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

TL;DR

The paper critiques a recent claim that classical action can exactly solve the Schrödinger equation, showing that the derivation contains a foundational error that reduces it to a semiclassical approximation.

Contribution

It identifies a key mathematical mistake in the previous work, clarifying that the proposed method is not exact but semiclassical, and explains why the examples do not demonstrate true exactness.

Findings

The derivation neglects the quantum potential by ignoring spatial derivatives of density.

The claimed exact solution is actually a semiclassical approximation.

Examples either have zero quantum potential or rely on imported eigenfunctions, masking the error.

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.