Madelung fluid

In a further extension [42] of the model the Madelung fluid is assumed to be some kind of physically real fluid with an embedded particle, which takes the form of a highly localized inhomogeneity that moves with the local... 

This model is not adequate in itself as it contains nothing to describe the actual location x(t) of the particle which is required for a causal interpretation of quantum theory. It is therefore necessary to postulate a particle that takes the form of a highly localized inhomogeneity that moves with the local fluid velocity v(x,t). The inhomogeneity could be of density close to that of the fluid, which is simply being carried along with the local velocity of the fluid. As in any macroscopic fluid random fluctuations are assumed [37] to occur in the Madelung fluid. It is shown that such fluctuations may lead to the statistical result, P = 2. 

In a refined form of the theory [37] the same quantity features as the well known quantum potential. The notion of a particle emerges in this theory in the form of a highly localized inhomogeneity that moves with the local fluid velocity, v(x, t), thus as a stable dynamic structure that exists in the fluid, for example, as a small stable vortex or a pulse-like distortion. To explain why the causal theory needs probability densities it is argued [37] that the Madelung fluid must experience more or less random fluctuations in its motion to account for irregular turbulence. The turbulence necessitates a wave theory to describe the motion of vortices embedded in the fluid. The particle velocity is therefore not exactly VS/m, nor is the density exactly... 

In writing the Lagrangean density of quantum mechanics in the modulus-phase representation, Eq. (140), one notices a striking similarity between this Lagrangean density and that of potential fluid dynamics (fluid dynamics without vorticity) as represented in the work of Seliger and Whitham [325]. We recall briefly some parts of their work that are relevant, and then discuss the connections with quantum mechanics. The connection between fluid dynamics and quantum mechanics of an electron was already discussed by Madelung [326] and in Holland s book [324]. However, the discussion by Madelung refers to the equations only and does not address the variational formalism which we discuss here. 

Madelung assumed that R2 represented the density p(x) of a continuous fluid with stream velocity v—VS/m. Putting... 

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... 

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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