncing wave packets, ehrenfest theorem, and uncertainty relation based upon a new concept for the momentum of a particle in a box, Annals of Physics 452 (2023) 169289. doi:https://doi.org/10.1016/j.aop.2023.169289. URLhttps://www.sciencedirect.com/science/ article/pii/S000349162300074X

    work page doi:10.1016/j.aop.2023.169289 2023

  30. [30]

    Mariani, U.-J

    A. Mariani, U.-J. Wiese, Self-adjoint momentum operator for a parti- cle confined in a multi-dimensional cavity, Journal of Mathematical Physics 65 (4) (2024) 042102.doi:10.1063/5.0178419. URLhttps://doi.org/10.1063/5.0178419 :Preprint submitted to Elsevier Page 11 of 12

    work page doi:10.1063/5.0178419 2024

  31. [31]

    Belloni, R

    M. Belloni, R. Robinett, The infinite well and dirac delta function potentials as pedagogical, mathematical and physical models in quantum mechanics, Physics Reports 540 (2) (2014) 25–122.doi: https://doi.org/10.1016/j.physrep.2014.02.005. URLhttps://www.sciencedirect.com/science/ article/pii/S037015731400043X

    work page doi:10.1016/j.physrep.2014.02.005 2014

  32. [32]

    N. C. Dias, J. N. Prata, Wigner functions with boundaries, Journal of Mathematical Physics 43 (10) (2002) 4602–4627.doi:10.1063/ 1.1504885. URLhttps://doi.org/10.1063/1.1504885

    work page doi:10.1063/1.1504885 2002

  33. [33]

    Gradshteyn, I

    I. Gradshteyn, I. Ryzhik, Table of Integrals, Series, and Prod- ucts, 8th Edition, Academic Press, 2015.doi:10.1016/ C2010-0-64839-5

    work page 2015

  34. [34]

    Andrews, Wave packets bouncing off walls, American Journal of Physics 66 (3) (1998) 252–254.doi:10.1119/1.18854

    M. Andrews, Wave packets bouncing off walls, American Journal of Physics 66 (3) (1998) 252–254.doi:10.1119/1.18854. URLhttps://doi.org/10.1119/1.18854

    work page doi:10.1119/1.18854 1998

  35. [35]

    E. A. González-Velasco, Fourier series, in: Fourier Analysis and Boundary Value Problems, Elsevier, 1995, pp. 23–83.doi:10. 1016/B978-012289640-8/50002-2

    work page 1995

  36. [36]

    Bateman, A

    H. Bateman, A. Erdeélyi, Higher transcendental functions, McGraw- Hill, 1953

    work page 1953

  37. [37]

    Lewin, Polylogarithms and associated functions, North Holland, 1981

    L. Lewin, Polylogarithms and associated functions, North Holland, 1981. :Preprint submitted to Elsevier Page 12 of 12

    work page 1981