Madelung hydrodynamic model

Despite his reservations about the self-energy of an electron Madelung did offer some guidance on the hydrodynamic model for many-electron systems. Of the three possibilities that... 

The causal interpretation of quantum theory as proposed by De Broglie and Bohm is an extension of the hydrodynamic model originally proposed by Madelung and further developed by Takabayasi [36]. In Madelung s original proposal R2 was interpreted as the density p(x) of a continuous fluid with stream velocity v= VS/rri. Equation (5) then expresses conservation of fluid, while (6) determines changes of the velocity potential S in terms of the classical potential V, and the quantum potential... 

Soon after first publication Schrodinger s wave-mechanical model was extended by Madelung [39] on the basis of the obvious correspondence with the classical theory of hydrodynamics, already pointed out in Schrodinger s original papers [34] (II,p.l7). Writing the time dependence of k in terms of an action function... 

The similarity in form between the two real equations implied by the single-body spin-0 Schrddinger equation in the position representation (wave mechanics) and the equations of fluid mechanics with potential flow in its Eulerian formulation was first pointed out by Madelung in 1926 [1]. In this analogy, the probability density is proportional to the fluid density, and the phase of the wave function is a velocity potential. A novel feature of the quantum fluid is the appearance of quantum stresses, which are usually represented through the quantum potential. To achieve mathematical equivalence of the models, the hydrodynamic variables have to satisfy... 

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