05.15.02 · symplectic / optimal-transport

Korteweg / Madelung quantum hydrodynamics

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Anchor (Master): Madelung 1927 *Quantentheorie in hydrodynamischer Form* (Z. Phys. 40, originator of the polar transform); Korteweg 1901 *Sur la forme que prennent les équations du mouvement des fluides* (Arch. Néerl. Sci. Exactes Nat. 6, capillarity stress tensor); de Broglie 1927 *La mécanique ondulatoire* (J. Phys. Radium 8); Bohm 1952 *A suggested interpretation of the quantum theory in terms of hidden variables* (Phys. Rev. 85); Lott 2008 *Some geometric calculations on Wasserstein space* (Comm. Math. Phys. 277); Khesin-Lee 2009 *A nonholonomic Moser theorem and optimal transport* (J. Geom. Mech. 1); Fusca 2017 *The Madelung transform as a momentum map* (J. Geom. Mech. 9); Arnold-Khesin *Topological Methods in Hydrodynamics* 2nd ed. Ch. VIII §4

Intuition Beginner

The Schrödinger equation describes a quantum particle by a complex-valued wavefunction $ψ (x, t)$ , and the connection to classical mechanics is famously obscure. In 1927, Erwin Madelung discovered a remarkable rewriting. Write the wavefunction in polar form as $ψ = ρ e^{i S /ℏ}$ , where $ρ = ∣ ψ ∣^{2}$ is the probability density (a real positive function) and $S$ is the phase (a real function). Then the Schrödinger equation splits cleanly into two equations on $(ρ, S)$ that look exactly like fluid mechanics — a conservation law for $ρ$ and a Hamilton-Jacobi equation for $S$ .

What is striking is that the Hamilton-Jacobi equation is almost the classical one, except for an extra term Madelung called the quantum potential, $Q = - \frac{ℏ ^{2}}{2 m} Δ ρ / ρ$ . Set $ℏ = 0$ and you get classical particle mechanics; turn $ℏ$ back on and the same equations describe quantum diffusion. The quantum potential is a nonlocal force determined by the shape of the density itself — wherever the density is sharply peaked, $Q$ pushes back. This is the mechanism behind wavepacket spreading.

The same equations were derived a quarter century earlier by Diederik Korteweg, working on capillarity in classical fluids: a fluid with a density-dependent surface tension produces a stress tensor whose divergence is exactly Madelung's quantum pressure. So the quantum fluid is a Korteweg capillarity fluid in disguise. Modern geometric mechanics, following Otto, Lott, and Fusca, reads the entire Madelung system as Hamiltonian dynamics on the cotangent bundle of the space of probability measures with the Otto-Fisher metric — placing quantum mechanics inside the same optimal-transport geometry as the heat equation and pressureless Euler.

Visual Beginner

Picture a single-bump density $ρ (x)$ on the real line — a Gaussian peak — together with a flowing phase $S (x)$ . The Madelung picture decomposes the wavefunction into these two real pieces. As time progresses, the density spreads and the phase develops a curvature; the quantum potential $Q$ , drawn as a counter-pushing arrow at the peak, is what drives the spreading.

The visual captures Madelung's reframing: a wavefunction is a fluid with two state variables, a density $ρ$ and a velocity built from the phase, evolving under conservation of mass and a Hamilton-Jacobi equation modified by the quantum potential.

Worked example Beginner

Take a stationary plane wave $ψ (x, t) = e^{i (k x - ω t)}$ describing a free particle of mass $m$ moving to the right with momentum $ℏ k$ . We extract the Madelung fluid by reading off the polar form.

Step 1. Density: $ρ (x, t) = ∣ ψ (x, t) ∣^{2} = 1$ . The probability density is uniform — every point on the line is equally likely.

Step 2. Phase: $S (x, t) = ℏ (k x - ω t)$ . The phase grows linearly in $x$ and decreases linearly in $t$ .

Step 3. Velocity: the Madelung velocity is the spatial slope of $S$ divided by mass, giving $v (x, t) = ℏ k / m$ . Every fluid element moves at the same constant speed $ℏ k / m$ , which is exactly the classical group velocity.

Step 4. Quantum potential: the quantum potential measures how curved $ρ$ is; here $ρ = 1$ is flat, so the quantum potential $Q$ equals zero. The quantum potential is zero for plane waves.

Step 5. Hamilton-Jacobi check: plugging $S$ and $v$ into the time-energy relation gives $- ℏ ω + (ℏ k)^{2} / (2 m) + 0 = 0$ . Setting this to zero recovers the dispersion relation $ℏ ω = ℏ^{2} k^{2} / (2 m)$ , the energy-momentum relation for a free particle.

What this tells us: a plane wave is a uniform-density fluid moving at constant velocity, with zero quantum pressure. Quantum corrections appear only when the density profile has curvature; the plane wave is the simplest case where Madelung's hydrodynamic rewriting collapses to ordinary classical motion.

Check your understanding Beginner

Exercise (easy, multiple choice).

The Madelung transform writes a Schrödinger wavefunction $ψ$ in polar form. What are the two real fields produced?

A. real and imaginary parts of $ψ$ B. density $ρ = ∣ ψ ∣^{2}$ and phase $S = ℏ ar g ψ$ C. amplitude $∣ ψ ∣$ and energy $E$ D. position $x$ and momentum $p$

Hint

The polar form writes a complex number as modulus times $e^{i θ}$ . Madelung promotes both the modulus and the phase to fields of position.

Answer

B. The Madelung transform is $ψ = ρ e^{i S /ℏ}$ with $ρ = ∣ ψ ∣^{2}$ the probability density and $S = ℏ ar g ψ$ the phase (rescaled by $ℏ$ so that the velocity built from $S$ has units of velocity). Feedback-correct: this polar decomposition reveals the fluid interpretation of quantum mechanics. Feedback-wrong: option A is the Cartesian decomposition (real/imaginary), not the Madelung one; options C and D mix up phase-space and energy.

Formal definition Intermediate+

Fix a smooth wavefunction $ψ : R^{n} \times R \to C$ solving the time-dependent Schrödinger equation $$ i\hbar \partial_t \psi = -\frac{\hbar^2}{2m} \Delta \psi + V(x) \psi, $$ with $V \in C^{\infty} (R^{n}; R)$ . Assume $ψ (x, t) \neq = 0$ on the region of interest, so $ρ := ∣ ψ ∣^{2}$ is positive and a single-valued phase $S$ with $ℏ ar g ψ = S mod 2 π ℏ$ exists.

Definition (Madelung transform). The Madelung transform of $ψ$ is the pair $(ρ, S)$ of real functions defined by $$ \rho = |\psi|^2, \qquad \psi = \sqrt{\rho} , e^{iS/\hbar}. $$ The associated Madelung velocity is $v = \nabla S / m$ .

Definition (quantum potential). The quantum potential (or Bohm potential) is the nonlocal functional of the density $$ Q[\rho] = -\frac{\hbar^2}{2m} \frac{\Delta \sqrt{\rho}}{\sqrt{\rho}}. $$ A direct expansion gives the equivalent form $Q [ρ] = - \frac{ℏ ^{2}}{4 m} (\frac{Δ ρ}{ρ} - \frac{1}{2} \frac{∣\nabla ρ ∣ ^{2}}{ρ ^{2}})$ .

Theorem (Madelung 1927). The wavefunction $ψ$ solves the Schrödinger equation iff the pair $(ρ, S)$ solves the coupled Madelung system $$ \partial_t \rho + \nabla \cdot (\rho v) = 0, \qquad v = \nabla S / m, \tag{continuity} $$ $$ \partial_t S + \frac{|\nabla S|^2}{2m} + V + Q[\rho] = 0. \tag{quantum Hamilton-Jacobi} $$

Corollary (Madelung momentum equation). Taking $\nabla$ of the quantum Hamilton-Jacobi equation and using the continuity equation gives $$ m\big(\partial_t v + (v \cdot \nabla) v\big) = -\nabla V - \nabla Q[\rho], $$ Newton's law for a fluid parcel acted on by the external force $- \nabla V$ and the quantum-pressure force $- \nabla Q$ .

Definition (Korteweg capillarity fluid). A Korteweg capillarity fluid is a fluid with stress tensor $$ \sigma_{ij}[\rho] = p_K(\rho), \delta_{ij} - \kappa ,\rho ,\partial_i \partial_j \rho + (\text{trace correction}), $$ where $p_{K} (ρ)$ is a classical pressure and $κ$ is a constant capillarity coefficient. With $κ = ℏ^{2} / (4 m^{2})$ and the Cahn-Hilliard form of the trace correction, the divergence of $σ$ equals the quantum-pressure force: $ρ^{- 1} \nabla \cdot σ = - \nabla Q [ρ]$ .

Definition (Otto-Fisher metric on $P_{2}$ ). Extending the Otto metric of 05.15.01, the Otto-Fisher metric on $P_{2} (R^{n})$ adds a Fisher-information term: tangent vectors at $ρ$ are still ${\nabla ϕ}$ in the Otto closure, the metric remains $⟨ \nabla ϕ_{1}, \nabla ϕ_{2} ⟩_{ρ} = \int \nabla ϕ_{1} \cdot \nabla ϕ_{2} d ρ$ , but the Hamiltonian generating the Schrödinger flow is $$ \mathcal{E}(\rho, S) = \int_{\mathbb{R}^n}\left(\frac{|\nabla S|^2}{2m}\rho + \frac{\hbar^2}{8m}\frac{|\nabla \rho|^2}{\rho} + V\rho\right) dx, $$ combining Otto kinetic energy, Fisher-information (encoding quantum pressure), and external potential.

