Madelung equations

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${\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }$ Schrödinger equation
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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.