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Intellectual Archive

The incompleteness of the Schrodinger equation in classical limit

Miroslav Pardy

Published Dec 18, 2022• 0 citations
Publisher

Abstract

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).

DOI
10.32370/ia_2022_12_2

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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).
The Schrodinger equation has been considered to be a postulate of quantum physics, but it is also perceived and derived heuristically as the quantum equivalent of the classical energy relation. We indicate that the Schrodinger equation cannot be a physical postulate, and we show that the heuristic derivations wrongly consider that the particle kinetic energy is proportional to the square of the momentum operator acting on the wavefunction. The correct term is proportional to the square of the momentum. Analyzing particle interactions, we realize that particles have multiple virtual motions and that each motion is accompanied by a wave that has constant amplitude. Accordingly, we define the wavefunction as the superposition of the virtual waves of the particle. As a result, quantum mechanics becomes a local and global theory, while in the traditional formulation it is a local theory. In several interaction settings we can tell what particle motions arise and explain the outcomes in simple and tangible terms. Most importantly, the mathematical foundation of quantum mechanics becomes clear and justified, and we derive the Schrodinger, Dirac, etc. equations as the conditions the wavefunction must satisfy at each space-time point in order to fulfill the respective total energy equation.
The introduction of nonlinearities in the Schrödinger equation has been considered in the literature as an effective manner to describe the action of external environments or mean fields. Here, in particular, we explore the nonlinear effects induced by subtracting a term proportional to Bohm’s quantum potential to the usual (linear) Schrödinger equation, which generates the so-called “classical” Schrödinger equation. Although a simple nonlinear transformation allows us to recover the well-known classical Hamilton–Jacobi equation, by combining a series of analytical results (in the limiting cases) and simulations (whenever the analytical treatment is unaffordable), we find an analytical explanation to why the dynamics in the nonlinear “classical” regime is still strongly nonclassical. This is even more evident by establishing a one-to-one comparison between the Bohmian trajectories associated with the corresponding wave function and the classical trajectories that one should obtain. Based on these observations, it is clear that the transition to a fully classical regime requires extra conditions in order to remove any trace of coherence, which is the truly distinctive trait of quantum mechanics. This behavior is investigated in three paradigmatic cases, namely, the dispersion of a free propagating localized particle, the harmonic oscillator, and a simplified version of Young’s two-slit experiment.
An exact classical field theory, for a recently proposed nonlinear generalization of the Schrödinger equation, is presented. In this generalization, a nonlinearity depending on an index q appears in the kinetic term, such that the free-particle linear Schrödinger equation is recovered in the limit q →1. It is shown that besides the usual , a new field must be introduced, which becomes only when q →1. In analogy to the linear case, is interpreted as the probability density for finding the particle at time t , in a given position inside an arbitrary finite volume V , for any q . The possible physical consequences are discussed, and, in particular, the important property that the fields and do not interact with light.