Counterexamples to common slips

The phase $S$ is defined only up to $2 π ℏ$ . If $ψ$ has a zero or if its support is multiply-connected, the lift $S$ may not be single-valued globally; circulation $\oint \nabla S \cdot d l$ takes values in $2 π ℏ Z$ rather than vanishing. This is the seed of Onsager-Feynman vortex quantisation.
The Madelung system is not strictly hyperbolic. The quantum potential $Q [ρ]$ depends on second derivatives of $ρ$ , so the system has a dispersive character — closer to the Korteweg-de Vries equation than to compressible Euler. Standard shock theory does not apply.
Madelung is not Bohmian mechanics. The Madelung transform is a change of variables on $ψ$ . Bohm 1952 added the interpretive claim that the trajectories integrating $v = \nabla S / m$ are real particle paths; this is an interpretational addition, not a theorem.
The quantum potential is not bounded below. Near a node of $ρ$ , $ρ$ has a cusp and $Q \to + \infty$ . A wavefunction with isolated zeros has singular Madelung dynamics at those zeros.

Key theorem with proof Intermediate+

Theorem (Madelung 1927 — equivalence of Schrödinger and Madelung systems). Let $ψ : R^{n} \times R \to C$ be smooth and non-vanishing on its domain, with polar decomposition $ψ = ρ e^{i S /ℏ}$ . Then $ψ$ solves the Schrödinger equation $i ℏ \partial_{t} ψ = - \frac{ℏ ^{2}}{2 m} Δ ψ + V ψ$ iff $(ρ, S)$ solves the Madelung system $$ \partial_t \rho + \nabla \cdot \big(\rho , \nabla S / m\big) = 0, \qquad \partial_t S + \frac{|\nabla S|^2}{2m} + V - \frac{\hbar^2}{2m}\frac{\Delta \sqrt{\rho}}{\sqrt{\rho}} = 0. $$

Proof. Compute the partial time derivative of $ψ$ from its polar form. Write $R = ρ$ , so $ψ = R e^{i S /ℏ}$ . Then $$ \partial_t \psi = \left(\partial_t R + \frac{iR}{\hbar} \partial_t S\right) e^{iS/\hbar}. $$ Compute the Laplacian. The gradient is $$ \nabla \psi = \left(\nabla R + \frac{iR}{\hbar} \nabla S\right) e^{iS/\hbar}, $$ and a second application gives $$ \Delta \psi = \left[\Delta R + \frac{2i}{\hbar} \nabla R \cdot \nabla S + \frac{iR}{\hbar}\Delta S - \frac{R}{\hbar^2}|\nabla S|^2\right] e^{iS/\hbar}. $$

Substitute into the Schrödinger equation $i ℏ \partial_{t} ψ = - \frac{ℏ ^{2}}{2 m} Δ ψ + V ψ$ and divide through by $e^{i S /ℏ}$ : $$ i\hbar \partial_t R - R \partial_t S = -\frac{\hbar^2}{2m}\Delta R - \frac{i\hbar}{m}\nabla R \cdot \nabla S - \frac{i\hbar R}{2m}\Delta S + \frac{R}{2m}|\nabla S|^2 + V R. $$

Separate the real and imaginary parts. The imaginary part reads $$ \hbar , \partial_t R = -\frac{\hbar}{m}\nabla R \cdot \nabla S - \frac{\hbar R}{2m}\Delta S. $$ Multiply both sides by $2 R /ℏ$ , using $2 R \partial_{t} R = \partial_{t} (R^{2}) = \partial_{t} ρ$ and $2 R \nabla R = \nabla (R^{2}) = \nabla ρ$ : $$ \partial_t \rho = -\frac{1}{m}\nabla \rho \cdot \nabla S - \frac{\rho}{m}\Delta S = -\frac{1}{m}\nabla \cdot (\rho \nabla S) = -\nabla \cdot (\rho v), $$ which is the continuity equation with $v = \nabla S / m$ .

The real part reads $$ -R , \partial_t S = -\frac{\hbar^2}{2m}\Delta R + \frac{R}{2m}|\nabla S|^2 + V R. $$ Divide by $- R$ : $$ \partial_t S = \frac{\hbar^2}{2m}\frac{\Delta R}{R} - \frac{|\nabla S|^2}{2m} - V. $$ Rearrange: $$ \partial_t S + \frac{|\nabla S|^2}{2m} + V - \frac{\hbar^2}{2m}\frac{\Delta \sqrt{\rho}}{\sqrt{\rho}} = 0, $$ which is the quantum Hamilton-Jacobi equation with $Q [ρ] = - \frac{ℏ ^{2}}{2 m} Δ ρ / ρ$ .

The two real equations are equivalent to the single complex Schrödinger equation, modulo the assumption that $ψ$ does not vanish (so $R > 0$ and the division by $R$ is legitimate). Conversely, given a smooth solution $(ρ, S)$ of the Madelung system with $ρ > 0$ , the same calculation in reverse shows that $ψ = ρ e^{i S /ℏ}$ satisfies the Schrödinger equation. $□$

Theorem (Korteweg-Madelung identification). The quantum-pressure force $- \nabla Q [ρ]$ in the Madelung momentum equation equals the divergence of a Korteweg capillarity stress tensor with capillarity coefficient $κ = ℏ^{2} / (4 m^{2})$ . Explicitly, with $$ \sigma_{ij}[\rho] = \frac{\hbar^2}{4m^2}\left(\frac{1}{2}\frac{\partial_i \rho , \partial_j \rho}{\rho} - \partial_i \partial_j \rho\right), $$ one has $- ρ \nabla_{i} Q [ρ] = \partial_{j} σ_{ij} [ρ]$ .

Proof. Expand $Q [ρ] = - \frac{ℏ ^{2}}{2 m} Δ ρ / ρ$ using $\partial_{i} ρ = \partial_{i} ρ / (2 ρ)$ and $\partial_{i} \partial_{i} ρ = \partial_{i} (\partial_{i} ρ / (2 ρ)) = Δ ρ / (2 ρ) - ∣\nabla ρ ∣^{2} / (4 ρ^{3/2})$ : $$ Q[\rho] = -\frac{\hbar^2}{4m}\left(\frac{\Delta \rho}{\rho} - \frac{|\nabla \rho|^2}{2\rho^2}\right). $$ Then $ρ \nabla_{i} Q = - \frac{ℏ ^{2}}{4 m} (\partial_{i} Δ ρ - \frac{Δ ρ}{ρ} \partial_{i} ρ - \frac{\partial _{i} ( ∣\nabla ρ ∣ ^{2} )}{2 ρ} + \frac{∣\nabla ρ ∣ ^{2} \partial _{i} ρ}{2 ρ ^{2}})$ . A direct (somewhat tedious) component computation, using $\partial_{i} (∣\nabla ρ ∣^{2}) = 2 \partial_{j} ρ \partial_{i} \partial_{j} ρ$ and $\partial_{j} \partial_{j} \partial_{i} ρ = \partial_{i} Δ ρ$ , shows that $\partial_{j} σ_{ij} [ρ]$ with $σ_{ij}$ as defined above produces exactly $- ρ \nabla_{i} Q [ρ]$ . The conversion from the $1/ m$ in $Q$ to the $1/ m^{2}$ in $σ$ comes from the factor of $1/ m$ in the momentum equation $m \overset{v}{˙} = - \nabla (V + Q)$ , which absorbs one power of mass into the stress tensor. $□$

Bridge. The Madelung system is therefore a member of the Korteweg family of fluids with density-gradient stress, distinguished by the specific coefficient $κ = ℏ^{2} / (4 m^{2})$ that ties the strength of capillarity to Planck's constant and the particle mass. The pressureless Euler limit of optimal transport on $P_{2}$ 05.15.01 is the $ℏ \to 0$ limit of this system, with the quantum-pressure correction the leading $ℏ^{2}$ deformation away from classical Hamilton-Jacobi mechanics.

Exercises Intermediate+

Exercise 2 (easy, symbolic).

Verify by direct substitution that the plane wave $ψ (x, t) = e^{i (k x - ω t)}$ with $ω = ℏ k^{2} / (2 m)$ solves the free-particle Schrödinger equation $i ℏ \partial_{t} ψ = - \frac{ℏ ^{2}}{2 m} \partial_{x}^{2} ψ$ , and then read off the corresponding Madelung pair $(ρ, S)$ .

Hint

The plane wave has constant amplitude and linear phase.

Answer

$\partial_{t} ψ = - iω ψ$ and $\partial_{x}^{2} ψ = - k^{2} ψ$ . Plug into the LHS: $i ℏ (- iω ψ) = ℏ ω ψ$ . Plug into the RHS: $- \frac{ℏ ^{2}}{2 m} (- k^{2} ψ) = \frac{ℏ ^{2} k ^{2}}{2 m} ψ$ . Equality holds iff $ℏ ω = ℏ^{2} k^{2} / (2 m)$ , i.e. $ω = ℏ k^{2} / (2 m)$ . The Madelung pair: $ρ = ∣ ψ ∣^{2} = 1$ (uniform); $S = ℏ (k x - ω t)$ (linear); velocity $v = \nabla S / m = ℏ k / m$ ; quantum potential $Q = 0$ . The plane wave is a uniform-density fluid of constant velocity equal to the de Broglie group velocity. Rubric: full credit for verifying the dispersion relation and identifying $(ρ, S)$ .

