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This paper proposes a nonlinear extension of the Madelung transformation that introduces a singular hyperbolic coupling between amplitude and phase, fundamentally altering the geometric structure of Bohmian hydrodynamics. By deriving generalized continuity equations and modified quantum force terms, the framework suggests that the Meissner effect in superconductors becomes sensitive to spatial density gradients. This theoretical development offers a novel perspective on amplitude-sensitive transport and dissipative wave propagation within quantum systems.
Amplitude-dependent quantum hydrodynamics from a \(\coth\)-Madelung ansatz
arXiv:2606.06561v1 Announce Type: new Abstract: We investigate a nonlinear extension of the Madelung transformation based on a hyperbolic phase--amplitude coupling of the form \[ \Psi = R e^{\imath\theta \coth R}, \] where \(R\) is a real amplitude field and \(\theta\) is an auxiliary phase coordinate governed by Schr\"odinger's equation. In contrast to the conventional polar decomposition, this construction imposes a singular hyperbolic relation between amplitude and phase, thereby endowing the Bohmian or hydrodynamic description with an intrinsically geometric structure. We show that the associated velocity field acquires a density-gradient contribution, producing generalized continuity equations and modified quantum force terms. When interpreted as a complex macroscopic order parameter, this generalized phase structure leads to modified superconducting electrodynamics; in particular, the London equations acquire additional contributions that render the Meissner response sensitive to spatial density gradients. The proposed framework is motivated by broader developments involving complex group velocities, dissipative wave propagation, and amplitude-sensitive transport in quantum systems.
Abstract:We investigate a nonlinear extension of the Madelung transformation based on a hyperbolic phase--amplitude coupling of the form \[ \Psi = R e^{\imath\theta \coth R}, \] where \(R\) is a real amplitude field and \(\theta\) is an auxiliary phase coordinate governed by Schrödinger's equation. In contrast to the conventional polar decomposition, this construction imposes a singular hyperbolic relation between amplitude and phase, thereby endowing the Bohmian or hydrodynamic description with an intrinsically geometric structure. We show that the associated velocity field acquires a density-gradient contribution, producing generalized continuity equations and modified quantum force terms. When interpreted as a complex macroscopic order parameter, this generalized phase structure leads to modified superconducting electrodynamics; in particular, the London equations acquire additional contributions that render the Meissner response sensitive to spatial density gradients. The proposed framework is motivated by broader developments involving complex group velocities, dissipative wave propagation, and amplitude-sensitive transport in quantum systems.
Quantum Physics (quant-ph); Superconductivity (cond-mat.supr-con); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
arXiv:2606.06561 [quant-ph]
(or arXiv:2606.06561v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2606.06561 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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