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## Madelung version of Schrödinger: a link between classical and quantum mechanics

This week I am at the Wolfgang Pauli Institute (WPI) in Vienna for the summer school on “Schrödinger equations”. Several interesting talks on or related to Schrödinger, including those of Y. Brenier on Madelung equations, F. Golse on mean field and classical limits of N-body quantum system, P. Germain on the derivation of the kinetic wave equation, C. Bardos on Maxwell-Boltzmann relation for electrons, F. Nier on Bosonic mean field dynamics, among others (still two days to go!). I also spoke on the Grenier’s iterative scheme, as discussed in my previous blog.

Yann Brenier (Ecole polytechnique) gave an interesting lecture on Madelung’s version of Schrodinger equations, giving a link between classical and quantum mechanics, which according to him is not so popular in the literature. Below are the materials taken from his lecture (misconceptions, if any, are mine, of course).

In 1925, Erwin Schrödinger wrote down the famous Schrödinger equations to model quantum mechanics. Almost immediately after that in 1926, Erwin Madelung (also Erwin!) established a surprising link between quantum mechanics and classical fluid mechanics. In its simplest form, the Schrödinger equation reads

$\displaystyle i \partial_t \psi + \Delta \psi = 0,$

for ${\psi = \psi(t,x) \in \mathbb{C}}$, ${t\ge 0, x\in \mathbb{R}^d}$. Madelung’s idea is to introduce the change of variables:

$\displaystyle \psi = \sqrt{\rho e^{i\theta}} = \sqrt \rho e^{\frac{i\theta}{2}}$

in which ${\rho\ge 0}$ is viewed as the density distribution of some fluid and ${v: = \nabla \theta}$ describes its velocity field in ${\mathbb{R}^d}$ (the quadratic form is also classical in a view of the Wigner’s transform). Let’s find the dynamics of the fluid ${(\rho, \rho v)}$. One computes

$\displaystyle \frac{\nabla \psi}{\psi} = \frac12 \frac{\nabla \rho}{\rho} + \frac i2 \nabla \theta,\qquad \frac{\partial_t\psi}{\psi} = \frac12 \frac{\partial_t\rho}{\rho} + \frac i2 \partial_t\theta .$

Together with ${\rho = \psi \bar \psi}$, the first identity immediately gives the momentum of the fluid:

$\displaystyle \rho \nabla \theta = \frac{1}{2i} \Big( \bar\psi \nabla \psi - \psi \nabla \bar \psi\Big)$

and so ${\nabla \cdot (\rho \nabla \theta) = \frac{1}{2i}( \bar\psi \Delta \psi - \psi \Delta \bar \psi) = - \partial_t \rho}$. Denoting ${v = \nabla \theta}$, this gives the continuity equation:

$\displaystyle \partial_t \rho + \nabla \cdot (\rho v) =0.\ \ \ \ \ (1)$

Similarly, direct calculations give the momentum equation:

$\displaystyle \partial_t (\rho v) + \nabla \cdot (\rho v \otimes v) + \nabla \cdot \Big( \frac{\nabla \rho \otimes \nabla \rho}{\rho} \Big) = \nabla \Delta \rho .\ \ \ \ \ (2)$

The last two terms are called quantum terms for the reason that when inserting the Planck constant ${\hbar}$ in the Schrödinger equations, they involve ${\hbar}$ and vanish in the classical limit ${\hbar \rightarrow 0}$. In the literature, the equations (1)(2) are used to describe Korteweg fluids, taking into account of the capillarity effect; see, for instance, the work by [Benzoni-Gavage, Danchin, Descombes, and Jamet, Interfaces Free Boundaries, 2005].

Remark: In case of cubic nonlinear Schrödinger equations (NLS), the nonlinearity contributes a pressure term ${p (\rho) = \beta \rho}$, where ${\beta = \pm}$ depending on the focusing or defocusing case. General nonlinearity contributes a more general pressure term. Note also that in the classical limit, the quantum terms vanish and one gets the shallow water equations from the cubic NLS.