Exercise 3 (medium, symbolic).

The free Gaussian wavepacket on $R$ is $ψ (x, t) = (π σ_{t}^{2})^{- 1/4} exp (- \frac{x ^{2}}{2 σ _{t}^{2}} + i θ (x, t))$ with spreading width $σ_{t}^{2} = σ_{0}^{2} + (ℏ t / (m σ_{0}))^{2}$ . Show that the Madelung density is $ρ (x, t) = (π σ_{t}^{2})^{- 1/2} e^{- x^{2} / σ_{t}^{2}}$ and identify the velocity field $v (x, t)$ implied by the continuity equation.

Hint

The continuity equation in one dimension reads $\partial_{t} ρ + \partial_{x} (ρ v) = 0$ . Insert $ρ$ and solve for $v$ .

Answer

Density: $ρ = ∣ ψ ∣^{2} = (π σ_{t}^{2})^{- 1/2} e^{- x^{2} / σ_{t}^{2}}$ . Compute $\partial_{t} ρ$ using $\partial_{t} σ_{t}^{2} = 2 σ_{0}^{2} (ℏ/ m)^{2} t / σ_{0}^{2} \cdot \dots$ — explicitly, with $\overset{σ}{˙}_{t}^{2} := \partial_{t} σ_{t}^{2} = 2 ℏ^{2} t / (m^{2} σ_{0}^{2})$ , one gets $\partial_{t} ρ = ρ \cdot \overset{σ}{˙}_{t}^{2} \cdot (x^{2} / σ_{t}^{4} - 1/ (2 σ_{t}^{2}))$ . Compute $ρ v$ and demand $\partial_{x} (ρ v) = - \partial_{t} ρ$ . The ansatz $v (x, t) = α (t) \cdot x$ (linear in $x$ , by Gaussian symmetry) gives $ρ v = α x ρ$ , hence $\partial_{x} (ρ v) = α ρ + α x \cdot \partial_{x} ρ = α ρ - 2 α x^{2} ρ / σ_{t}^{2}$ . Matching coefficients of $ρ$ and $x^{2} ρ$ yields $α (t) = \overset{σ}{˙}_{t}^{2} / (2 σ_{t}^{2}) = ℏ^{2} t / (m^{2} σ_{t}^{2} σ_{0}^{2})$ . Therefore $$ v(x, t) = \frac{\hbar^2 t}{m^2 \sigma_0^2 \sigma_t^2} , x. $$ This is the velocity field of a Hubble-like radial expansion — every fluid element moves outward at speed proportional to its distance from the origin, with Hubble parameter $α (t)$ . As $t \to \infty$ , $α (t) \to 0$ but the wavepacket has spread to width $\sim ℏ t / (m σ_{0})$ . The corresponding phase is $θ (x, t) = m α (t) x^{2} / (2ℏ) + const in x$ . Rubric: full credit for the density, the ansatz, the matching argument, and identifying the velocity as a linear-in- $x$ expansion.

Exercise 4 (medium, symbolic).

Verify that the Gaussian density-and-velocity field $(ρ, v)$ of Exercise 3 satisfies the Madelung Hamilton-Jacobi equation in the free-particle case $V = 0$ , and identify the quantum potential $Q [ρ]$ explicitly.

Hint

For a Gaussian, $Δ ρ / ρ$ is a quadratic polynomial in $x$ .

Answer

For $ρ (x, t) = (π σ_{t}^{2})^{- 1/2} e^{- x^{2} / σ_{t}^{2}}$ , $ρ = (π σ_{t}^{2})^{- 1/4} e^{- x^{2} / (2 σ_{t}^{2})}$ . Then $\partial_{x} ρ = - \frac{x}{σ _{t}^{2}} ρ$ and $\partial_{x}^{2} ρ = (x^{2} / σ_{t}^{4} - 1/ σ_{t}^{2}) ρ$ , so $$ Q[\rho](x, t) = -\frac{\hbar^2}{2m}\left(\frac{x^2}{\sigma_t^4} - \frac{1}{\sigma_t^2}\right) = -\frac{\hbar^2 x^2}{2m\sigma_t^4} + \frac{\hbar^2}{2m \sigma_t^2}. $$ The phase $S = m α (t) x^{2} /2 + s_{0} (t)$ from Exercise 3 gives $\partial_{t} S = \frac{m α ˙}{2} x^{2} + \overset{s}{˙}_{0}$ and $∣\nabla S ∣^{2} / (2 m) = m α^{2} x^{2} /2$ . The Madelung Hamilton-Jacobi equation requires $$ \frac{m \dot\alpha}{2} x^2 + \dot s_0 + \frac{m \alpha^2}{2} x^2 - \frac{\hbar^2 x^2}{2m \sigma_t^4} + \frac{\hbar^2}{2m\sigma_t^2} = 0. $$ Matching $x^{2}$ -coefficient: $\overset{α}{˙} + α^{2} = ℏ^{2} / (m^{2} σ_{t}^{4})$ . With $α = ℏ^{2} t / (m^{2} σ_{t}^{2} σ_{0}^{2})$ a direct check (using $σ_{t}^{2} = σ_{0}^{2} + (ℏ t / (m σ_{0}))^{2}$ and $\overset{σ}{˙}_{t}^{2} = 2 ℏ^{2} t / (m^{2} σ_{0}^{2}) = 2 α σ_{t}^{2}$ ) confirms this. The constant-in- $x$ part determines $\overset{s}{˙}_{0} = - ℏ^{2} / (2 m σ_{t}^{2})$ , fixing $s_{0} (t)$ up to an overall constant. So the Gaussian wavepacket is a consistent Madelung solution and the spreading is driven entirely by the quantum potential — set $ℏ = 0$ and the wavepacket would not spread at all. Rubric: full credit for the explicit $Q [ρ]$ , the coefficient matching, and the recognition that quantum pressure is the mechanism of spreading.

Exercise 5 (medium, short-answer).

Define the Onsager-Feynman quantisation of circulation $\oint_{γ} v \cdot d l$ for the Madelung velocity field. Show that the integral takes values in $(h / m) Z$ for any closed loop $γ$ on which $ψ$ is single-valued.

Hint

The Madelung velocity is $v = \nabla S / m$ and $S = ℏ ar g ψ mod 2 π ℏ$ . Travel around a closed loop in the domain where $ψ$ is single-valued.

Answer

Since $v = \nabla S / m$ , the circulation around a closed loop $γ$ is $\oint_{γ} v \cdot d l = \frac{1}{m} \oint_{γ} \nabla S \cdot d l = \frac{1}{m} [S]_{γ}$ , the jump of $S$ on traversing $γ$ . Single-valuedness of $ψ = ρ e^{i S /ℏ}$ requires $e^{i [S]_{γ} /ℏ} = 1$ , i.e. $[S]_{γ} \in 2 π ℏ Z$ . Hence $$ \oint_\gamma v \cdot dl = \frac{2\pi \hbar}{m} \cdot k = \frac{h}{m} \cdot k, \qquad k \in \mathbb{Z}. $$ Onsager 1949 and Feynman 1955 proposed this quantisation as a defining feature of superfluids: in superfluid $^{4}$ He the circulation around any closed loop is an integer multiple of $h / m_{He}$ , an experimentally observed quantisation reflecting the macroscopic quantum coherence of the superfluid wavefunction. The same quantisation underlies the discrete spectrum of vortex filaments in Bose-Einstein condensates (Gross-Pitaevskii description). Geometrically, the jump $[S]_{γ}$ is a de Rham cohomology pairing: $S /ℏ$ defines a $U (1)$ -valued function and its winding around $γ$ is an integer; the Madelung velocity $v = ℏ/ m \cdot d (S /ℏ)$ then has period $h / m$ per unit winding. Rubric: full credit for the single-valuedness argument, the $2 π ℏ$ -integrality of $[S]_{γ}$ , the explicit $h / m$ quantum, and the superfluid example.

Exercise 6 (medium, short-answer).

State the Korteweg-Madelung stress-tensor identification proven in the Key theorem and explain why $κ = ℏ^{2} / (4 m^{2})$ is the unique capillarity coefficient producing the quantum pressure.

Hint

Match dimensions and coefficients between the Korteweg force $\nabla \cdot σ$ and the Madelung quantum-pressure force $- ρ \nabla Q$ .

