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arXiv:1611.07518 · Journal of Mathematical Physics

Recursive formulation of Madelung continuity equation leads to propagation equation

D. M. Heim

Published Dec 1, 2018• 2 citations
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Abstract

We apply a recursive approach to the continuity equation of the Madelung fluid resulting in a propagation equation for particle probability densities. This propagation equation can be used to propagate particle distributions in the presence of a Madelung pressure field. We show that the derived propagation equation goes over into the guidance equation of the de Broglie-Bohm theory in the limit of well located single particles. As an example, we propagate particles that enter the lower slit of a double-slit experiment, while the Madelung fluid enters both slits.

DOI
10.1063/1.5051340

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We apply a recursive approach to the continuity equation of the Madelung fluid resulting in a propagation equation for particle probability densities. This propagation equation can be used to propagate particle distributions in the presence of a Madelung pressure field. We show that the derived propagation equation goes over into the guidance equation of the de Broglie-Bohm theory in the limit of well located single particles. As an example, we propagate particles that enter the lower slit of a double-slit experiment, while the Madelung fluid enters both slits.
We describe a series of experimental analogies between fluid mechanics and quantum mechanics recently discovered by a team of physicists. These analogies arise in droplet systems guided by a surface (or pilot) wave. We argue that these experimental facts put ancient theoretical work by Madelung on the analogy between fluid and quantum mechanics into new light. After re-deriving Madelung’s result starting from two basic fluid mechanical equations (the Navier–Stokes equation and the continuity equation), we discuss the relation with the de Broglie–Bohm theory. This allows to make a direct link with the droplet experiments. It is argued that the fluid mechanical interpretation of quantum mechanics, if it can be extended to the general N-particle case, would have a considerable advantage over the Bohm interpretation: it could rid Bohm’s theory of its non-local character.
The quantumlike formalism in the form of Madelung fluid has been applied to describe the collective dynamics of a mesoscopic aggregate of particles in a potential field. The scheme has been specialized to the one-degree-of-freedom dynamics of a charged particle beam in an accelerator in the presence of high-order multipole nonlinearities. A hierarchy of recursive equations satisfied by the moments of the density distribution has been obtained. It has been shown that the recursion relations can be in principle solved if a finite number of the first several moments is known. Beam dynamics in the presence of octupole and decapole nonlinearities has been studied as an example, demonstrating the above scheme. It has been shown that an appropriate mechanism can be introduced to control stability and coherence of states when high-order nonlinearities have to be taken into account.
ABSTRACT The probability continuity equation is solved within the Madelung–Bohm framework, assuming a separable phase expressed as . Using operator methods, the amplitude of the wave function is reformulated in a form that is more practical for application to any given initial condition. This results in the function , which characterizes the amplitude of the wave function, with being the transformation of the position operator via a squeeze‐like operator. To demonstrate how this result can be applied, two scenarios are examined: one where the external potential is zero, causing the dynamics to originate only from the Bohm potential, and another where the form of mirrors the characteristics of wave propagation within optical waveguide arrays. In the second scenario, the amplitude, phase, and potentials might become complex, yet still comply with the Schrödinger equation.
Quantum Fluids follow Quantum Dynamical Equation(s), which were not known till date. There exist a set of two equations that is semi classical approach to Quantum Fluids called Madelung’s Equations. But a new fully quantum variant of Madelung’s Equations when embedded in the Schrodinger Equation is gives full description of evolution of Quantum fluid with respect to time and position. The equation presented in this article has two unknown variables, one is density and other is velocity field as a function of spatial and time coordinates. The equation presented in this article, is derived from Schrodinger Equation, obeying Continuity equation, and Navier Strokes Equation. Bohm’s potential was externally added in Madeline’s equation. But the new equation which is fully quantum mechanical in nature; Bohm’s potential appears out of the equation, which is interesting to observe. Astrophysical cold stellar dynamics and condensed fluids have the main application of this equation. Quantum fluids show strange behaviour when compared to normal fluids. It is also shown that quantum fluid also has spins which has no classical analog.