Madelung equations

The Madelung equations or the equations of quantum hydrodynamics are Erwin Madelung's equivalent alternative formulation of the Schrödinger equation, 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.

The Madelung equations are quantum Euler equations:

where ${\displaystyle {\mathbf {u}}}$ is the flow velocity, ${\displaystyle \rho _{m}=m\rho =m|\psi |^{2}}$ is the mass density, ${\displaystyle Q=-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}{\sqrt {\rho }}}{\sqrt {\rho }}}=-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}{\sqrt {\rho _{m}}}}{\sqrt {\rho _{m}}}}}$ is the Bohm quantum potential, and V is the potential from the Schroedinger equation. The circulation of the flow velocity field along any closed path obeys the auxiliary condition ${\displaystyle {\begin{matrix}\Gamma \doteq \oint {m\mathbf {u} \cdot d\mathbf {l} }=2\pi n\hbar ,&n\in \mathbb {Z} \\\end{matrix}}}$.

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