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arXiv:2312.06381 · Modern Physics Letters B

Instability and quantization in quantum hydrodynamics

Yakir Aharonov, Tomer Shushi

Published Mar 7, 2024• 3 citations
arXivHTMLPDF

Abstract

We show how the quantum hydrodynamical formulation of quantum mechanics converts the nonlocality in the standard wave-like description of quantum systems by an instability of the quantum system, which opens the door to a new way for studying quantum systems based on known methodologies for studying the stability of fluids. As a second result, we show how the Madelung equations describe quantized energies without any external quantization conditions.

DOI
10.1142/s0217984924502683

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We show how the quantum hydrodynamical formulation of quantum mechanics converts the nonlocality in the standard wave-like description of quantum systems by an instability of the quantum system, which opens the door to a new way for studying quantum systems based on known methodologies for studying the stability of fluids. As a second result, we show how the Madelung equations describe quantized energies without any external quantization conditions.
In the paper with the above-noted title, T. C. Wallstrom claims that the description of the particle's motion as a certain "conservative" diffusion is not equivalent to quantum mechanics in spite of the fact that the Madelung "hydrodynamic" equations, which provide the description of such a diffusion, can be converted to the Schroedinger equation. He pointed out that such a stochastic theory can be regarded as equivalent to conventional quantum mechanics only if they can derive from it not just the Madelung equations but also the condition that the circulation of the "probability fluid" is always quantized, which is equivalent the condition for the single-valuedness of the wave function. We, however, show that the single-valuedness of the wave function required in quantum mechanics, is not an auxiliary condition but a property of all local solutions of the Schrodinger equation. Based on the one-to-one correspondence between local solutions of the Schroedinger and the Madelung equations this means that the quantization of the circulation of the "probability fluid" is a property of all solutions of the Madelung equations.
This work identifies a “chronological fallacy” at the heart of quantum foundations: the axiomatization of quantum mechanics by von Neumann (1932) predated the rigorous formulation of classical probability by Kolmogorov (1933), creating an artificial schism between physics and statistics. To resolve the “spectral divergence” between the continuous ontology of hydrodynamic quantum models and the discrete phenomenology of spectroscopy, the hydrodynamic stability hypothesis is proposed. By synthesizing Madelung hydrodynamics, Radon-Nikodým measure theory, and dynamical systems theory, it is demonstrated that “quantization” is not an intrinsic property of the operator algebra, but an emergent dynamical stability phenomenon triggered by the measurement interaction. Eigenstates of the Hamiltonian act as basins of attraction for the probability fluid, making the “quantum jump” a continuous, deterministic relaxation process. This framework restores a Kolmogorovian probability structure to quantum mechanics, resolving the measurement problem without abandoning local realism or invoking discontinuous collapse.
A nonlinear interaction term is added to the Schr\"odinger equation which makes it exactly equivalent to classical mechanics. In the hydrodynamic formulation of this equation, the usual classical theory is obtained. This theory is a Lagrangian density formulation of hydrodynamics and is used to calculate the density profile of superfluid helium in a line vortex. The theory can be canonically quantized, in the sense of quantum field theory, in terms of the density and velocity fields. The quantized theory is not equivalent to that which one would obtain if the quantization were done in the usual way in terms of the field operations ${\ensuremath{\psi}}^{*}$ and ($\ensuremath{\psi}$). The Schr\"odinger equation, with this interaction, has localized solutions with a definite energy and momentum and does not give quantized energy levels for a one-dimensional harmonic oscillator.
By differentiating the Schr\"odinger equation and separating the real amd imaginary parts, one obtains the Madelung hydrodynamic equations, which have inspired numerous classical interpretations of quantum mechanics. Such interpretations frequently assume that these equations are equivalent to the Schr\"odinger equation, and thus provide an alternative basis for quantum mechanics. This paper proves that this is incorrect: to recover the Schr\"odinger equation, one must add by hand a quantization condition, as in the old quantum theory. The implications for various alternative interpretations of quantum mechanics are discussed.
The quantization process was always tightly connected to the Hamiltonian formulation of classical mechanics. For non-Hamiltonian systems, traditional quantization algorithms turn out to be unsuitable. Numerous attempts to quantize non-Hamiltonian systems have shown that this problem is nontrivial and requires the development of new approaches. In this paper, we present the quantization methods that do not depend upon the Hamiltonian formulation of classical mechanics. Two approaches to the quantization of mechanical systems are considered: axiomatic and hydrodynamic. It is shown that the formal application of these approaches to the classical Hamilton-Jacobi theory allows obtaining the wave equation for the corresponding quantum system in natural way. Examples are considered that show the effectiveness of the proposed approaches, both for Hamiltonian and non-Hamiltonian systems. The spinor form of the relativistic Hamilton-Jacobi theory for classical particles is considered. It is shown that it naturally leads to the Dirac equation for the corresponding quantum particle and to its non-Hamiltonian generalization, the bispinor relativistic Kostin equation.
The hydrodynamical form of Wigner–Dunkl quantum mechanics has been formulated. The main step to get such formulation was to derive a pair of Madelung-like equations (called Madelung–Wigner–Dunkl equations) from the Schrödinger equation appearing in Wigner–Dunkl quantum mechanics. Some aspects of both equations contained in the pair, i.e. the inhomogeneous continuity equation and the quantum Hamilton–Jacobi equation, have been investigated and discussed. With certain assumption, the relation between the expectation value of the quantum potential appearing in the quantum Hamilton–Jacobi equation and the so-called Fisher information has been derived. Then an uncertainty relation between momentum and position which may be tighter than the standard uncertainty relation has been derived by using Cramér–Rao inequality and Fisher information.