Madelung equations

In theoretical physics, the Madelung equations, or the equations of quantum hydrodynamics, are Erwin Madelung's alternative formulation of the Schrödinger equation for a spinless non relativistic particle, written in terms of hydrodynamical variables, similar to the Navier–Stokes equations of fluid dynamics. The derivation of the Madelung equations is similar to the de Broglie–Bohm formulation, which represents the Schrödinger equation as a quantum Hamilton–Jacobi equation. In both cases the hydrodynamic interpretations are not equivalent to Schrodinger's equation without the addition of a quantization condition.

Recently, the extension to the relativistic case with spin was done by having the Dirac equation written with hydrodynamic variables. In the relativistic case, the Hamilton–Jacobi equation is also the guidance equation, which therefore does not have to be postulated.

In the fall of 1926, Erwin Madelung reformulated Schrödinger's quantum equation in a more classical and visualizable form resembling hydrodynamics. His paper was one of numerous early attempts at different approaches to quantum mechanics, including those of Louis de Broglie and Earle Hesse Kennard. The most influential of these theories was ultimately de Broglie's through the 1952 work of David Bohm now called Bohmian mechanics.

In 1994 Timothy C. Wallstrom showed that an additional ad hoc quantization condition must be added to the Madelung equations to reproduce Schrodinger's work. His analysis paralleled earlier work by Takehiko Takabayashi on the hydrodynamic interpretation of Bohmian mechanics. The mathematical foundations of the Madelung equations continue to be a topic of research.

The Madelung equations are the quantum mechanical counterpart of the Euler equations:^{[citation needed]}

$${\displaystyle {\begin{aligned}&{\frac {\partial \rho _{m}(\mathbf {x} ,t)}{\partial t}}+\nabla \cdot (\rho _{m}(\mathbf {x} ,t)\mathbf {v} (\mathbf {x} ,t))=0,\\[4pt]&{\frac {D\mathbf {v} (\mathbf {x} ,t)}{Dt}}={\frac {\partial \mathbf {v} (\mathbf {x} ,t)}{\partial t}}+\mathbf {v} (\mathbf {x} ,t)\cdot \nabla \mathbf {v} (\mathbf {x} ,t)=-{\frac {1}{m}}\nabla (Q(\mathbf {x} ,t)+V(\mathbf {x} ,t)),\end{aligned}}}$$

where

The Madelung equations answer the question of whether ${\displaystyle \mathbf {v} (\mathbf {x} ,t)}$ obeys the continuity equations of hydrodynamics, and, if so, what plays the role of the stress tensor.