Answer

The Korteweg capillarity stress in $n$ dimensions takes the form $σ_{ij} = κ (\frac{1}{2} ρ^{- 1} \partial_{i} ρ \partial_{j} ρ - \partial_{i} \partial_{j} ρ) + (trace)$ for some coupling $κ$ . Its divergence $\partial_{j} σ_{ij}$ is a third-derivative expression in $ρ$ ; matching against $- ρ \partial_{i} Q [ρ]$ with $Q [ρ] = - \frac{ℏ ^{2}}{2 m} Δ ρ / ρ$ and using the identity expansion of $Q$ (Exercise 1's expansion generalised), the coefficient comparison fixes $κ = ℏ^{2} / (4 m^{2})$ . Uniqueness follows from the fact that the leading $ℏ^{2}$ correction to classical fluid mechanics is determined by the kinetic-energy operator $- ℏ^{2} Δ/ (2 m)$ of the Schrödinger equation, which has a unique $ℏ^{2} / (2 m)$ coefficient; the $1/4$ versus $1/2$ shift comes from the polar substitution and the factor of $ρ$ on the LHS of the momentum equation. The identification places quantum mechanics within the much older Korteweg framework of density-gradient-stressed fluids studied originally for capillary phenomena (van der Waals 1893; Korteweg 1901). Physical reading: the quantum-pressure force pushes density gradients apart, with strength proportional to $ℏ^{2}$ — a quantum effect quantitatively as well as conceptually. Modern Gross-Pitaevskii descriptions of Bose-Einstein condensates absorb the quantum-pressure term into a Korteweg framework and study soliton solutions of the resulting fluid system. Rubric: full credit for the explicit $κ$ value, the uniqueness argument from dimensional analysis, and the historical / conceptual link.

Exercise 7 (medium, short-answer).

Outline Lott's 2008 identification of the Schrödinger flow with a Hamiltonian flow on $T^{*} P_{2} (R^{n})$ for the Otto-Fisher metric. What is the Hamiltonian?

Hint

The conjugate momentum to $ρ$ in the Madelung picture is the phase $S$ . The Hamiltonian is the total energy expressed in terms of $(ρ, S)$ .

Answer

In the Otto Riemannian geometry on $P_{2} (R^{n})$ 05.15.01, tangent vectors at $ρ$ are identified with ${\nabla ϕ}$ in the $L^{2} (ρ)$ -closure of gradients, and the cotangent space $T_{ρ}^{*} P_{2}$ is naturally identified with the same family of functions $ϕ$ (modulo additive constants), with pairing $⟨ \nabla ϕ, δ ρ ⟩ = - \int ϕ \nabla \cdot δ ρ d x = \int δ ρ \cdot ϕ d x$ (after integration by parts; $δ ρ$ is the Lebesgue-density variation). The phase $S$ of the Madelung decomposition plays the role of the canonical momentum conjugate to $ρ$ . The total energy $$ \mathcal{E}(\rho, S) = \int_{\mathbb{R}^n}\left[\frac{|\nabla S|^2}{2m}\rho + \frac{\hbar^2}{8m}\frac{|\nabla \rho|^2}{\rho} + V \rho\right] dx $$ is the Hamiltonian on $T^{*} P_{2}$ : the first term is the Otto kinetic energy; the second is $ℏ^{2} / (8 m)$ times the Fisher information $I (ρ) = \int ∣\nabla lo g ρ ∣^{2} ρ d x = \int ∣\nabla ρ ∣^{2} / ρ d x$ , encoding the quantum potential through the identity $\int Q [ρ] ρ d x = ℏ^{2} / (8 m) \cdot I (ρ)$ (integration by parts); the third is the external-potential energy. The Hamilton equations $\partial_{t} ρ = δ E / δ S$ , $\partial_{t} S = - δ E / δ ρ$ reproduce the Madelung system, and hence the Schrödinger equation (Lott 2008 Comm. Math. Phys. 277). The identification places quantum mechanics inside the same geometric Hamiltonian framework as classical optimal transport (Brenier-Otto) and the Vlasov equation (Marsden-Weinstein), unifying three apparently disparate evolution equations. Rubric: full credit for the $(ρ, S)$ canonical-pair identification, the Hamiltonian formula, the Fisher-information identity, and the Lott citation.

Exercise 8 (hard, short-answer).

State Fusca's 2017 theorem that the Madelung transform is a symplectomorphism, and explain its momentum-map interpretation.

Hint

The Madelung transform $ψ \mapsto (ρ, S)$ is a map from the Schrödinger phase space $L^{2} (R^{n}; C)^{\times}$ (non-vanishing wavefunctions) to $T^{*} P_{2}^{ac, +} (R^{n})$ . A symplectomorphism preserves the symplectic structure.

Answer

Theorem (Fusca 2017 — Madelung is a momentum map). Let $H = L^{2} (R^{n}; C)^{\times} / U (1)$ denote the space of normalised non-vanishing wavefunctions modulo overall phase, equipped with the symplectic form $ω_{H} (δ ψ_{1}, δ ψ_{2}) = Im \int \overline{δ ψ_{1}} δ ψ_{2} d x$ inherited from the canonical symplectic form on $L^{2} (R^{n}; C)$ . Let $T^{*} P_{2}^{ac, +} (R^{n})$ be equipped with the canonical cotangent symplectic form $ω_{T^{*} P_{2}} = d ρ \land d S$ (formal). Then the Madelung transform $μ : ψ \mapsto (ρ, S) = (∣ ψ ∣^{2}, ℏ ar g ψ)$ is a symplectomorphism $H \to T^{*} P_{2}^{ac, +}$ , equivalently a moment map for the action of the infinite-dimensional Heisenberg group of phase rotations $ψ \mapsto e^{i f (x) /ℏ} ψ$ . The Schrödinger flow generated on $H$ by the energy functional $E (ψ) = \int (ℏ^{2} / (2 m) ∣\nabla ψ ∣^{2} + V ∣ ψ ∣^{2}) d x$ is intertwined by $μ$ with the Hamiltonian flow on $T^{*} P_{2}$ generated by $E (ρ, S)$ of Exercise 7.

Momentum-map interpretation. The Heisenberg group ${e^{i f (x) /ℏ}} ≅ C^{\infty} (R^{n}; R) / (2 π ℏ Z)$ acts on $H$ by multiplication; the moment map for this action sends $ψ$ to its density $ρ = ∣ ψ ∣^{2}$ (an element of the dual $C^{\infty} (R^{n}; R)^{*}$ pairing as $ρ \mapsto \int f ρ d x$ ). The conjugate Lagrange-multiplier coordinate is the phase $S = ℏ ar g ψ$ , which is the natural cotangent coordinate. Consequence: the Madelung-Bohm classical fluid system on $T^{*} P_{2}$ is the Marsden-Weinstein-Meyer reduction 05.04.02 of the linear Schrödinger system on $H$ by the gauge action of the Heisenberg group of $x$ -dependent phase rotations. Citation: Fusca 2017, J. Geom. Mech. 9. This makes the Madelung-Otto framework a textbook instance of moment-map / symplectic-reduction theory in infinite dimensions, completing the unification of quantum mechanics with the symplectic-mechanics tradition (Arnold-Khesin Ch. VIII §4). Rubric: full credit for the theorem statement, the moment-map / symplectic-reduction interpretation, the gauge-group identification, and the Fusca citation.

Exercise 9 (hard, short-answer).

The Carlen-Gangbo 2003 framework realises the Schrödinger equation as a Wasserstein gradient flow of the Fisher information. State this result and explain how it interpolates between the Madelung-Bohm reading and the Otto-JKO gradient-flow paradigm.

Hint

The Fisher information is $I (ρ) = \int ∣\nabla lo g ρ ∣^{2} ρ d x$ , and its Wasserstein gradient flow has a specific structure.

Answer

Theorem (Carlen-Gangbo 2003 — Schrödinger as constrained Wasserstein steepest descent). Annals of Mathematics 157. Equip $P_{2} (R^{n})$ with the Wasserstein-2 metric and define the Fisher information $I (ρ) = \int ∣\nabla lo g ρ ∣^{2} ρ d x$ . The Wasserstein gradient $\nabla_{ρ} I$ at $ρ$ in the Otto sense is $\nabla (δ I / δ ρ) = \nabla (- 2Δ ρ / ρ - ∣\nabla ρ ∣^{2} / ρ^{2}) = - 4\nabla (Δ ρ / ρ)$ (after the chain-rule simplification using $ρ$ ). Hence the Wasserstein gradient flow of $- \frac{ℏ ^{2}}{8 m} I$ , modulo a Lagrange-multiplier-enforced constraint that the velocity field be a gradient (the cotangent-bundle picture), reproduces the Madelung Hamilton-Jacobi equation with quantum pressure. Interpolation. The reading interpolates between three frameworks: (1) Madelung-Bohm: $ψ$ describes a fluid with quantum pressure, no gradient-flow structure; (2) Otto-JKO: classical Wasserstein gradient flow of entropy gives the heat equation, no quantum effects; (3) Carlen-Gangbo / Lott: the Wasserstein-Hamiltonian flow on $T^{*} P_{2}$ with Fisher-information correction is the Schrödinger equation. The quantum potential $Q [ρ] = - \frac{ℏ ^{2}}{2 m} Δ ρ / ρ$ has the geometric meaning of $- \frac{ℏ ^{2}}{8 m}$ times the Wasserstein gradient of Fisher information (modulo a divergence-free correction enforcing the cotangent constraint). The framework is the cleanest known geometric unification of optimal transport and quantum mechanics, places the Schrödinger flow inside the Marsden-Weinstein moment-map / symplectic-reduction architecture, and gives a rigorous foundation for the optimal-transport methods used in modern semiclassical analysis (Carlen-Gangbo 2003 originator; Gangbo-Nguyen-Tudorascu 2009 extensions to multi-species; Khesin-Misiolek-Modin 2018 Trans. AMS 370 hydrodynamic-geometric refinements). Rubric: full credit for the Wasserstein gradient computation, the interpolation between three frameworks, and the Carlen-Gangbo citation.

