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Hidden Properties of Quantum Mechanics

Michail A. Zak

Published Apr 1, 2014• 0 citations
OpenAlex

Abstract

Two unsolved problems in quantum mechanics are addressed: a source of randomness and an origin of entanglement. These problems being hidden in the Schrodinger equations became transparent in its Madelung version. Special attention is concentrated on equivalence between the Schrodinger and the Madelung equations. It has been demonstrated that randomness in quantum mechanics has the same mathematical source as that in turbulence and chaos, and the origin of entanglement is the global constraint imposed by the normalization constraint of the probability density that becomes an additional variable in the Madelung version of the Schrodinger equation.

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Two unsolved problems in quantum mechanics are addressed: a source of randomness and an origin of entanglement. These problems being hidden in the Schrodinger equations became transparent in its Madelung version. Special attention is concentrated on equivalence between the Schrodinger and the Madelung equations. It has been demonstrated that randomness in quantum mechanics has the same mathematical source as that in turbulence and chaos, and the origin of entanglement is the global constraint imposed by the normalization constraint of the probability density that becomes an additional variable in the Madelung version of the Schrodinger equation.
It has been shown by us recently that Einstein was right in his controversy with Bohr or that the so called hidden-variable theory should be preferred to the Copenhagen quantum mechanics. In the following paper the corresponding arguments will be shortly repeated. The main attention will be then devoted to explaining main differences between these two quantum alternatives, differing in the access to the problem of chance, causality and locality of microscopic objects. The actual meaning of the mentioned hidden variables will be discussed, too, the essence of which remained practically unclear during all past discussions. It will be shown that the theory of hidden variables (or Schrodinger equation alone) is able to represent the properties of the whole known physical reality.
In light of a recently discovered general solution to the Schrödinger equation in an arbitrary integer-dimensional real domain with arbitrary initial conditions, boundary conditions, and quantum mechanical potential via a modified Cole-Hopf transform, a solution of the Madelung equations of equal generality is derived in this work. After a sufficient degree of mathematical context is established by a brief recapitulation of the solution of the Schrödinger equation, the derivation of the Madelung equations from first principles proves that a general Madelung flow field is implied by a general Schrödinger wave function through illustration of the mathematical essence of the quantities in the Madelung equations. The solution follows by a decomposition of the newly found general wave function into its polar form and its substitution into the definitions of the Madelung flow velocity and density, and a transfer from wave function initial and boundary conditions to density and velocity boundary conditions. The transfer between the complex scalar-valued wave function and the real vector-valued velocity field is achieved by a conditional inversion of the gradient operator. Plots of the velocity and vorticity take mere milliseconds to generate on consumer hardware via an enclosed GNU Octave script and show highly turbulent patterns consistent with the laws of quantum mechanics, showing that this analytical Madelung solution possesses unprecedented ability to deterministically replicate quantum turbulence without any numerical methods at all, effectively democratizing the study of quantum turbulence with its internal structure and mathematical origin. This Madelung solution paves the way to equally general solutions of unsolved problems in classical fluid mechanics including the Euler and Navier-Stokes equations, whose methods and analyses including existence and smoothness are expounded in works currently in production by the author.
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