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

Abstract: A recent article by Lohmiller & Slotine (Proc.\ R.\ Soc.\ A \textbf{482}: 20250413) claims that the Schrödinger 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.

• The paper demonstrates that the claimed exact derivation of quantum wave functions from classical action neglects critical spatial derivatives, invalidating its general applicability.
• It reveals that assuming a classical density without gradients dismisses the quantum potential, effectively reducing the formulation to a semiclassical limit.
• The critique underscores that special-case successes arise from initial condition choices rather than truly deriving quantum evolution from classical mechanics.

Critical Analysis of the Mathematical Foundations in On computing quantum waves exactly from classical action

Overview

The paper "Comment on `On computing quantum waves exactly from classical action'" (2605.02621) provides a rigorous critique of the central claims presented by Lohmiller and Slotine, who argue that quantum wave functions can be constructed exactly using only the classical action and density, thus purportedly bypassing semiclassical approximations. Vattay demonstrates that this assertion is mathematically unsound, rooted in a conflation of trajectory-based and spatial derivatives, which results in the systematic omission of the quantum potential—a key non-classical term. The analysis clarifies why examples that appear to validate the original framework are in fact special cases or employ circular logic, ultimately reinforcing the boundaries of semiclassical and exact quantum formulations.

Mathematical Structure and Core Error

Lohmiller and Slotine introduce the phase–amplitude ansatz $\psi_j = \sqrt{\rho_j}\, e^{i\varphi_j/\hbar}$, with $\rho_j$ and $\varphi_j$ taken from classical mechanics, claiming this form solves the Schrödinger equation exactly. The core error identified in the comment is a foundational one: treating the density $\rho_j$ as possessing only pathwise (total) time dependence, ignoring its spatial derivatives. The kinetic term in the time-dependent Schrödinger equation includes $\nabla^2 \psi$, which generically leads (via product and chain rules) to the so-called quantum potential,

$$Q = -\frac{\hbar^2}{2M}\frac{\nabla^2 \sqrt{\rho_j}}{\sqrt{\rho_j}}.$$

Neglecting $\nabla \sqrt{\rho_j}$ eliminates $Q$ and reduces the quantum Hamilton–Jacobi equation to its classical analogue. This is not an exact quantum result but the well-understood WKB/semi-classical limit, as exhaustively detailed in the literature [Berry & Mount, Cvitanović et al., Gutzwiller, Brack & Bhaduri]. When the quantum potential is non-zero, this reduction cannot recover true quantum dynamics.

Case Analysis of Examples and Apparent Successes

The comment addresses why the original paper reproduces correct quantum-mechanical behavior in certain examples:

• Trivial Classical Density: For scenarios such as the infinite square well or plane-wave tunneling, $\rho_j$ is spatially uniform, yielding $\nabla\sqrt{\rho_j}=0$ and thus $\rho_j$0. In these highly symmetric geometries, the quantum corrections vanish identically, and classical propagation aligns with quantum results. For the double-slit diffraction pattern, the Laplacian of the relevant radial decay also vanishes (save for measure-zero singularities), again nullifying the quantum potential.
• Importation of Quantum Structure via Initial Conditions: In systems where the quantum potential must contribute—such as the harmonic oscillator and hydrogen atom—the original framework relies on summation over initial conditions expressed in quantum eigenfunction bases (Hermite polynomials for the oscillator, radial/spherical harmonics for the hydrogen atom). This choice trivially reconstructs the quantum solution; propagation of the phase is then a mere assignment of time evolution. Therefore, quantum structure is not derived from classical mechanics but assumed at the foundation, producing a circular argument.

Theoretical and Practical Implications

The critique emphasizes that connections between classical action and quantum wave functions are exhaustively codified within semiclassical mechanics, where neglecting the quantum potential is strictly the limit of validity and not the general case [Berry & Mount 1972, Cvitanović et al. 2020, Gutzwiller 1990]. Madelung's hydrodynamical approach and Bohmian mechanics both centrally feature the quantum potential as the generator of non-classical interference and bound state behavior. The error in the original claim—removal of spatial derivatives from the probability amplitude—amounts to an assertion that quantum mechanics generically reduces to classical flow, which contradicts both established theory and experimental data except in the aforementioned exceptional cases.

Practically, this critique reinforces the standard methodology: exact quantum evolution requires retaining the full spatial structure of the amplitude, and wave-packet spreading, tunneling corrections, and interference effects all derive from nonzero $\rho_j$1. For future quantum–classical correspondence studies or quantum trajectory simulations, these results further underscore the necessity of including amplitude gradients and the limits on direct reduction of quantum dynamics to classical mechanisms.

Conclusion

The analysis provided in (2605.02621) conclusively demonstrates that the claim of exact quantum solutions via classical action and density, absent the quantum potential, is unfounded. The reviewed framework is, in general, limited to the semiclassical regime. This clarification is vital for theoretical rigor in semiclassical and quantum trajectory methods, and serves as an instructive example concerning the pitfalls of overextending classical analogies in quantum physics. Further developments in quantum-classical correspondence must retain careful attention to the mathematical structure of the quantum potential and avoid conflating special-case results with general capabilities.