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

This paper refutes the claim that the Schrödinger equation can be solved exactly using only classical action and fluid density by demonstrating that the authors' derivation contains a foundational error which neglects the quantum potential, thereby reducing their proposed method to a standard semiclassical approximation rather than an exact solution.

The Gist

Imagine you are trying to predict the path of a swarm of fireflies in the dark.

A recent paper by scientists Lohmiller and Slotine claimed they found a "magic shortcut." They said you could predict exactly where every firefly would be and how they would move using only the rules of classical physics (like how a single ball rolls down a hill) and the density of the swarm, without needing the complex, weird rules of quantum mechanics. They claimed this method was "exact" and didn't need any approximations.

This new paper, written by Gábor Vattay, is a polite but firm "stop the presses" letter. Vattay argues that the magic shortcut isn't magic at all; it's actually a known, simplified version of physics that only works in very specific, rare situations.

Here is the breakdown of the argument using simple analogies:

1. The Missing Ingredient: The "Ghost Force"

In quantum mechanics, particles don't just act like solid balls; they act like waves. To describe this, physicists use a formula that has two parts:

• The Phase: Like the rhythm or timing of a wave.
• The Amplitude (Density): How "thick" or concentrated the wave is in a specific spot.

Lohmiller and Slotine tried to build the whole wave using only the rhythm (derived from classical paths) and the density. However, Vattay points out that they made a math error: they treated the density as if it were perfectly smooth and unchanging, like a flat sheet of water.

In reality, the density of a quantum wave is often bumpy and changing. When you have these bumps, a special "ghost force" appears, called the Quantum Potential.

• The Analogy: Imagine driving a car on a road. Lohmiller and Slotine calculated the car's speed based only on the engine (classical action) and the traffic density, assuming the road was perfectly flat. Vattay says, "You forgot the potholes!" Those potholes are the Quantum Potential. If you ignore them, your calculation is only an approximation, not an exact solution.

2. Why Did Their Examples Work?

You might ask, "If their math was wrong, why did their examples (like the double-slit experiment or a particle in a box) look correct?" Vattay explains that they got lucky because they picked two types of tricks:

Trick A: The "Flat Road" Illusion
In some specific scenarios (like a particle bouncing between two walls or passing through a slit), the "bumps" in the wave are so perfectly arranged that the "ghost force" (Quantum Potential) cancels out to zero.

• The Analogy: It's like saying, "I can predict the weather exactly by ignoring wind." This works perfectly if you are standing in a room with no windows and no fans (no wind). But it fails the moment you step outside. Lohmiller and Slotine picked examples where the wind happened to be zero, so their "no-wind" formula looked perfect, even though it's not a general rule.

Trick B: Cheating with the Starting Line
For more complex problems (like an atom or a vibrating spring), the "ghost force" is definitely not zero. So, how did they get the right answer?

• The Analogy: Imagine they claimed they could predict the outcome of a soccer game using only the rules of running. But, to make their prediction work, they secretly started the game with the players already arranged in the exact winning formation.
• Vattay shows that in these examples, Lohmiller and Slotine didn't actually derive the quantum behavior from classical rules. Instead, they started with the initial conditions (the starting position of the particles) and secretly used the known quantum answers (the "winning formation") to set them up. They then used classical physics just to spin the players around. They didn't discover the quantum rules; they just hid the quantum answers inside the starting line.

The Bottom Line

Vattay concludes that the relationship between classical physics and quantum waves is a well-known field called semiclassical approximation. It is a useful tool, but it is an approximation, not an exact replacement.

The paper claims that Lohmiller and Slotine didn't find a new way to solve quantum mechanics exactly. Instead, they accidentally rediscovered a standard approximation method, and their examples only worked because they either chose problems where the approximation is perfect by luck, or they secretly built the quantum answer into the problem from the start.

Technical Summary

1. Problem Statement

The paper addresses a recent claim by Lohmiller & Slotine (Proc. R. Soc. A 482: 20250413) asserting that the Schrödinger equation can be solved exactly using only classical least action and classical fluid density. Their proposed formulation constructs a quantum wave function $\psi_j = \sqrt{\rho_j} e^{i\phi_j/\hbar}$ from a single classical path $j$, claiming this yields an exact solution without semiclassical approximations.

Vattay argues that this claim is mathematically flawed. The core issue is that Lohmiller & Slotine conflate the total derivative along a trajectory with the spatial partial derivatives required by the quantum kinetic energy operator. By treating the probability density $\rho_j$ as having no spatial variation along the path, the authors inadvertently eliminate the quantum potential, reducing their "exact" formulation to the standard semiclassical (WKB) approximation.

