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Doklady Mathematics

Schrödinger Equation as a Self-Consistent Field

Victor Valentinovich Vedenyapin, Татьяна Семеновна Казакова, V. Ya. Kiselevskaya-Babinina, Boris Nikolaevich Chetverushkin

Published May 1, 2018• 12 citations
Publisher

Abstract

It is well known that the Schrödinger equation can be reduced to the Hamilton–Jacobi equation in Bohmian mechanics. Corresponding new equations of the Vlasov and Lamb types are derived, and their stationary solutions are investigated.
DOI
10.1134/s1064562418030122

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•Feb 16, 2026•0 citations
We find the form of the potential depending on the coordinates and the time such that a solution, SSS, of the Hamilton--Jacobi equation yields an exact solution, exp⁡(iS/ℏ)\exp ({\rm i} S/\hbar)exp(iS/ℏ), of the corresponding Schrödinger equation.
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 dynamics