1. Further on the Madelung’s transformation

On the second part of his lecture, Brenier gave an intriguing chain of transforming one equation to another in different contexts. For instance, how one goes from the Euler equations of isothermal flows (written down by Euler in 1755) to the Heat conduction problem (Fourier, 1807), then to Schrödinger – Madelung equations (discovered in 1925 – 1926), and finally to Quantum diffusion equations.

One starts by taking ${(\tilde \rho,\tilde v)}$ to be the solution to the compressible Euler equations with pressure ${p(\tilde\rho) =\tilde \rho }$ (i.e., isothermal flows; in general, ${p = p(\tilde \rho)}$ is referred to isentropic flows); namely,

\displaystyle \begin{aligned} \partial_t \tilde \rho + \nabla \cdot (\tilde \rho \tilde v) &=0 \\ \partial_t (\tilde \rho \tilde v) + \nabla \cdot (\tilde \rho \tilde v \otimes \tilde v) + \nabla \tilde \rho &=0 \end{aligned}\ \ \ \ \ (3)

Let ${\theta = \frac{t^2}{2}}$. One makes a change of variables:

$\displaystyle \tilde \rho (t,x) = \rho (\theta,x),\qquad \tilde v(t,x) = \sqrt{2\theta} v(\theta,x)= v(\theta,x) \frac{d\theta}{dt} .$

The first equation in (3) becomes ${\partial_\theta \rho + \nabla \cdot (\rho v) =0}$, while the second equation now reads

$\displaystyle 2\theta \Big[ \partial_\theta( \rho v) + \nabla \cdot (\rho v \otimes v) \Big] + \rho v + \nabla \rho =0.$

Thus, in the limit ${\theta \rightarrow \infty}$, the pair ${(\rho,v)}$ solves the pressureless Euler, while the limit ${\theta \rightarrow 0}$ yields ${\rho v = - \nabla \rho}$, which together with the continuity equation yields the heat equation: ${\partial_\theta \rho = \Delta \rho}$. Hence, Euler degenerates to Heat.

Next, observe that solutions to the Euler equations are critical points ${(\rho, m=\rho v)}$ of the Lagrangian ${(L = T-V)}$ functional:

$\displaystyle \iint \Big[ \frac{|m|^2}{2\rho} - \rho \log \rho \Big] \; dxdt$

for space-time compactly supported perturbations, subject to the differential constraints: ${\partial_t \rho + \nabla \cdot m=0}$. At the same time, the solutions to Euler enjoy Hamiltonian structures; in particular, one has the invariant of the kinetic + potential energy:

$\displaystyle \frac{d}{dt} \int \Big[ \frac{|m|^2}{2\rho} + \rho \log \rho \Big] \; dx =0.$

In the ${\theta = t^2/2}$ variable, the invariant reads

$\displaystyle \frac{d}{d\theta} \int \rho \log \rho \; dx + 2\theta \frac{d}{d\theta} \int \frac{|m|^2}{2\rho} \; dx = - \int \frac{|m|^2}{\rho}\; dx.$

Hence, again in the limit ${\theta \rightarrow 0}$, one gets ${m = -\nabla \rho}$ from the previous calculation and so

$\displaystyle \frac{d}{d\theta}\int \rho \log \rho \; dx = - \int \frac{|\nabla \rho |^2}{\rho}\; dx.$

It then turns out that Madelung equation of Schrödinger enjoys a least action principle, having the Lagrangian function with the above new potential energy; precisely, Madelung equation can be obtained as the Lagrangian of the functional

$\displaystyle \iint \Big[ \frac{|m|^2}{2\rho} - \frac{|\nabla \rho|^2}{\rho} \Big] \; dxdt .$

With the same calculations as those from Euler to Heat, one can obtain the quantum diffusion equations from the Madelung equations. Brenier goes on with some interesting explicit calculations for some ODEs, and discusses further possible applications of this chain in some other contexts.