arxiv: 2605.02621 · v1 · submitted 2026-05-04 · 🪐 quant-ph · hep-th· math-ph· math.MP· nlin.CD· physics.hist-ph

Recognition: 2 theorem links

· Lean Theorem

Comment on `On computing quantum waves exactly from classical action'

Gabor Vattay

Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:10 UTC · model grok-4.3

classification 🪐 quant-ph hep-thmath-phmath.MPnlin.CDphysics.hist-ph

keywords Schrödinger equationquantum potentialsemiclassical approximationMadelung fluidBohm interpretationclassical actionprobability density amplitude

0 comments

The pith

Neglecting spatial derivatives of the probability density omits the quantum potential and reduces the claimed exact quantum solution to the semiclassical approximation.

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

This comment paper shows that a recent proposal to compute quantum waves exactly from classical action and fluid density contains a key error. The derivation neglects terms that produce the quantum potential, which is essential for capturing full quantum behavior beyond the semiclassical regime. As a result, the method does not provide an exact solution to the Schrödinger equation but instead reproduces the standard semiclassical approximation. The original paper's examples either involve geometries where this potential vanishes or rely on quantum eigenfunctions supplied through initial conditions, hiding the approximation. Readers interested in the interface between classical and quantum mechanics should note this clarification of what counts as an exact derivation.

Core claim

The paper demonstrates that by neglecting the spatial derivatives of the probability density amplitude, the authors of the original work omit the quantum potential identified by Madelung and Bohm. This omission means their formulation is equivalent only to the semiclassical approximation rather than an exact quantum solution. Consequently, the illustrative examples either belong to classes where the quantum potential vanishes due to problem geometry or recover quantum results by importing quantum eigenfunctions via initial conditions.

What carries the argument

The quantum potential arising from spatial derivatives of the probability density amplitude in the Madelung fluid formulation of the Schrödinger equation.

If this is right

The claimed exact equivalence holds only in the semiclassical limit where the quantum potential can be neglected.
Examples with vanishing quantum potential due to geometry do not validate general exactness.
Importing quantum eigenfunctions through initial conditions supplies quantum information and conceals the semiclassical character.
The method cannot claim to avoid semiclassical approximations without the full quantum potential.

Where Pith is reading between the lines

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

Any formulation deriving quantum dynamics solely from classical action must retain the quantum potential to reach exact results.
The same omission may appear in other fluid-based reformulations of quantum mechanics.
Numerical tests on systems with nonzero quantum potential, such as particle tunneling, would show where the approximation breaks.

Load-bearing premise

That the original derivation achieves an exact quantum solution without semiclassical approximations or by embedding quantum information in the initial conditions.

What would settle it

A term-by-term comparison of the derived equations against the full Madelung equations to determine whether the quantum potential term is absent or retained.

read the original abstract

A recent article by Lohmiller \& Slotine (Proc.\ R.\ Soc.\ A \textbf{482}: 20250413) claims that the Schr\"odinger equation can be solved exactly using only classical least action and classical fluid density, asserting that this formulation avoids semiclassical approximations. We show that their mathematical derivation contains a foundational error. By neglecting the spatial derivatives of the probability density amplitude, the authors inadvertently omit the quantum potential -- the term originally identified by Madelung and later emphasised by Bohm. Consequently, their proposed equivalence is not exact but rather constitutes the standard semiclassical approximation. We further demonstrate that each of the paper's illustrative examples either belongs to a class where the quantum potential vanishes identically due to the geometry of the problem, or recovers the correct quantum result by importing quantum eigenfunctions through the initial conditions, thereby concealing the error.

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 comment correctly shows that the Lohmiller-Slotine paper drops the quantum potential and ends up with the standard semiclassical fluid equations rather than an exact quantum solution.

read the letter

This comment correctly shows that the Lohmiller-Slotine paper drops the quantum potential and ends up with the standard semiclassical fluid equations rather than an exact quantum solution. The derivation neglects spatial derivatives of the amplitude, which removes the term Madelung and Bohm identified long ago. The result is classical Euler dynamics plus continuity, not the full Schrödinger equation. The comment then checks the original examples and finds that each either has vanishing quantum potential by symmetry or geometry, or recovers the right answer only because the initial conditions already contain the quantum eigenfunctions. That specific check is the new piece of work here. It is not present in the cited literature and makes the critique concrete instead of abstract. The paper does this part cleanly and stays within established quantum hydrodynamics without adding extra assumptions. The argument has independent grounding in the Madelung equations, so there is no circularity. A minor limitation is the narrow scope. The comment stops once the error is shown and does not explore whether a corrected version of the approach could still be useful or how it sits next to other semiclassical methods. That is typical for a comment, but it means the reader has to do the next step. This is for people working on quantum fluid formulations, quantum chaos, or action-based derivations in quantum mechanics. Anyone who needs to keep exact and approximate results separate will find the distinction useful. The paper deserves a serious referee because the central claim rests on standard theory and the example analysis adds verifiable detail. I recommend sending it for peer review.

Referee Report

0 major / 1 minor

Summary. This comment manuscript critiques Lohmiller & Slotine (Proc. R. Soc. A 482: 20250413), who claim an exact solution of the Schrödinger equation via classical least action and fluid density without semiclassical approximations. The authors show that the derivation neglects spatial derivatives of the probability density amplitude R = √ρ, thereby omitting the quantum potential term Q = −(ℏ²/2m)(∇²R/R) originally identified by Madelung and emphasized by Bohm. This reduces the dynamics to the classical Euler equations plus continuity, constituting the standard semiclassical limit. They further demonstrate that each illustrative example either has identically vanishing Q due to symmetry or geometry, or recovers the correct quantum result only by importing quantum eigenfunctions through the initial conditions.

