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Proceedings of the National Academy of Sciences

Solitary waves and Bohmian mechanics

Alberto Abbondandolo, Vieri Benci

Published Oct 23, 2002• 7 citations
Publisher

Abstract

We study a Schrödinger-like equation with a nonlinear term. This nonlinearity has the effect of allowing the existence of highly concentrated stable solitary waves of a topological nature. Such solitary waves tend to move according to Bohmian mechanics. Therefore our model can be considered a nonsingular realization of de Broglie pilot wave theory.

DOI
10.1073/pnas.222494299

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We study a Schrödinger-like equation with a nonlinear term. This nonlinearity has the effect of allowing the existence of highly concentrated stable solitary waves of a topological nature. Such solitary waves tend to move according to Bohmian mechanics. Therefore our model can be considered a nonsingular realization of de Broglie pilot wave theory.
Bohmian mechanics, also known as pilot-wave theory or de Broglie-Bohm theory, is a formulation of quantum mechanics whose fundamental axioms are not about what observers will see if they perform an experiment but about what happens in reality. It is therefore called a "quantum theory without observers," alongside with collapse theories and many-worlds theories and in contrast to orthodox quantum mechanics. It follows from these axioms that in a universe governed by Bohmian mechanics, observers will see outcomes with exactly the probabilities specified by the usual rules of quantum mechanics for empirical predictions. Specifically, Bohmian mechanics asserts that electrons and other elementary particles have a definite position at every time and move according to an equation of motion that is one of the fundamental laws of the theory and involves a wave function that evolves according to the usual Schrödinger equation. Bohmian mechanics is named after David Bohm (1917-1992), who was, although not the first to consider this theory, the first to realize (in 1952) that it actually makes correct predictions.
Abstract Bohmian mechanics supplements the quantum wavefunction with deterministic particle trajectories, offering an alternate, dynamical language for quantum theory. However, the Bohmian wavefunction evolves independently of these trajectories, and is thus unaffected by the observable properties of the system. While this property is widely assumed necessary to ensure agreement with quantum mechanics, much work has recently been dedicated to understanding classical pilot-wave systems, which feature a two-way coupling between particle and wave. These systems—including the “walking droplet” system of Couder and Fort (Couder and Fort (2006) Phys. Rev. Lett. 97:154101) and its various abstractions (Dagan and Bush (2020) CR Mecanique 348:555–571; Durey and Bush (2020) Front. Phys. 8:300; (2021) Chaos 31:033136; Darrow and Bush (2024) Symmetry 16:149)—allow us to investigate the limits of classical systems and offer a touchstone between quantum and classical dynamics. In this work, we present a general result that bridges Bohmian mechanics with this classical pilot-wave theory. Namely, Darrow and Bush ((2024) Symmetry 16:149) recently introduced a Lagrangian pilot-wave framework to study quantum-like behaviours in classical systems; with a particular choice of particle-wave coupling, they recover key dynamics hypothesised in de Broglie’s early double-solution theory (de Broglie (1970) Foundations Phys. 1:5–15). We here show that, with a different choice of coupling, their de Broglie-like system reduces exactly to single-particle Bohmian mechanics in the non-relativistic limit. Our result clarifies that, while multi-particle entanglement is impossible to replicate in general with local, classical theories, no such restriction exists for single-particle quantum mechanics. Moreover, connecting with the previous work of Darrow and Bush, our work demonstrates that de Broglie’s and Bohm’s theories can be connected naturally within a single Lagrangian framework. Finally, we present an application of the present work in developing a single-particle analogue for position measurement in a de Broglie-like setting.
Bohmian mechanics, also known as pilot-wave theory or de Broglie-Bohm theory, is a formulation of mechanics whose fundamental axioms are not about what observers will see if they perform an experiment but about what happens in reality. It is therefore called a quantum theory without observers, alongside with collapse theories and many-worlds theories and in contrast to orthodox mechanics. It follows from these axioms that in a universe governed by Bohmian mechanics, observers will see outcomes with exactly the probabilities specified by the usual rules of mechanics for empirical predictions. Specifically, Bohmian mechanics asserts that electrons and other elementary particles have a definite position at every time and move according to an equation of motion that is one of the fundamental laws of the theory and involves a wave function that evolves according to the usual Schr\odinger equation. Bohmian mechanics is named after David Bohm (1917-1992), who was, although not the first to consider this theory, the first to realize (in 1952) that it actually makes correct predictions.
Bohmian mechanics, also known as pilot-wave theory or de Broglie–Bohm theory, is a formulation of quantum mechanics whose fundamental axioms are not about what observers will see if they perform an experiment but about what happens in reality. It is therefore called a “quantum theory without observers,” alongside with collapse theories and many-worlds theories and in contrast to orthodox quantum mechanics. It follows from these axioms that in a universe governed by Bohmian mechanics, observers will see outcomes with exactly the probabilities specified by the usual rules of quantum mechanics for empirical predictions. Specifically, Bohmian mechanics asserts that electrons and other elementary particles have a definite position at every time and move according to an equation of motion that is one of the fundamental laws of the theory and involves a wavefunction that evolves according to the usual Schr\\"odinger equation. Bohmian mechanics is named after David Bohm (1917–1992), who was, although not the first to consider this theory, the first to realize (in 1952) that it actually makes correct predictions.