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

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.