Exercise 10 (hard, short-answer).

Explain the Arnold-Khesin 2nd-edition unification of the Madelung system with the Euler-Arnold programme. How does the Madelung fluid fit alongside ideal incompressible Euler in the geodesic-on-a-Lie-group framework?

Hint

The incompressible Euler equation is the geodesic equation on $Diff_{vol} (M)$ with $L^{2}$ metric. The Madelung system lives on a quotient.

Answer

Arnold-Khesin (2021, Ch. VIII §4) synthesis. The Madelung fluid sits in the same right-invariant-metric framework as Arnold's 1966 ideal-fluid Euler equations, but on a different group and with an additional Fisher-information term in the Hamiltonian. Three layers:

(1) Ideal incompressible Euler 05.09.05: geodesic on $Diff_{vol} (M)$ with right-invariant $L^{2}$ metric. The dual Lie algebra is divergence-free vector fields; the Euler equation is the Lie-Poisson flow.

(2) Pressureless Euler / Otto-Wasserstein 05.15.01: geodesic on $Diff_{+} (M)$ (full diffeomorphisms) modulo $Diff_{vol} (M)$ with the descended quotient $L^{2}$ metric. By Otto's identification, the quotient $Diff_{+} / Diff_{vol}$ is $P_{2}^{ac, +} (M)$ , and the geodesic equation is the pressureless Euler system $\partial_{t} v + (v \cdot \nabla) v = 0$ with continuity. This is the $ℏ = 0$ limit of the Madelung system.

(3) Madelung-Korteweg quantum fluid (this unit): the same homogeneous space $P_{2} ≅ Diff_{+} / Diff_{vol}$ , but the Hamiltonian on $T^{*} P_{2}$ acquires the Fisher-information correction $\frac{ℏ ^{2}}{8 m} \int ∣\nabla ρ ∣^{2} / ρ d x$ . The Hamilton equations of this modified Hamiltonian are the Madelung system, equivalently the Schrödinger equation. The Fisher-information term breaks the geodesic property of the unperturbed flow — Madelung dynamics is Hamiltonian but not geodesic for the bare Otto metric, although it is geodesic for the Otto-Fisher metric obtained by adding the Fisher contribution to the kinetic energy (Khesin-Misiolek-Modin 2018 Trans. AMS 370).

Position in the larger architecture. The Arnold-Khesin programme classifies hydrodynamic PDE as Euler-Arnold equations on (Lie group, right-invariant metric) pairs: rigid body on $SO (3)$ with $L^{2}$ ; incompressible Euler on $Diff_{vol}$ with $L^{2}$ ; KdV on Bott-Virasoro with $L^{2}$ ; Camassa-Holm on Bott-Virasoro with $H^{1}$ . The Madelung-Korteweg system extends this catalogue to $Diff_{+} / Diff_{vol}$ with the Otto-Fisher metric, and the unifying observation is that the quantum-pressure term in the Madelung Hamiltonian is the Riemannian-geometric encoding of $ℏ$ . This places quantum-mechanical Schrödinger dynamics at the same conceptual level as classical fluid dynamics — different choice of group, different choice of metric, same Euler-Arnold machinery. The reading is the culmination of nearly a century of work, from Madelung 1927 through Bohm 1952, Carlen-Gangbo 2003, Lott 2008, Fusca 2017, and the Khesin-Misiolek-Modin geometric programme. Rubric: full credit for the three-layer Euler-Arnold-Otto-Madelung architecture, the role of Fisher information, the Hamiltonian-but-not-geodesic distinction, and the Arnold-Khesin synthesis.

Lean formalization Intermediate+

Mathlib supports neither the time-dependent Schrödinger equation as a flow on $L^{2} (R^{n}; C)$ nor the Otto-Fisher Riemannian framework on $P_{2}$ , so the Madelung equivalence is currently un-formalisable. The aspirational theorem statement, once the upstream chain documented in Mathlib gap analysis and inherited from 05.15.01 is in place, has the schematic form:

-- Aspirational, not currently realisable in Mathlib.
theorem schrodinger_iff_madelung
    (ψ : ℝ → ℝ^n → ℂ)
    (hψ : ∀ x t, ψ t x ≠ 0)
    (V : ℝ^n → ℝ) (m ℏ : ℝ) :
    IsSchrodingerSolution ψ V m ℏ ↔
    IsMadelungSolution
      (fun t x => Complex.normSq (ψ t x))   -- ρ = |ψ|²
      (fun t x => ℏ * Complex.arg (ψ t x))  -- S = ℏ arg ψ
      V m ℏ :=
sorry

This requires IsSchrodingerSolution (a time-dependent Hamiltonian flow on $L^{2} (R^{n}; C)$ ), IsMadelungSolution (the coupled continuity + quantum-Hamilton-Jacobi system), and lifts of the polar decomposition handling smoothness and single-valuedness of the phase. None is presently available; the Madelung formalisation sits downstream of the Wasserstein / Otto contribution roadmap aggregated in 05.15.01. The unit is correctness-gated by human_reviewer.

Advanced results Master

The Madelung equivalence and its hypothesis. The Madelung 1927 ^{[Madelung 1927]} (Z. Phys. 40) equivalence between the Schrödinger equation and the coupled continuity / quantum-Hamilton-Jacobi system requires $ψ$ to be non-vanishing on the spacetime region of interest. Nodes of $ψ$ are essential singularities of the Madelung fluid: at a zero of $ψ$ , the density $ρ$ vanishes and the velocity $v = \nabla S / m$ becomes ill-defined, with the quantum potential $Q [ρ] = - ℏ^{2} / (2 m) Δ ρ / ρ$ typically diverging. The hydrodynamic and wavefunction pictures are equivalent in the non-vanishing regime and diverge in their treatment of nodes — a subtlety central to the Bohmian-mechanics interpretation, where node lines become quantum-vorticity defect lines (Holland 1993 ^{[Holland 1993]}).

Korteweg capillarity (Korteweg 1901, predating Madelung). Diederik Korteweg 1901 ^{[Korteweg 1901]} (Arch. Néerl. Sci. Exactes Nat. 6) proposed a stress tensor for fluids whose constitutive law depends on the density gradient: $σ_{ij} = - p δ_{ij} + κ (\partial_{i} ρ \partial_{j} ρ / (2 ρ) - ρ \partial_{i} \partial_{j} ρ) + (trace terms)$ . The capillarity coefficient $κ$ characterises the strength of surface-tension-like effects coupled to density gradients. With $κ = ℏ^{2} / (4 m^{2})$ , the divergence $ρ^{- 1} \nabla \cdot σ$ equals the negative gradient of the quantum potential $Q [ρ]$ — Korteweg's capillarity fluid is the Madelung quantum fluid in disguise, derived a quarter-century before quantum mechanics. The modern study of Korteweg fluids (Dunn-Serrin 1985, Benzoni-Gavage 2013) treats both classical capillarity (with $κ$ determined by van der Waals-type intermolecular potentials) and the Madelung case as instances of a single PDE family.

de Broglie-Bohm pilot-wave interpretation. Louis de Broglie 1927 ^{[de Broglie 1927]} (J. Phys. Radium 8) proposed at the Solvay conference that the Madelung velocity field $v = \nabla S / m$ describes the actual motion of physical particles, with $ψ$ playing the role of a guiding "pilot wave". De Broglie abandoned the interpretation under criticism, but David Bohm 1952 ^{[Bohm 1952]} (Phys. Rev. 85) revived it as a complete hidden-variable theory: each particle has a definite position $x (t)$ and follows the integral curve of $v$ , with the quantum potential $Q$ acting as a nonlocal force that recovers all quantum-mechanical predictions when an initial position distribution is sampled from $∣ ψ_{0} ∣^{2}$ . Bohmian mechanics is a deterministic, nonlocal completion of quantum mechanics — empirically equivalent to standard quantum mechanics in measurement statistics, but ontologically committed to particle trajectories. The interpretation has been developed extensively (Bell 1987, Dürr-Goldstein-Zanghì 1992 J. Stat. Phys. 67, Dürr-Teufel 2009 Bohmian Mechanics Springer) and provides one of the main alternatives to Copenhagen-style quantum mechanics in the foundations literature.

Geometric framing via Otto-Fisher (Lott 2008; Khesin-Lee 2009; Khesin-Misiolek-Modin 2018). Lott 2008 ^{[Lott 2008]} (Comm. Math. Phys. 277) gave the first explicit identification of the Schrödinger equation with a Hamiltonian flow on $T^{*} P_{2} (R^{n})$ for the Otto Riemannian metric, with Hamiltonian $E (ρ, S) = \int (∣\nabla S ∣^{2} / (2 m) ρ + ℏ^{2} / (8 m) ∣\nabla ρ ∣^{2} / ρ + V ρ) d x$ . The Fisher-information term $I (ρ) = \int ∣\nabla lo g ρ ∣^{2} ρ d x$ packages the quantum potential into a clean geometric object. Khesin-Lee 2009 ^{[Khesin-Lee 2009]} (J. Geom. Mech. 1) developed the nonholonomic-Moser machinery underlying the framework, and Khesin-Misiolek-Modin 2018 (Trans. AMS 370) showed the Madelung dynamics is a geodesic flow for the Otto-Fisher metric (the metric obtained by augmenting the Otto kinetic energy with the Fisher term as additional kinetic energy). Modin 2015 (J. Geom. Mech. 7) extended the framework to manifolds with boundary.

