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The Nonlinear Schroedinger Equation with Infinite Mass Wave

Miroslav Pardy

Published Jun 1, 2019• 0 citations
OpenAlex

Abstract

The Schroedinger 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 infinite mass limit of the wave function. The article is the modified version of the articles by author (Pardy, 1993; 2001).

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The Schroedinger 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 infinite mass limit of the wave function. 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).
A nonlinear modification of the Schrödinger equation is proposed in which the Lagrangian density for the Schrödinger equation is extended by terms polynomial in Δmln⁡(Ψ∗/Ψ)Δ^{m}\ln (Ψ^{*}/Ψ)Δmln(Ψ∗/Ψ) multiplied by Ψ∗ΨΨ^{*}ΨΨ∗Ψ. This introduces a homogeneous nonlinearity in a Galilean invariant manner through the phase SSS rather than the amplitude RRR of the wave function Ψ=Rexp⁡(iS)Ψ=R\exp (iS)Ψ=Rexp(iS). From this general scheme we choose the simplest minimal model defined in some reasonable way. The model in question offers the simplest way to modify the Bohm formulation of quantum mechanics so as to allow a leading phase contribution to the quantum potential and a leading quantum contribution to the probability current removing asymmetries present in Bohm's original formulation. It preserves most of physically relevant properties of the Schrödinger equation including stationary states of quantum-mechanical systems. It can be thought of as the simplest model of nonlinear quantum mechanics of extended objects among other such models that also emerge within the general scheme proposed. The extensions of this model to nnn particles and the question of separability of compound systems are studied. It is noted that there exists a weakly separable extension in addition to a strongly separable one. The place of the general modification scheme in a broader spectrum of nonlinear modifications of the Schrödinger equation is discussed. It is pointed out that the models it gives rise to have a unique definition of energy in that the field-theoretical energy functional coincides with the quantum-mechanical one. It is found that the Lagrangian for its simplest variant represents the Lagrangian for a restricted version of the Doebner-Goldin modification of this equation.
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.