2. Methodology

Vattay employs a rigorous mathematical re-derivation to test the validity of Lohmiller & Slotine's central lemma (Lemma 3.1). The methodology involves:

• Ansatz Substitution: Substituting the wave function ansatz $\psi_j = \sqrt{\rho_j} e^{i\phi_j/\hbar}$ into the time-dependent Schrödinger equation for a scalar potential $V(x)$.
• Calculus Verification: Applying standard product and chain rules to compute the spatial gradient ($\nabla \psi_j$) and the Laplacian ($\nabla^2 \psi_j$).
• Separation of Real and Imaginary Parts: Decomposing the resulting equation to isolate the continuity equation (imaginary part) and the energy conservation equation (real part).
• Comparative Analysis: Comparing the derived real-part equation against the classical Hamilton–Jacobi equation to identify discrepancies, specifically focusing on the term representing the quantum potential.
• Case Study Review: Analyzing the specific examples provided in the original paper (particle in a box, double-slit, harmonic oscillator, hydrogen atom) to determine why they appeared to yield correct results despite the theoretical error.

3. Key Contributions and Mathematical Findings

A. Identification of the Foundational Error

Vattay demonstrates that the Schrödinger equation requires the Laplacian to act on the full spatial dependence of $\psi_j$, including the amplitude $\sqrt{\rho_j}$.

• The Derivation:
  $$\nabla^2 \psi_j = \left[ \nabla^2\sqrt{\rho_j} + \frac{2i}{\hbar}\nabla\sqrt{\rho_j} \cdot \nabla\phi_j + \frac{i}{\hbar}\sqrt{\rho_j}\nabla^2\phi_j - \frac{1}{\hbar^2}\sqrt{\rho_j}(\nabla\phi_j)^2 \right] e^{i\phi_j/\hbar}$$
• The Resulting Real Equation:
  $$\frac{\partial\phi_j}{\partial t} + \frac{1}{2M}(\nabla\phi_j)^2 + V - \underbrace{\frac{\hbar^2}{2M}\frac{\nabla^2\sqrt{\rho_j}}{\sqrt{\rho_j}}}_{Q} = 0$$
  The term $Q$ is the quantum potential, originally identified by Madelung and emphasized by Bohm.
• The Error: Lohmiller & Slotine implicitly assume $\nabla\sqrt{\rho_j} = 0$ (i.e., the density has no spatial gradient). This assumption artificially sets $Q=0$, causing the equation to collapse into the classical Hamilton–Jacobi equation. Vattay asserts this is the definition of the semiclassical limit, not an exact solution.

B. Explanation of "Successful" Examples

Vattay explains why the examples in the original paper appeared to work, categorizing them into two distinct types where the error is masked:

1. Trivial Density (Quantum Potential Vanishes):

   • Cases: Particle in a box, quantum tunneling, and the double-slit experiment (in free space).
   • Reasoning: In these scenarios, the classical density $\rho$ is either spatially constant (plane waves) or varies as $1/r$ (spherical waves). In 3D, $\nabla^2(1/r) = 0$ everywhere except the singularity. Consequently, $Q=0$ identically. The error is invisible because the omitted term is naturally zero due to the problem's geometry.

2. Circular Reasoning (Importing Quantum Solutions):

   • Cases: Harmonic oscillator and Hydrogen atom.
   • Reasoning: For these bound states, the density varies spatially, and $Q \neq 0$. However, the original authors did not use a single classical branch. Instead, they summed over an infinite ensemble of initial conditions.
   • The Flaw: The expansion coefficients used for the initial density (e.g., Hermite polynomials for the oscillator) were chosen specifically because they correspond to quantum eigenfunctions.
   • Conclusion: The authors did not derive quantum mechanics from classical mechanics; they assumed the quantum basis functions in the initial conditions and used the classical action merely to propagate the phase. This is a circular argument that masks the single-branch error.

4. Results

• The claim that the Schrödinger equation can be solved exactly using only classical action and density on a single path is false.
• The formulation presented by Lohmiller & Slotine is mathematically equivalent to the standard semiclassical (WKB) approximation, which neglects the quantum potential.
• The "exact" results obtained in the illustrative examples are artifacts of either:
  1. Special geometries where the quantum potential vanishes naturally.
  2. The implicit injection of quantum eigenfunctions into the initial conditions, rendering the derivation tautological.

5. Significance

• Correction of Literature: The paper corrects a significant misunderstanding in the literature regarding the relationship between classical action and quantum wave functions. It clarifies that classical trajectories alone cannot generate exact quantum wave functions without accounting for the quantum potential (spatial dispersion).
• Clarification of Semiclassical Limits: It reinforces the established understanding (citing Berry & Mount, Madelung, and Bohm) that the transition from classical to quantum mechanics requires the inclusion of the quantum potential term to account for wave interference and dispersion.
• Methodological Warning: It serves as a caution against "back-door" derivations where quantum solutions are smuggled in via initial condition parameterizations, mistaking the result for a derivation from first principles.
• Impact on Quantum Hydrodynamics: The comment reaffirms the necessity of Madelung's hydrodynamic formulation, which naturally includes the quantum potential, as the correct framework for linking fluid density to quantum mechanics.