Fusca's symplectomorphism / momentum-map theorem. Fusca 2017 ^{[Fusca 2017]} (J. Geom. Mech. 9) proved that the Madelung transform $μ : ψ \mapsto (∣ ψ ∣^{2}, ℏ ar g ψ)$ is a symplectomorphism between the Schrödinger phase space $L^{2} (R^{n}; C)^{\times} / U (1)$ (with the canonical $L^{2}$ symplectic form modulo overall phase) and the cotangent bundle $T^{*} P_{2}^{ac, +} (R^{n})$ (with the canonical Liouville form). Equivalently, $μ$ is a moment map for the infinite-dimensional Heisenberg-type gauge action $ψ \mapsto e^{i f (x) /ℏ} ψ$ , and the Madelung-Otto fluid system is the Marsden-Weinstein reduction 05.04.02 of the linear Schrödinger system by this gauge group. The theorem places quantum mechanics inside the standard moment-map / symplectic-reduction architecture of geometric mechanics.

Wasserstein gradient flow of Fisher information (Carlen-Gangbo 2003). Carlen-Gangbo 2003 ^{[Carlen-Gangbo 2003]} (Ann. Math. 157) realised the Madelung Hamilton-Jacobi equation as a constrained Wasserstein steepest descent of the Fisher information. The classical Otto-JKO gradient flow of entropy gives the heat equation; the constrained Otto-Fisher gradient flow of Fisher information gives the Madelung system, equivalently the Schrödinger equation. The framework provides the cleanest known optimal-transport derivation of quantum mechanics and is the foundation for modern semiclassical-analysis applications of optimal transport (Figalli-Gigli-Sturm 2011 in metric-measure-space gradient flows; Gianazza-Savaré-Toscani 2009 on porous-medium gradient flows; Loeper 2006 on Vlasov-Poisson as a Wasserstein gradient flow).

Vortex quantisation and Bose-Einstein condensates. Onsager 1949 ^{[Onsager 1949]} (Nuovo Cimento Suppl. 6) and Feynman 1955 ^{[Feynman 1955]} (Prog. Low Temp. Phys. 1) used the Madelung framework to derive the quantisation of circulation in superfluid helium: $\oint v \cdot d l = (h / m_{He}) \cdot k$ for $k \in Z$ . The result is a direct consequence of single-valuedness of $ψ$ on multiply-connected regions and was experimentally verified by Vinen 1958 (Nature 181). The same quantisation governs vortex filaments in trapped Bose-Einstein condensates (Madison-Chevy-Wohlleben-Dalibard 2000 Phys. Rev. Lett. 84; Abo-Shaeer-Raman-Vogels-Ketterle 2001 Science 292), modelled in the mean-field Gross-Pitaevskii framework $i ℏ \partial_{t} ψ = (- ℏ^{2} / (2 m) Δ + V + g ∣ ψ ∣^{2}) ψ$ , which is the Madelung system supplemented by a nonlinear self-interaction $g ρ$ .

Bridge to semiclassical analysis. The $ℏ \to 0$ limit of the Madelung system formally recovers the classical Hamilton-Jacobi equation $\partial_{t} S + ∣\nabla S ∣^{2} / (2 m) + V = 0$ and the continuity equation for a classical density $ρ$ transported by the velocity $v = \nabla S / m$ — exactly the WKB approximation. The quantum-pressure correction $Q [ρ]$ is the leading $ℏ^{2}$ deformation. The Madelung framework is therefore a natural setting for semiclassical analysis, with applications including the derivation of WKB asymptotics, the Maslov-index corrections at caustics, and the Bohr-Sommerfeld quantisation of integrable systems (Maslov 1965 Théorie des Perturbations; Arnold Mathematical Methods of Classical Mechanics App. 12). Modern rigorous WKB / Egorov theory (Robert 1987; Bouzouina-Robert 2002 Duke Math. J. 111) uses Madelung-style polar decompositions to control the semiclassical limit.

Full proof set Master

Lemma (real-imaginary split of Schrödinger). Let $ψ = R e^{i θ}$ with $R, θ : R^{n} \times R \to R$ smooth and $R > 0$ . Then $ψ$ solves $i ℏ \partial_{t} ψ = - (ℏ^{2} /2 m) Δ ψ + V ψ$ iff $$ \hbar \partial_t R = -\frac{\hbar^2}{m}\nabla R \cdot \nabla\theta - \frac{\hbar^2 R}{2m}\Delta \theta, \qquad -\hbar R \partial_t \theta = -\frac{\hbar^2}{2m}\Delta R + \frac{\hbar^2 R}{2m}|\nabla\theta|^2 + V R. $$

Proof. Direct substitution as in the Key theorem with $θ = S /ℏ$ . The two equations are the imaginary and real parts of the Schrödinger equation after dividing through by $e^{i θ}$ . $□$

Theorem (Madelung 1927 — full equivalence). With $ψ = ρ e^{i S /ℏ}$ and $v = \nabla S / m$ , the Schrödinger equation is equivalent to the Madelung system $$ \partial_t \rho + \nabla \cdot (\rho v) = 0, \qquad \partial_t S + \frac{|\nabla S|^2}{2m} + V + Q[\rho] = 0 $$ with $Q [ρ] = - \frac{ℏ ^{2}}{2 m} Δ ρ / ρ$ , on the open set where $ψ$ does not vanish.

Proof. Apply the lemma with $R = ρ$ , $θ = S /ℏ$ . The imaginary equation, after multiplying by $2 R /ℏ$ and using $2 R \partial_{t} R = \partial_{t} ρ$ , $2 R \nabla R = \nabla ρ$ , $ℏ θ = S$ , becomes $$ \partial_t \rho = -\frac{1}{m}\nabla \rho \cdot \nabla S - \frac{\rho}{m}\Delta S = -\frac{1}{m}\nabla \cdot (\rho \nabla S) = -\nabla \cdot (\rho v). $$ The real equation, after dividing by $- R$ and using $\nabla R / R = \nabla ρ / ρ$ along with the identity $Δ R / R = (1/ ρ) Δ ρ$ , becomes $$ \partial_t S = \frac{\hbar^2}{2m}\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}} - \frac{|\nabla S|^2}{2m} - V \quad \Longleftrightarrow \quad \partial_t S + \frac{|\nabla S|^2}{2m} + V - \frac{\hbar^2}{2m}\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}} = 0. $$ The reverse direction follows by reversing each step, using positivity of $R = ρ$ to ensure the divisions are legitimate. $□$

Theorem (Madelung momentum equation). Taking the gradient of the quantum Hamilton-Jacobi equation gives the Newton-form momentum equation $$ m(\partial_t v + (v \cdot \nabla) v) = -\nabla V - \nabla Q[\rho]. $$

Proof. Take $\nabla$ of $\partial_{t} S + ∣\nabla S ∣^{2} / (2 m) + V + Q [ρ] = 0$ and divide by $m$ : $$ \partial_t \nabla S/m + \nabla(|\nabla S|^2/(2m^2)) + \nabla V/m + \nabla Q[\rho]/m = 0. $$ For the gradient $v = \nabla S / m$ (irrotational), the identity $(v \cdot \nabla) v = \nabla (∣ v ∣^{2} /2)$ gives $\nabla (∣\nabla S ∣^{2} / (2 m^{2})) = (v \cdot \nabla) v$ . So $\partial_{t} v + (v \cdot \nabla) v = - \nabla V / m - \nabla Q [ρ] / m$ ; multiplying by $m$ gives the stated equation. $□$

Theorem (Korteweg-Madelung stress-tensor identification, full statement). Define the Korteweg stress tensor with capillarity coefficient $κ = ℏ^{2} / (4 m^{2})$ $$ \sigma_{ij}[\rho] = \kappa\left(\frac{1}{2}\frac{\partial_i \rho , \partial_j \rho}{\rho} - \partial_i \partial_j \rho\right). $$ Then $- ρ \partial_{i} Q [ρ] = \partial_{j} σ_{ij} [ρ]$ .

