Madelung equations

The Madelung equations are Erwin Madelung's alternative formulation of the Schrödinger equation.

[edit] Equations

The first equation has the form of a continuity equation.

$\partial_t \rho +\frac{1}{m}\nabla(\rho\nabla S)= 0$

$\partial_t S +\frac{1}{2m}(\nabla S)^2 +V(x)- \frac{\hbar^2}{2m} \frac{\Delta \sqrt{\rho}}{\sqrt{\rho}}= 0$

where $\nabla S$ is interpreted as an impulse.