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Intellectual Archive

De Broglie pilot wave from the nonlinear Schrodinger equation

Miroslav Pardy

Published Sep 21, 2026• 0 citations
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Abstract

The Schrodinger equation with the logarithmic nonlinear term is derived by the natural generalization of the hydrodynamical model of quantum mechanics. The nonlinear term appears to be logically necessary because it enables explanation of the classical motion for the infinite mass limit. The article is the modified version of the articles by author (Pardy, 1993; 2001).

DOI
10.32370/ia_2026_03_1

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The Schrodinger equation with the logarithmic nonlinear term is derived by the natural generalization of the hydrodynamical model of quantum mechanics. The nonlinear term appears to be logically necessary because it enables explanation of the classical motion for the infinite mass limit. The article is the modified version of the articles by author (Pardy, 1993; 2001).
The Schrodinger equation with the nonlinear term is derived in the framework of the Dirac heuristics. The particle behaves classically in case the mass of it is infinite. The nonlinear term is crucial and involves new physical constant b. The constant can be measured by the same methods that were used in the case of the Casimir effect (Spaarnay, 1958; Tabor et al., 1969). Of course, the experimental procedure is based on well educated experimenters. The new experiments, different from the Zeilinger ones, are proposed, with Faraday simplicity, for the determination of this new very small constant b. The article is the extended and perfectionized version of the articles by author (Pardy, 1993; 1994; 2001).
Via Nelson's stochastic mechanics, a method for the solution of the hydrodynamical version of the logarithmic nonlinear Schr\"odinger equation (LNLSE), with a time-dependent forced-harmonic-oscillator potential, is presented. Based on a new interpretation of the interplay between dispersion and nonlinearity, a revealing general spreading-wave-packet solution is found. The complete stochastic process associated with the LNLSE is also derived and decomposed into underlying processes of independent nature (classical and quantum). Physical consequences and conditions which allow the existence of solitonlike (nonspreading) solutions are described: Among them, it is found that the zero-point energy is given by ${\ensuremath{\epsilon}}_{0}$=\ensuremath{\Elzxh}${\ensuremath{\Omega}}_{0}$/2, where ${\ensuremath{\Omega}}_{0}$=\ensuremath{\lambda}/2+[(\ensuremath{\lambda}/2)+${\ensuremath{\omega}}_{0}^{2}$] 1/2^{1/2}1/2 with ${\ensuremath{\omega}}_{0}$ and \ensuremath{\lambda} being the harmonic frequency and the nonlinearity strength, respectively.
It is known since Madelung that the Schrödinger equation can be thought of as governing the evolution of an incompressible fluid, but the current theory fails to mathematically express this incompressibility in terms of the wavefunction without facing problem. In this paper after showing that the current definition of quantum-mechanical momentum as a linear operator is neither the most general nor a necessary result of the de Broglie hypothesis, a new definition is proposed that can yield both a meaningful mathematical condition for the incompressibility of the Madelung fluid, and nonlinear generalisations of Schrödinger and Klein-Gordon equations. The derived equations satisfy all conditions that are expected from a proper generalisation: simplification to their linear counterparts by a well-defined dynamical condition; Galilean and Lorentz invariance (respectively); and signifying only rays in the Hilbert space.