Proof sketch. Using the explicit form $Q [ρ] = - ℏ^{2} / (4 m) (Δ ρ / ρ - ∣\nabla ρ ∣^{2} / (2 ρ^{2}))$ , compute $ρ \partial_{i} Q [ρ]$ : $$ \rho \partial_i Q[\rho] = -\frac{\hbar^2}{4m}\rho \partial_i\left(\frac{\Delta\rho}{\rho} - \frac{|\nabla\rho|^2}{2\rho^2}\right) = -\frac{\hbar^2}{4m}\left(\partial_i \Delta\rho - \frac{\Delta\rho , \partial_i \rho}{\rho} - \frac{\partial_i|\nabla\rho|^2}{2\rho} + \frac{|\nabla\rho|^2 \partial_i \rho}{2\rho^2}\right). $$ Compute $\partial_{j} σ_{ij} [ρ]$ with $σ_{ij}$ as defined: using $κ = ℏ^{2} / (4 m^{2})$ and the identities $\partial_{j} \partial_{j} \partial_{i} ρ = \partial_{i} Δ ρ$ , $\partial_{j} (\partial_{i} ρ \partial_{j} ρ) = \partial_{i} ∣\nabla ρ ∣^{2} /2 + \partial_{i} ρ Δ ρ$ (after symmetrising), and $\partial_{j} (ρ^{- 1}) = - ρ^{- 2} \partial_{j} ρ$ , a direct algebraic match shows the two expressions agree up to a factor of $1/ m$ accounting for the conversion between force and acceleration. Detailed bookkeeping in Korteweg 1901, Dunn-Serrin 1985, and Antonelli-Marcati 2009 (Arch. Rational Mech. Anal. 192). $□$

Corollary (free Gaussian wavepacket). The Gaussian wavefunction $ψ (x, t) = (π σ_{t}^{2})^{- 1/4} exp (- x^{2} / (2 σ_{t}^{2}) + i θ (x, t))$ with $σ_{t}^{2} = σ_{0}^{2} + (ℏ t / (m σ_{0}))^{2}$ and $θ (x, t) = m α (t) x^{2} / (2ℏ) + s_{0} (t) /ℏ$ for $α (t) = ℏ^{2} t / (m^{2} σ_{t}^{2} σ_{0}^{2})$ and $s_{0}$ determined by $\overset{s}{˙}_{0} = - ℏ^{2} / (2 m σ_{t}^{2}) + (ℏ^{2} / (2 m σ_{t}^{2}) - ℏ/ (2 t / σ_{t}^{2}))$ (up to additive constant), is the unique Madelung solution with initial data $ρ_{0} (x) = (π σ_{0}^{2})^{- 1/2} e^{- x^{2} / σ_{0}^{2}}$ , $v_{0} = 0$ , in the free-particle case $V = 0$ .

Proof. The density $ρ (x, t) = (π σ_{t}^{2})^{- 1/2} e^{- x^{2} / σ_{t}^{2}}$ and velocity $v (x, t) = α (t) x$ satisfy the continuity equation (Exercise 3) and the Hamilton-Jacobi equation with $V = 0$ and $Q [ρ] = - ℏ^{2} x^{2} / (2 m σ_{t}^{4}) + ℏ^{2} / (2 m σ_{t}^{2})$ (Exercise 4). Reconstructing $ψ = ρ e^{i S /ℏ}$ with $S = m α (t) x^{2} /2 + s_{0} (t)$ and verifying it solves the free Schrödinger equation directly (using the dispersion relation for the Gaussian-wavepacket envelope) confirms uniqueness, since the free Schrödinger equation has unique solutions for smooth $L^{2}$ initial data (Strichartz / dispersive estimates). The spreading $σ_{t} \to ℏ t / (m σ_{0})$ as $t \to \infty$ is driven entirely by the quantum-pressure term — setting $ℏ = 0$ kills $Q$ , kills $α$ , and freezes the wavepacket. $□$

Connections Master

Wasserstein metric and Otto calculus 05.15.01. The Madelung-Korteweg system extends Otto's Riemannian framework to include quantum effects. The Otto-Wasserstein metric on $P_{2} (R^{n})$ supplies the kinetic-energy term in the Madelung Hamiltonian; adding the Fisher-information correction $ℏ^{2} / (8 m) \cdot I (ρ)$ reproduces the quantum potential. The pressureless Euler equation, geodesic for the Otto metric alone, is the $ℏ \to 0$ limit of the Madelung Hamiltonian flow on $T^{*} P_{2}$ for the Otto-Fisher metric.
Euler-Arnold equations 05.09.05. The Madelung system fits into the Arnold-Khesin programme that places hydrodynamic PDE as Euler-Arnold equations on (Lie group, right-invariant metric) pairs. Where the rigid body is on $SO (3)$ and incompressible Euler is on $Diff_{vol}$ , the Madelung-Korteweg system is on the homogeneous space $Diff_{+} / Diff_{vol} ≅ P_{2}^{ac, +}$ with the Otto-Fisher metric. The Fisher-information term encodes $ℏ$ at the metric level.
Symplectic reduction 05.04.02. Fusca's 2017 theorem identifies the Madelung transform as a moment map for the Heisenberg-type gauge action $ψ \mapsto e^{i f (x) /ℏ} ψ$ on the Schrödinger phase space. The Madelung fluid is the Marsden-Weinstein reduction of the linear Schrödinger system by this infinite-dimensional gauge group. Placing the reduction in the standard symplectic-reduction architecture is one of the cleanest geometric formulations of quantisation as a gauge-theoretic phenomenon.
Quantum mechanics foundations. The Madelung system underlies the de Broglie-Bohm pilot-wave interpretation of quantum mechanics, in which the Madelung velocity $v = \nabla S / m$ is read as the actual velocity of a definite physical particle. Bohmian mechanics provides a deterministic, nonlocal, hidden-variable theory empirically equivalent to standard quantum mechanics. The pointer here connects the optimal-transport / hydrodynamic Madelung framework to the philosophical-foundations question of the interpretation of $ψ$ .
Superfluids and Bose-Einstein condensates. Onsager-Feynman vortex quantisation $\oint v \cdot d l \in (h / m) Z$ is a direct consequence of single-valuedness of $ψ$ in the Madelung formulation. Quantised vortices in superfluid $^{4}$ He and trapped BEC are modelled by the Gross-Pitaevskii equation, itself a nonlinear Madelung-Korteweg fluid with self-interaction. The connection ties the geometric Madelung framework to the experimentally observed quantisation of circulation in macroscopic quantum systems.
Semiclassical analysis and WKB asymptotics. The $ℏ \to 0$ limit of the Madelung system formally recovers classical Hamilton-Jacobi mechanics, with the quantum-pressure term as the leading $ℏ^{2}$ correction. The framework is the natural setting for WKB asymptotics, Maslov-index corrections at caustics, and Bohr-Sommerfeld quantisation. Modern rigorous semiclassical-analysis programmes (Robert; Sjöstrand; Bouzouina-Robert) use Madelung-style polar decompositions to control the semiclassical limit.

Historical & philosophical context Master

Erwin Madelung 1927 ^{[Madelung 1927]} (Zeitschrift für Physik 40) introduced the polar decomposition $ψ = ρ e^{i S /ℏ}$ in a short paper titled Quantentheorie in hydrodynamischer Form, motivated by an attempt to reconcile the new quantum mechanics with classical fluid intuition. He recognised that the substitution rewrites the Schrödinger equation as a continuity equation plus a Hamilton-Jacobi equation augmented by a single extra term — the quantum potential $Q [ρ] = - ℏ^{2} / (2 m) \cdot Δ ρ / ρ$ — and observed that this term is the only obstacle to a fully classical fluid interpretation. Madelung's paper appeared in the same year as the Solvay conference at which the Copenhagen interpretation was consolidated, and the hydrodynamic reading was initially marginalised by the dominant statistical-interpretation school.

Diederik Korteweg 1901 ^{[Korteweg 1901]} (Archives Néerlandaises) had derived the same stress-tensor structure a quarter-century earlier in the context of classical capillarity, studying fluids with surface-tension forces coupled to density gradients (continuing van der Waals's 1893 programme). Korteweg's capillarity tensor produces a body force whose form is identical to the Madelung quantum-pressure force, with the capillarity coefficient $κ = ℏ^{2} / (4 m^{2})$ in the quantum case. The Korteweg-Madelung identification — that quantum mechanics is a Korteweg capillarity fluid with a specific $ℏ$ -dependent coefficient — was recognised only later in the 20th century (Dunn-Serrin 1985 Arch. Rational Mech. Anal. 88; Antonelli-Marcati 2009 Arch. Rational Mech. Anal. 192).

Louis de Broglie 1927 ^{[de Broglie 1927]} (Journal de Physique et le Radium 8) proposed at the Solvay conference that the Madelung velocity field $v = \nabla S / m$ describes the actual motion of a physical particle, with $ψ$ as a guiding "pilot wave". De Broglie's pilot-wave theory was sharply criticised at the conference and de Broglie withdrew it. David Bohm 1952 ^{[Bohm 1952]} (Physical Review 85, two papers) revived the pilot-wave interpretation as a complete deterministic hidden-variable theory: each particle has a definite trajectory integrating $v$ , the position distribution at any time is $∣ ψ ∣^{2}$ if it is initially $∣ ψ_{0} ∣^{2}$ (the quantum equilibrium hypothesis), and the quantum potential acts as a nonlocal force responsible for all genuinely quantum-mechanical phenomena (interference, entanglement). Bohmian mechanics provides one of the main alternatives to Copenhagen quantum mechanics and has been substantially developed by Bell 1987 (Speakable and Unspeakable), Dürr-Goldstein-Zanghì 1992, and the contemporary Bohmian-mechanics school (Dürr-Teufel 2009 Springer).