Significance. If the central claim holds, the comment is significant for correcting the record on quantum-classical correspondences in hydrodynamics. It reinforces that exact equivalence to the Schrödinger equation requires the quantum potential and shows how the original examples either evade or conceal this requirement. The grounding in standard Madelung fluid equations and Bohmian quantum potential provides independent, non-circular support for the critique.

minor comments (1)

The abstract and introduction could include an explicit equation reference (e.g., the precise form of the neglected ∇²R term) from the original Lohmiller & Slotine derivation to aid readers who have not yet consulted the target paper.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation to accept. The referee's summary accurately captures the core argument of our comment.

read point-by-point responses

Referee: This comment manuscript critiques Lohmiller & Slotine (Proc. R. Soc. A 482: 20250413), who claim an exact solution of the Schrödinger equation via classical least action and fluid density without semiclassical approximations. The authors show that the derivation neglects spatial derivatives of the probability density amplitude R = √ρ, thereby omitting the quantum potential term Q = −(ℏ²/2m)(∇²R/R) originally identified by Madelung and emphasized by Bohm. This reduces the dynamics to the classical Euler equations plus continuity, constituting the standard semiclassical limit. They further demonstrate that each illustrative example either has identically vanishing Q due to symmetry or geometry, or recovers the correct quantum result only by importing quantum eigenfunctions through the initial conditions.

Authors: We agree with the referee's summary. Our derivation shows that neglecting the spatial derivatives of R omits the quantum potential, reducing the claimed exact equivalence to the standard semiclassical (Madelung) fluid equations. The examples in the original work either have vanishing Q by symmetry or import quantum information via initial conditions. revision: no

Circularity Check

0 steps flagged

No circularity; critique rests on independent Madelung-Bohm derivation

full rationale

The comment derives its central claim by direct substitution into the Madelung fluid equations obtained from the Schrödinger equation, showing that dropping spatial derivatives of R removes the quantum potential term Q. This step is a standard algebraic identity, not a self-definition or fitted prediction. All supporting references (Madelung 1927, Bohm 1952) are external and predate the work; no self-citation chain, ansatz smuggling, or uniqueness theorem imported from the authors appears. The examination of the original paper's examples follows the same external hydrodynamic structure and does not reduce to the comment's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The comment relies on the established framework of quantum hydrodynamics without introducing new free parameters or entities.

axioms (1)

domain assumption The quantum potential arises precisely from the spatial derivatives of the probability density amplitude in the Madelung formulation of the Schrödinger equation.
This is a standard result in quantum mechanics as identified by Madelung and emphasized by Bohm.

pith-pipeline@v0.9.0 · 5451 in / 1160 out tokens · 55552 ms · 2026-05-08T18:10:48.665711+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 7 canonical work pages

[1]

2026 On computing quantum waves exactly from classical action

Lohmiller W, Slotine J-JE. 2026 On computing quantum waves exactly from classical action. Proc. R. Soc. A482, 20250413. (doi:10.1098/rspa.2025.0413)

work page doi:10.1098/rspa.2025.0413 2026
[2]

1972 Semiclassical approximations in wave mechanics.Rep

Berry MV , Mount KE. 1972 Semiclassical approximations in wave mechanics.Rep. Prog. Phys. 35, 315–397. (doi:10.1088/0034-4885/35/1/306)

work page doi:10.1088/0034-4885/35/1/306 1972
[3]

2020Chaos: Classical and Quantum

Cvitanovi´ c P , Artuso R, Mainieri R, Tanner G, Vattay G. 2020Chaos: Classical and Quantum. Copenhagen: Niels Bohr Institute. (ChaosBook.org)
[4]

Gutzwiller,Chaos in Classical and Quantum Mechanics, vol

Gutzwiller MC. 1990Chaos in Classical and Quantum Mechanics. New York: Springer-Verlag. (doi:10.1007/978-1-4612-0983-6)

work page doi:10.1007/978-1-4612-0983-6
[5]

2003Semiclassical Physics

Brack M, Bhaduri RK. 2003Semiclassical Physics. Boulder, CO: Westview Press
[6]

Quantentheorie in hydrodynamischer Form

Madelung E. 1927 Quantentheorie in hydrodynamischer Form.Z. Phys.40, 322–326. (doi:10.1007/BF01400372)

work page doi:10.1007/bf01400372 1927
[7]

A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables

Bohm D. 1952 A suggested interpretation of the quantum theory in terms of “hidden” variables. I.Phys. Rev.85, 166–179. (doi:10.1103/PhysRev.85.166)

work page doi:10.1103/physrev.85.166 1952
[8]

2018Introduction to Quantum Mechanics, 3rd edn

Griffiths DJ, Schroeter DF. 2018Introduction to Quantum Mechanics, 3rd edn. Cambridge, UK: Cambridge University Press
[9]

1979 Solution of the path integral for theH-atom.Phys

Duru IH, Kleinert H. 1979 Solution of the path integral for theH-atom.Phys. Lett. B84, 185–

1979
[10]

(doi:10.1016/0370-2693(79)90280-6)

work page doi:10.1016/0370-2693(79)90280-6
[11]

1926 An undulatory theory of the mechanics of atoms and molecules.Phys

Schrödinger E. 1926 An undulatory theory of the mechanics of atoms and molecules.Phys. Rev.28, 1049–1070. (doi:10.1103/PhysRev.28.1049)

work page doi:10.1103/physrev.28.1049 1926