Quantum Potential from the Material Derivative of the Osmotic Velocity: A Two-Fluid Madelung Framework

TL;DR

The paper addresses deriving the quantum potential and Madelung dynamics from a purely kinematic origin, by linking the material derivative of the osmotic velocity to quantum forces. It introduces a two-fluid Madelung framework and shows how, with $D = \hbar/(2m)$, the quantum potential emerges and leads to the Schrödinger equation via $\Psi = R e^{i S/\hbar}$, while also incorporating electromagnetic coupling. The approach is further extended to the relativistic Klein–Gordon regime, yielding a self-consistent relativistic Madelung description and EM interactions. By unifying hydrodynamic, kinematic, and electromagnetic perspectives, the framework provides a versatile tool for modeling complex quantum fluids in both nonrelativistic and relativistic settings.

Abstract

We derive the quantum potential directly from the material derivative of the osmotic velocity and formulate a two-fluid model that reproduces the Madelung equations. Interactions between the fluids are included but remain secondary. The framework is generalized to incorporate electromagnetic fields, yielding self-consistent description of both the Schrodinger and Klein-Gordon equations. Extenstion to the relativistic Klein-Gordon case demonstrates the model's flexibility and applicability to spinless relativistic quantum systems. This approach unifies hydrodynamic, kinematic, and electromagnetic perspectives, providing a clear physical interpretation of quantum potentials and forces and offering a versatile platform for modeling complex quantum systems in both non-relativistic and relativistic regimes.

Figures (1)

• Figure 1: Densities $\rho_a$ of the two fluids ($a=1,2$). Two time scales are considered: short intervals $\delta t_\alpha$ ($\alpha=1,2,\dots,N$) and a longer interval $\Delta t$, with $\delta t_\alpha \ll \Delta t$. At the end of each short interval $\delta t_\alpha$, a sudden jump enforces $\rho_2 = \rho_1$, while between jumps $\rho_2$ evolves according to the simple diffusion law. Over the longer time $\Delta t$, the density $\rho_1$ changes by $\Delta \rho_1$, with $|\Delta \rho_1| \ll \rho_1$. The figure is schematic: in reality, $\rho_2$ never deviates significantly from $\rho_1$.