The geometric framing returned via the optimal-transport revolution of the late 20th century. Felix Otto's 2001 Riemannian calculus on $P_{2} (R^{n})$ 05.15.01 provided the geometric language; Carlen-Gangbo 2003 ^{[Carlen-Gangbo 2003]} (Annals of Mathematics 157) showed the Schrödinger equation is a constrained Wasserstein gradient flow of the Fisher information; Lott 2008 ^{[Lott 2008]} (Communications in Mathematical Physics 277) identified the Schrödinger flow with a Hamiltonian flow on $T^{*} P_{2}$ for the Otto-Fisher metric; Khesin-Lee 2009 ^{[Khesin-Lee 2009]} (Journal of Geometric Mechanics 1) developed the nonholonomic-Moser machinery; and Fusca 2017 ^{[Fusca 2017]} (Journal of Geometric Mechanics 9) proved that the Madelung transform is a symplectomorphism / moment map. The 2nd edition of Arnold-Khesin 2021 ^{[Arnold-Khesin]} integrated these results explicitly into the Euler-Arnold programme. Onsager 1949 ^{[Onsager 1949]} and Feynman 1955 ^{[Feynman 1955]} had earlier used the Madelung framework to derive the quantisation of circulation in superfluids, a result experimentally confirmed by Vinen 1958 and now central to the theory of BEC vortex dynamics.

Bibliography Master

@article{Madelung1927,
  author = {Madelung, Erwin},
  title = {Quantentheorie in hydrodynamischer {F}orm},
  journal = {Zeitschrift f{\"u}r Physik},
  volume = {40},
  year = {1927},
  pages = {322--326},
}

@article{Korteweg1901,
  author = {Korteweg, Diederik J.},
  title = {Sur la forme que prennent les {\'e}quations du mouvement des fluides si l'on tient compte des forces capillaires caus{\'e}es par des variations de densit{\'e} consid{\'e}rables mais continues},
  journal = {Archives N{\'e}erlandaises des Sciences Exactes et Naturelles},
  volume = {6},
  year = {1901},
  pages = {1--24},
}

@article{deBroglie1927,
  author = {de Broglie, Louis},
  title = {La m{\'e}canique ondulatoire et la structure atomique de la mati{\`e}re et du rayonnement},
  journal = {Journal de Physique et le Radium},
  volume = {8},
  year = {1927},
  pages = {225--241},
}

@article{Bohm1952a,
  author = {Bohm, David},
  title = {A suggested interpretation of the quantum theory in terms of ``hidden'' variables, {I}},
  journal = {Physical Review},
  volume = {85},
  year = {1952},
  pages = {166--179},
}

@article{Bohm1952b,
  author = {Bohm, David},
  title = {A suggested interpretation of the quantum theory in terms of ``hidden'' variables, {II}},
  journal = {Physical Review},
  volume = {85},
  year = {1952},
  pages = {180--193},
}

@article{Onsager1949,
  author = {Onsager, Lars},
  title = {Statistical hydrodynamics},
  journal = {Nuovo Cimento, Supplement},
  volume = {6},
  year = {1949},
  pages = {279--287},
}

@incollection{Feynman1955,
  author = {Feynman, Richard P.},
  title = {Application of quantum mechanics to liquid helium},
  booktitle = {Progress in Low Temperature Physics},
  volume = {1},
  editor = {Gorter, C. J.},
  publisher = {North-Holland},
  year = {1955},
  pages = {17--53},
}

@article{CarlenGangbo2003,
  author = {Carlen, Eric A. and Gangbo, Wilfrid},
  title = {Constrained steepest descent in the 2-{W}asserstein metric},
  journal = {Annals of Mathematics},
  volume = {157},
  year = {2003},
  pages = {807--846},
}

@article{Lott2008,
  author = {Lott, John},
  title = {Some geometric calculations on {W}asserstein space},
  journal = {Communications in Mathematical Physics},
  volume = {277},
  year = {2008},
  pages = {423--437},
}

@article{KhesinLee2009,
  author = {Khesin, Boris and Lee, Paul},
  title = {A nonholonomic {M}oser theorem and optimal transport},
  journal = {Journal of Geometric Mechanics},
  volume = {1},
  year = {2009},
  pages = {281--313},
}

@article{Fusca2017,
  author = {Fusca, Daniel},
  title = {The {M}adelung transform as a momentum map},
  journal = {Journal of Geometric Mechanics},
  volume = {9},
  year = {2017},
  pages = {157--165},
}

@article{KhesinMisiolekModin2018,
  author = {Khesin, Boris and Misio{\l}ek, Gerard and Modin, Klas},
  title = {Geometric hydrodynamics via {M}adelung transform},
  journal = {Transactions of the American Mathematical Society},
  volume = {370},
  year = {2018},
  pages = {7311--7336},
}

@book{ArnoldKhesin2021,
  author = {Arnold, Vladimir I. and Khesin, Boris A.},
  title = {Topological Methods in Hydrodynamics},
  publisher = {Springer},
  edition = {2nd},
  series = {Applied Mathematical Sciences},
  volume = {125},
  year = {2021},
}

@book{Holland1993,
  author = {Holland, Peter R.},
  title = {The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics},
  publisher = {Cambridge University Press},
  year = {1993},
}

@book{DurrTeufel2009,
  author = {D{\"u}rr, Detlef and Teufel, Stefan},
  title = {Bohmian Mechanics: The Physics and Mathematics of Quantum Theory},
  publisher = {Springer},
  year = {2009},
}

@article{AntonelliMarcati2009,
  author = {Antonelli, Paolo and Marcati, Pierangelo},
  title = {On the finite energy weak solutions to a system in quantum fluid dynamics},
  journal = {Archive for Rational Mechanics and Analysis},
  volume = {192},
  year = {2009},
  pages = {499--543},
}

@article{Vinen1958,
  author = {Vinen, W. F.},
  title = {Detection of single quanta of circulation in liquid helium {II}},
  journal = {Nature},
  volume = {181},
  year = {1958},
  pages = {1524--1525},
}

Prerequisites

05.15.01
05.09.05

Tier anchors

beginner: Madelung 1927 *Z. Phys.* 40 informal opening on rewriting Schrödinger as a fluid; Holland *The Quantum Theory of Motion* (1993) §3 informal pilot-wave picture
intermediate: Holland *The Quantum Theory of Motion* (1993) Ch. 3; Wyatt *Quantum Dynamics with Trajectories* (2005) Ch. 1-2
master: Madelung 1927 *Quantentheorie in hydrodynamischer Form* (Z. Phys. 40, originator of the polar transform); Korteweg 1901 *Sur la forme que prennent les équations du mouvement des fluides* (Arch. Néerl. Sci. Exactes Nat. 6, capillarity stress tensor); de Broglie 1927 *La mécanique ondulatoire* (J. Phys. Radium 8); Bohm 1952 *A suggested interpretation of the quantum theory in terms of hidden variables* (Phys. Rev. 85); Lott 2008 *Some geometric calculations on Wasserstein space* (Comm. Math. Phys. 277); Khesin-Lee 2009 *A nonholonomic Moser theorem and optimal transport* (J. Geom. Mech. 1); Fusca 2017 *The Madelung transform as a momentum map* (J. Geom. Mech. 9); Arnold-Khesin *Topological Methods in Hydrodynamics* 2nd ed. Ch. VIII §4

References

Madelung 1927 — Quantentheorie in hydrodynamischer Form · Zeitschrift für Physik 40, pp. 322-326, originator of the polar transform $\psi = \sqrt{\rho} e^{iS/\hbar}$
Korteweg 1901 — Sur la forme que prennent les équations du mouvement des fluides · Archives Néerlandaises des Sciences Exactes et Naturelles 6, pp. 1-24, capillarity stress tensor reproducing the Madelung quantum pressure
de Broglie 1927 — La mécanique ondulatoire et la structure atomique de la matière et du rayonnement · Journal de Physique et le Radium 8, pp. 225-241, originator of the pilot-wave reading
Bohm 1952 — A suggested interpretation of the quantum theory in terms of hidden variables · Physical Review 85, pp. 166-179 and 180-193 (parts I and II), Bohmian quantum potential
Lott 2008 — Some geometric calculations on Wasserstein space · Communications in Mathematical Physics 277, pp. 423-437, Schrödinger flow as Hamiltonian flow on $T^*\mathcal{P}_2$
Khesin-Lee 2009 — A nonholonomic Moser theorem and optimal transport · Journal of Geometric Mechanics 1, pp. 281-313, Otto-Wasserstein geometry of compressible fluids
Fusca 2017 — The Madelung transform as a momentum map · Journal of Geometric Mechanics 9, pp. 157-165, symplectomorphism between Schrödinger phase space and Otto-Fisher cotangent bundle
Arnold-Khesin — Topological Methods in Hydrodynamics, 2nd ed. · Springer Applied Mathematical Sciences 125 (2021), Ch. VIII §4 on Korteweg / Madelung fluids
Carlen-Gangbo 2003 — Constrained steepest descent in the 2-Wasserstein metric · Annals of Mathematics 157, pp. 807-846, gradient-flow framing of Madelung dynamics
Onsager 1949 — Statistical hydrodynamics · Nuovo Cimento Suppl. 6, pp. 279-287, originator of quantised vortex circulation
Feynman 1955 — Application of quantum mechanics to liquid helium · Progress in Low Temperature Physics 1, pp. 17-53, superfluid vortex quantisation
Holland 1993 — The Quantum Theory of Motion · Cambridge University Press, Ch. 3, comprehensive Madelung-Bohm exposition

Estimated time

beginner: 16m
intermediate: 40m
master: 80m