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Theoretical Note: Derivation of the Schrödinger Equation from the Chronodynamic Euler State

Ambiel da Silva, Marcelo

Published Nov 25, 2025• 0 citations
Publisher

Abstract

In the framework of Chronodynamic Cosmology (CDC), the vacuum is defined as a physical superfluid (Psi). If this premise is correct, the fundamental equations of Quantum Mechanics must be derivable from Hydrodynamics. In this note, we perform the Madelung Transformation to demonstrate that the Schrödinger Equation is mathematically equivalent to the Euler Equation for an inviscid, compressible fluid subjected to a specific internal pressure potential. This derivation suggests that "Quantum Potential" (Q) is the hydrostatic pressure of the Time-Fluid, and that "Wavefunction Collapse" is a phase transition from an inviscid (superfluid) state to a viscous (classical) state.

DOI
10.5281/zenodo.17715834

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In the framework of Chronodynamic Cosmology (CDC), the vacuum is defined as a physical superfluid (Psi). If this premise is correct, the fundamental equations of Quantum Mechanics must be derivable from Hydrodynamics. In this note, we perform the Madelung Transformation to demonstrate that the Schrödinger Equation is mathematically equivalent to the Euler Equation for an inviscid, compressible fluid subjected to a specific internal pressure potential. This derivation suggests that "Quantum Potential" (Q) is the hydrostatic pressure of the Time-Fluid, and that "Wavefunction Collapse" is a phase transition from an inviscid (superfluid) state to a viscous (classical) state.
Here we present a transformation that maps the Schrodinger equation of quantum mechanics to the incompressible Euler equations of fluid mechanics. The transformation provides a wave solution and a potential function based on fluid properties that satisfy the Schrodinger equation given that the fluid velocity potential and pressure satisfy the Euler equations. Interestingly, in our transformation, the equivalent of quantum potential becomes the physical surface tension. This is contrary to the Madelung transformation that maps the Schrodinger equation to the compressible Euler equations where there is no physical counterpart for the quantum potential. Lastly, we show that using this transformation, the Bohm equation can be mapped to a particle's equation of motion moving on the free surface of the fluid.
This paper develops a hydrodynamic derivation of nonrelativistic quantum mechanics within the Hydrodynamic Quantum Gravity (HQG) programme. Using the Madelung transformation, the Schrödinger equation is written in amplitude–phase form and shown to be exactly equivalent to (i) a continuity equation for a conserved density and (ii) an Euler-type momentum equation augmented by the “quantum potential” term. In this interpretation, the quantum potential is treated as an effective dispersive stress associated with structured vacuum dynamics, rather than as a purely formal mathematical artefact. The paper reviews how standard quantum features appear naturally in the hydrodynamic variables: interference and diffraction as phase-driven flow, stationary states as time-harmonic solutions, and quantisation as a consequence of single-valued phase (circulation) constraints. It also discusses how pilot-wave style dynamics can be expressed in this language, and surveys laboratory analogues (including walking-droplet experiments) as phenomenological motivation and intuition-building, without claiming identity between the analogue systems and the quantum vacuum. Interpretation note: “Superfluid/condensate” language is used here as an effective mathematical/analogue description of vacuum behaviour; the paper does not assert a material substrate or an operationally accessible preferred rest frame. This paper is part of a series. Companion preprints develop the gravitational and electromagnetic components of the HQG framework and discuss additional open problems and testable predictions. Discussion and feedback: https://github.com/Gptham123456/Hydrodynamic-Quantum-Gravity/discussions
The Madelung equations express the Schrödinger equation as a continuity equation and modified Hamilton–Jacobi equation. These equations are equivalent to the Euler equations for a compressible, potential flow, when classical pressure per unit density is replaced by the quantum potential per unit mass. We extend this hydrodynamic interpretation by quantising a single, spinless, non-relativistic particle constrained to a surface wave with small slope. The wave is distinct from the wave function and, in order to reproduce the Schrödinger equation, it must satisfy the kinematic boundary condition for a free surface advected by twice the Madelung velocity field.
This monograph provides a deterministic, hydrodynamic foundation for Quantum Mechanics, replacing the probabilistic abstractions of the Copenhagen interpretation. By redefining the cosmic vacuum as a tangible, compressible Bose-Einstein Condensate (BEC), the abstract Schrödinger equation is mathematically mapped directly to the classical Euler fluid momentum equations via the Madelung Transformation. Through this hydrodynamic lens, wave-particle duality is physically resolved: a localized quantum droplet is mechanically steered by the acoustic pressure gradient of its own wake. The Heisenberg Uncertainty Principle is explicitly recontextualized as the irreducible variance of convective diffusion—a microscopic buoy subjected to the continuous turbulence of the vacuum's kinematic viscosity. Furthermore, quantum entanglement is modeled not as an abstract, non-local phenomenon, but as continuous longitudinal phononic resonance. Scaling these sub-atomic continuum mechanics to astrophysics, this framework replaces collisionless Dark Matter and stochastic planetary accretion with macroscopic acoustic nodes, viscous fluid drag, and deterministic tidal thresholds—recently validated by 2026 Chandra observations of pre-ignition EUV thermalization. Ultimately, this demonstrates that quantum mechanics is simply fluid dynamics operating at extreme limits of scale and viscosity.
The mass and momentum transfer phenomena in a compressible fluid represented by the Navier–Stokes equations are shown to convert into the Schrödinger equation for quantum mechanics. The complete Navier–Stokes equations render into an extended generalized version of Schrödinger equation. These results complement the Madelung’s (Zeitschrift für Physik 40 (3–4), pp. 322–326, 1926–1927) derivations that show how Schrödinger’s equation in quantum mechanics can be converted into the Euler equations for irrotational compressible flow. The theoretical results presented here join the classical Madelung paper to suggest the possibility that quantum effects at sub-atomic levels deal with a compressible fluid susceptible to wave propagation, rather than a particle. The link between such a fluid and the “quantum particle” is under current investigation.
The Madelung equations are two equations that are equivalent to the one-body time-dependent Schroedinger equation. In this paper, the Madelung equation, whose gradient is an Euler equation, is refined by introducing interpretations of functions that are shown to depend only on the real-part of the complex-valued wavefunction. These interpretations are extensions of functions from the recently derived generalized Bernoulli equation, applicable to real-valued quantum-mechanical stationary states. In particular, the velocity and pressure definitions are extended so that they depend on the real-part of a time-dependent complex-valued wavefunction. The Bohn quantum potential is then interpreted as the sum of two terms, one involving the kinetic energy and the other involving the pressure. Substituting the interpreted quantum-potential into the Madelung equation gives a refined equation containing two kinetic energy terms, a pressure term, and the external potential. It is easily demonstrated that the refined Madelung equation, applied to the hydrogen atom states with a nonzero magnetic quantum number, gives a fluid velocity that contains both a radial component and a free vortex. Hence, the fluid particles have angular momentum and move on streamlines that terminate at infinity. It is also demonstrated that the two velocities from the refined Madelung equation are related: One is the real component and the other is the imaginary component of a complex velocity. Furthermore, an Euler equation for quantum mechanical systems is derived by taking the gradient of the refined Madelung equation.