aqora

Fisher information and the complex nature of the Schr�dinger wave equation

B. Roy Frieden

Foundations of PhysicsJul 1, 199118 citations

Hydrodynamical formulation of Wigner–Dunkl quantum mechanics

Muhammad Farchani Rosyid

The hydrodynamical form of Wigner–Dunkl quantum mechanics has been formulated. The main step to get such formulation was to derive a pair of Madelung-like equations (called Madelung–Wigner–Dunkl equations) from the Schrödinger equation appearing in Wigner–Dunkl quantum mechanics. Some aspects of both equations contained in the pair, i.e. the inhomogeneous continuity equation and the quantum Hamilton–Jacobi equation, have been investigated and discussed. With certain assumption, the relation between the expectation value of the quantum potential appearing in the quantum Hamilton–Jacobi equation and the so-called Fisher information has been derived. Then an uncertainty relation between momentum and position which may be tighter than the standard uncertainty relation has been derived by using Cramér–Rao inequality and Fisher information.The hydrodynamical form of Wigner–Dunkl quantum mechanics has been formulated. The main step to get such formulation was to derive a pair of Madelung-like equations (called Madelung–Wigner–Dunkl equations) from the Schrödinger equation appearing in Wigner–Dunkl quantum mechanics. Some aspects of both equations contained in the pair, i.e. the inhomogeneous continuity equation and the quantum Hamilton–Jacobi equation, have been investigated and discussed. With certain assumption, the relation between the expectation value of the quantum potential appearing in the quantum Hamilton–Jacobi equation and the so-called Fisher information has been derived. Then an uncertainty relation between momentum and position which may be tighter than the standard uncertainty relation has been derived by using Cramér–Rao inequality and Fisher information.

Modern Physics Letters AOct 10, 20231 citation

More on the Schrödinger Equation

Jochen Pade

Undergraduate lecture notes in physicsNov 7, 20130 citations

Emergence of the Schrödinger Equation from Reversible Probability Dynamics

AI Trinity Labo

This work presents a reconstruction of quantum mechanics in which Schrödinger-type dynamics emerges from reversible probabilistic evolution rather than being postulated. Starting from a normalized probability density, we impose reversible probability dynamics, Fisher information geometry, and classical correspondence. These compatibility conditions yield coupled real equations equivalent to the continuity equation and the Hamilton–Jacobi equation with an additional information curvature (quantum potential) term. By combining density and phase into a complex amplitude, the Schrödinger equation appears as a compact encoding of reversible statistical flow. The result is mathematically equivalent to the Madelung hydrodynamic formulation and structurally connected to Bohmian mechanics, but derived without assuming Hilbert space, operator axioms, or measurement postulates. The work can therefore be viewed as an information-geometric derivation of quantum dynamics: quantum theory arises as the unique consistent representation of reversible probability flow. Keywords: quantum foundations, quantum reconstruction, emergent quantum mechanics, derivation of the Schrödinger equation, information geometry, Fisher information, Hamilton–Jacobi theory, Madelung formulation, Bohmian mechanics, reversible probability dynamicsThis work presents a reconstruction of quantum mechanics in which Schrödinger-type dynamics emerges from reversible probabilistic evolution rather than being postulated. Starting from a normalized probability density, we impose reversible probability dynamics, Fisher information geometry, and classical correspondence. These compatibility conditions yield coupled real equations equivalent to the continuity equation and the Hamilton–Jacobi equation with an additional information curvature (quantum potential) term. By combining density and phase into a complex amplitude, the Schrödinger equation appears as a compact encoding of reversible statistical flow. The result is mathematically equivalent to the Madelung hydrodynamic formulation and structurally connected to Bohmian mechanics, but derived without assuming Hilbert space, operator axioms, or measurement postulates. The work can therefore be viewed as an information-geometric derivation of quantum dynamics: quantum theory arises as the unique consistent representation of reversible probability flow. Keywords: quantum foundations, quantum reconstruction, emergent quantum mechanics, derivation of the Schrödinger equation, information geometry, Fisher information, Hamilton–Jacobi theory, Madelung formulation, Bohmian mechanics, reversible probability dynamics

Feb 16, 20260 citations

A new derivation of the schrödinger equation

H. N. Ioannidou

Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimentoAug 1, 19827 citations

Hydrodynamical formulation of quantum mechanics, Kahler structure, and Fisher information

Marcel Reginatto

The Schrodinger equation can be derived using the minimum Fisher information principle. I discuss why such an approach should work, and also show that the Kahler and Hilbert space structures of quantum mechanics result from combining the symplectic structure of the hydrodynamical model with the Fisher information metric.

arXiv:quant-ph/9909065Sep 21, 19992 citations

Toward a Thermo-hydrodynamic Like Description of Schrödinger Equation via the Madelung Formulation and Fisher Information

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